Progress in Mathematical Physics [630798]

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Progress in Mathematical Physics [630798]

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Progress in Mathematical Physics
V olume 37
Editors-in-Chief
Anne Boutet de Monvel, Université Paris VII Denis Diderot
Gerald Kaiser, The Virginia Center for Signals and Waves
Editorial Board
D. Bao, University of Houston
C. Berenstein, University of Maryland, College Park
P . Blanchard, Universi tät Bielefeld
A.S. Fokas, Imperial College of Science, Technology and Medicine
C. Tracy, Universit y of California, Davis
H. van den Berg, Wageningen U niversity

Birkhäuser V erlag
Basel · Boston · BerlinAlfredo Bermúdez de Castro
Continuum
Thermomechanics

Author:
Alfredo Bermúdez de Castro
Facultad de Matemáticas
Universidade de Santiago de Compostela
Campus Universitario Sur
15782 Santiago de Compostela
Spain
e-mail : [anonimizat]
2000 Mathematics Subject Classification 74A, 74J, 76A, 76N, 80A
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C.,
USA
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed biblio-
graphic data is available in the Internet at <http://dnb.ddb.de>.
ISBN 3-7643-7265-6 Birkhäuser Verlag, Basel – Boston – Berlin
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© 2005 Birkhäuser Verlag, P .O. Box 133, CH-4010 Basel, Switzerland
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ISBN-10: 3-7643-7265-6
ISBN-13: 978-3-7643-7265-1
9 8 7 6 5 4 3 2 1 www.birkhauser.ch

To my wife, Ana

Contents
Preface xi
1 General Deﬁnitions, Conservation Laws 1
1 . 1 M o t i o n o f a B o d y………………………. 1
1 . 2 C o n s e r v a t i o n o f M a s s …………………….. 4
1.3 Balance of Linear and Angular Momentum . . . . . . . . . . . . . 5
1.4 Balance of Energy. First Principle of Thermodynamics . . . . . . . 71.5 Second Principle of Thermodynamics. The Clausius-Duhem
I n e q u a l i t y………………………….. 1 1
2 Lagrangian Coordinates 13
2 . 1 T h e P i o l a – K i r c h h o ﬀ S t r e s s T e n s o r s……………… 1 3
2.2 The Conservation Equations in Lagrangian Coordinates . . . . . . 14
3 Constitutive Laws 17
3 . 1 T h e r m o d y n a m i c P r o c e s s . M a t e r i a l B o d y…………… 1 7
3 . 2 C o l e m a n – N o l l M a t e r i a l s …………………… 1 8
4 The Principle of Material Frame-Indiﬀerence 27
4.1 Change in the Observer. The Indiﬀerence Principle . . . . . . . . . 274.2 Consequences for Coleman-Noll Materials . . . . . . . . . . . . . . 28
5 Replacing Entropy with Temperature 33
5.1 The Conservation Equations in Terms of Temperature . . . . . . . 33
6I s o t r o p y 3 7
6 . 1 T h e E x t e n d e d S y m m e t r yG r o u p ………………. 3 7
6 . 2 I s o t r o p i c B o d i e s ……………………….. 3 9
7 Equations in Lagrangian Coordinates 43

viii Contents
8 Linearized Models 47
8.1 Linear Approximation of the Motion Equation . . . . . . . . . . . 47
8.2 Linear Approximation of the Energy Equation . . . . . . . . . . . 528 . 3 I s o t r o p i c L i n e a r T h e r m o v i s c o e l a s t i c i t y……………. 5 4
9 Quasi-static Thermoelasticity 57
9 . 1 S t a t e m e n t o f t h e E q u a t i o n s …………………. 5 79 . 2 T i m e D i s c r e t i z a t i o n …………………….. 5 79 . 3 A P a r t i c u l a r C a s e ………………………. 5 9
10 Fluids 61
1 0 . 1T h e C o n c e p t o f F l u i d , F i r s t P r o p e r t i e s……………. 6 1
10.2 Motion Equation. Thermodynamic Pressure . . . . . . . . . . . . . 63
1 0 . 3E n e r g y E q u a t i o n , E n t h a l p y …………………. 6 41 0 . 4T h e r m o d y n a m i c C o e ﬃ c i e n t s a n d E q u a l i t i e s…………. 6 61 0 . 5G i b b sF r e e E n e r g y ……………………… 7 71 0 . 6S t a t i c so f F l u i d s ………………………. 7 7
10.7 The Boussinesq Approximation, Natural Convection . . . . . . . . 78
11 Linearized Models for Fluids, Acoustics 81
1 1 . 1G e n e r a l E q u a t i o n s , D i s s i p a t i v e A c o u s t i c s ………….. 8 1
11.2 The Isentropic Case, Non-Dissipative Acoustics . . . . . . . . . . . 8511.3 Linearized Models under Gravity . . . . . . . . . . . . . . . . . . . 87
12 Perfect Gases 93
1 2 . 1D e ﬁ n i t i o n , G e n e r a l P r o p e r t i e s………………… 9 31 2 . 2E n t r o p y a n d F r e e E n e r g y ………………….. 9 41 2 . 3T h e C o m p r e s s i b l e N a v i e r – S t o k e s E q u a t i o n s …………. 9 7
1 2 . 4T h e C o m p r e s s i b l e E u l e r E q u a t i o n s……………… 9 8
13 Incompressible Fluids 101
1 3 . 1I s o c h o r i c P r o c e s s e s……………………… 1 0 1
1 3 . 2N e w t o n i a n F l u i d s………………………. 1 0 11 3 . 3I d e a l F l u i d s…………………………. 1 0 3
14 Turbulent Flow of Incompressible Newtonian Fluids 105
1 4 . 1T u r b u l e n c e M o d e l s……………………… 1 0 514.2 The k−/epsilon1M o d e l………………………. 1 0 7
15 Mixtures of Coleman-Noll Fluids 109
1 5 . 1G e n e r a l D e ﬁ n i t i o n s……………………… 1 0 91 5 . 2M i x t u r e o f P e r f e c t G a s e s …………………… 1 1 3

Contents ix
16 Chemical Reactions in a Stirred Tank 119
1 6 . 1C h e m i c a l K i n e t i c s . T h e M a s s A c t i o n L a w ………….. 1 1 91 6 . 2C o n s e r v a t i o n o f C h e m i c a l E l e m e n t s……………… 1 2 21 6 . 3R e a c t i n g M i x t u r e o f P e r f e c t G a s e s ……………… 1 2 3
17 Chemical Equilibrium of a Reacting Mixture of Perfect Gases
in a Stirred Tank 125
17.1 The Least Action Principle for the Gibbs Free Energy . . . . . . . 125
17.2 Equilibrium for a Set of Reversible Reactions,
Equilibrium Constants . . . . . . . . . . . . . . . . . . . . . . . . . 126
1 7 . 3T h e S t o i c h i o m e t r i c M e t h o d …………………. 1 3 1
18 Flow of a Mixture of Reacting Perfect Gases 135
1 8 . 1M a s s C o n s e r v a t i o n E q u a t i o n s………………… 1 3 5
1 8 . 2M o t i o n E q u a t i o n………………………. 1 3 71 8 . 3E n e r g y C o n s e r v a t i o n E q u a t i o n ……………….. 1 3 71 8 . 4C o n s e r v a t i o n o f E l e m e n t s ………………….. 1 4 0
18.5 Equilibrium Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . 141
1 8 . 6T h e C a s e o f L o w M a c h N u m b e r……………….. 1 4 2
19 The Method of Mixture Fractions 145
1 9 . 1G e n e r a l E q u a t i o n s ……………………… 1 4 51 9 . 2E x a m p l e s ………………………….. 1 4 8
1 9 . 3T h e A d i a b a t i c C a s e …………………….. 1 4 9
19.4 The Case of Equilibrium Chemistry . . . . . . . . . . . . . . . . . 149
20 Turbulent Flow of Reacting Mixtures of Perfect Gases,
The PDF Method 153
20.1 Elements of Probability . . . . . . . . . . . . . . . . . . . . . . . . 1532 0 . 2T h e M i x t u r e F r a c t i o n / P D F M e t h o d …………….. 1 5 5
A Vector and Tensor Algebra 161
A . 1 V e c t o r S p a c e . B a s i s ……………………… 1 6 1
A . 2 I n n e r P r o d u c t………………………… 1 6 3
A . 3 T e n s o r s …………………………… 1 6 4A . 4 T h e A ﬃ n e S p a c e………………………. 1 7 0
B Vector and Tensor Analysis 173
B . 1 D i ﬀ e r e n t i a l O p e r a t o r s ……………………. 1 7 3
B.2 Curves and Curvilinear Integrals . . . . . . . . . . . . . . . . . . . 175
B.3 Gauss’ and Green’s Formulas. Stokes’ Theorem . . . . . . . . . . . 177B . 4 C h a n g e o f V a r i a b l e i n I n t e g r a l s……………….. 1 7 8B . 5 T r a n s p o r t T h e o r e m s …………………….. 1 7 8
B . 6 L o c a l i z a t i o n T h e o r e m ……………………. 1 7 9

x Contents
B . 7 D i ﬀ e r e n t i a l O p e r a t o r s i n C o o r d i n a t e s …………….. 1 7 9
B . 7 . 1 C a r t e s i a n C o o r d i n a t e s………………… 1 7 9
B . 7 . 2 C y l i n d r i c a l C o o r d i n a t e s ……………….. 1 8 2B . 7 . 3 S p h e r i c a l C o o r d i n a t e s ………………… 1 8 4
C Some Equations of Continuum Mechanics in Curvilinear Coordinates 189
C . 1 M a s s C o n s e r v a t i o n E q u a t i o n ………………… 1 8 9
C . 2 M o t i o n E q u a t i o n………………………. 1 9 0
C.3 Constitutive Law for Newtonian Viscous Fluids in Cooordinates . 191
D Arbitrary Lagrangian-Eulerian (ALE) Formulations of the Conservation
Equations 195
D . 1 A L E C o n ﬁ g u r a t i o n……………………… 1 9 5D.2 Conservative ALE Form of Conservation Equations . . . . . . . . . 197
D.2.1 Mixed Conservative ALE Form of the Conservation
E q u a t i o n s………………………. 1 9 9
D.3 Mixed Nonconservative Form of ALE Conservation Equations . . . 200
Bibliography 203Index 205

Preface
This book is intended to be an extension of Gurtin’s book on continuum mechan-
ics [5] by including the laws of thermodynamics and thus making it possible to
study the mechanical behaviour of material bodies, the response of which involves
variables such as entropy or temperature. In order to do that our departure point
is Coleman and Noll’s article [3] on the thermodynamics of elastic materials withheat conduction and viscosity which has been extended for the purpose at hand
to the case of nonhomogeneous materials.
The present book has been used for many years as a textbook for gradu-
ate and undergraduate mathematics students at the University of Santiago de
Compostela.
The ﬁrst Chapter revisits the conservation principles of continuum thermo-
mechanics, that is, the conservation of mass, linear and angular momentum balanceand the ﬁrst two principles of thermodynamics: namely, energy conservation and
entropy inequality. All principles are introduced in integral form and in Eulerian
coordinates. Local forms consisting of partial diﬀerential equations are then ob-
tained. Writing these local equations in Lagrangian coordinates is the subject of
Chapter 2.
Chapter 3 deals with the constitutive laws of continuum thermomechanics.
First the notion of a material body characterised by its constitutive class is given.
Then we introduce a general material body deﬁned by Coleman and Noll in the
above referenced article. By imposing the second principle of thermodynamics,
we prove some relations to be satisﬁed by the response functions of such a mate-
rial. Then, in Chapter 4, the principle of material frame-indiﬀerence is introducedand its consequences for the response functions of the Coleman-Noll materials are
established. In Chapter 5, the partial diﬀerential equations governing a thermo-
dynamic process are written replacing entropy with temperature.
Chapter 6 is devoted to isotropy. By using the representation theorems for
isotropic tensor and vector-valued functions, we obtain simple forms for the re-
sponse functions of Coleman-Noll materials. In Chapter 7, the equations satisﬁedby each thermodynamic process of these materials are written in Lagrangian co-
ordinates. We also show that inviscid Coleman-Noll materials are hyperelastic.
The linear approximations of these equations about a static reference state
are deduced in Chapter 8, assuming that the gradient of the displacement and the
diﬀerence of temperature with respect to a reference state are both small. This
is rigorously done through careful computation of the derivatives of the responsefunctions. Isotropic materials are speciﬁcally considered. Thus, we obtain the par-
tial diﬀerential system for linear thermoviscoelasticity; its numerical solution by
incremental methods, in the inviscid quasi-static case, is addressed in Chapter 9.
Fluids are the subject of Chapter 10 where they are introduced as partic-
ular Coleman-Noll materials when the extended symmetry group is the unimod-
ular group. We deﬁne the classical thermodynamic variables like speciﬁc heat,sound speed, volumetric thermal expansion, and write the conservation equations

xii Preface
in terms of them. We also include a section dealing with ﬂuid statics under a grav-
ity ﬁeld, and ﬁnally the approximate Boussinesq model for natural convection is
introduced.
The linearized models obtained for general Coleman-Noll materials in Chap-
ter 8 are specialized for ﬂuids in Chapter 11. Thus, the standard models for dissipa-
tive and non-dissipative acoustics are properly deduced. Assuming that the body
force is gravity force, we also deduce the equations for internal gravity waves.
Particular ﬂuids called perfect gases are studied in Chapter 12 where the
compressible Navier-Stokes and Euler syst ems of partial diﬀerential equations are
deduced. Incompressible ﬂuids are brieﬂy examined in Chapter 13, but since these
materials are not Coleman-Noll materials, they need to be studied separately. Inparticular, the incompressible Navier-Stokes and Euler equations are introduced.
A quick overview of turbulence models is the goal of Chapter 14.
Thermodynamics of mixtures of Coleman-Noll ﬂuids and, more speciﬁcally,
of mixtures of perfect gases is the subject of Chapter 15. In Chapters 16 and
17 some concepts of chemical kinetics are given, with special emphasis placedon modelling the chemical equilibrium by using Gibbs free energy. By using this
methodology we study the ﬂow of a mixture of reacting perfect gases in Chapter
18. In particular, the standard equations for modelling combustion are properlywritten both for ﬁnite-rate and equilibrium chemistry.
Chapter 19 deals with a computational method widely used in combustion:
the mixture fraction method. The important case of equilibrium chemistry is
speciﬁcally considered.
Finally, in Chapter 20, we give a short introduction to the interesting subject
of turbulent reacting ﬂows by recalling the fundamentals of the probability density
function (PDF) method.
The appendices include some mathematical background in tensor algebra and
analysis and then the formulation of conservation equations in so-called arbitrary
Lagangian-Eulerian (ALE) coordinates is given. From a computational point of
view, this formulation is very useful when dealing with free boundary ﬂows or
ﬂuid-structure interaction problems.
Acknowledgements
I wish to thank R. Ohayon and P. G. Ciarlet for their encouragement to publish the
book, and P. Quintela and O. L. Pouso for their careful reading of the manuscript
and their many useful suggestions.

Chapter 1
General Deﬁnitions,
Conservation Laws
In this chapter we recall some notations and results from continuum mechanics
which will be extensively used along these notes. Further details can be found, for
instance, in [5] (see also Appendices A and B).
1.1 Motion of a Body
LetEbe an aﬃne space on a three-dimensional Euclidean vector space V.L e tu s
denote by Linthe vector space of endomorphisms of V(which is isomorphic to
the space of second order tensors) and by Symthe subspace of those which are
symmetric.
Deﬁnition 1.1.1. AbodyBis a regular region of the Euclidean space E. Sometimes
we will refer to Bas the reference conﬁguration .E l e m e n t si n Bare called material
points .
Deﬁnition 1.1.2. Adeformation ofBis a smooth one-to-one mapping fwhich
mapsBonto a closed region in Eand satisﬁes
det(∇f)>0. (1.1)
Deﬁnition 1.1.3. Amotion ofBis a class C3mapping
X:B×R→E (1.2)
withX(·,t), a deformation of Bfor each ﬁxed t.

2 Chapter 1. General Deﬁnitions, Conservation Laws
We refer to x=X(p,t)a st h e place occupied by the material point pat time
twhile
Bt=X(B,t) (1.3)
is the region of space occupied by the body at time t(see Figure 1.1).
X(, ) tBBt
P(, ) t
Figure 1.1: Motion and reference map.
Deﬁnition 1.1.4. We denote by trajectory of the motion the set
T={(x, t):x∈Bt,t∈R}.
The vector u(p,t)=X(p,t)−prepresents the displacement ofpat time t
while the gradient of the motion is the tensor ﬁeld deﬁned by
F(p,t): =∇X(p,t), (1.4)
so that we have F(p,t)=I+∇u(p,t). The gradient gives information on the
local deformation of the body. In continuum mechanics other tensor ﬁelds are also
used to represent the state of local deformation. Let us mention C:=FtFand
B:=FFt,which are called the right and left Cauchy-Green strain tensors, re-
spectively, and the Green-Lagrange (sometimes called Green-Saint Venant )s t r a i n
tensor deﬁned by
G=1
2(C−I)=1
2(FtF−I). (1.5)
It is straightforward to see that
G=1
2(∇u+∇ut+∇ut∇u). (1.6)
The linear part of this expression, namely
E=∇u+∇ut
2, (1.7)
is called inﬁnitesimal strain tensor.
Moreover, we recall that, from the Polar Decomposition Theorem ,w ec a n
factorize Fas
F=RU=VR , (1.8)

1.1. Motion of a Body 3
B×R TΨ
W
–
@
@
@@R

φm φTB × RΨ−1
W
–
@
@
@@R

ϕs ϕ
Figure 1.2: Material and spatial descriptions.
where Ris a rotation tensor ﬁeld (i.e., orthogonal with determinant equal to 1)
andUandVare symmetric positive deﬁnite tensor ﬁelds given by
U=B1/2,V =C1/2. (1.9)
LetP(·,t):Bt→Bbe the inverse mapping of X(·,t). Then the mapping
P:T→ B (1.10)
gives the material point which occupies the place xat time t. It is called the refer-
ence map of the motion. We call ˙X(p,t)t h evelocity and¨X(p,t)t h eacceleration
of the material point pat time t.
Fields deﬁned in Tare called spatial (or Eulerian) ﬁelds while those deﬁned
inB×Rare called material (or Lagrangian) ﬁelds .
LetΨ:B×R→Tbe the mapping given by
Ψ(p,t): =(X(p,t),t), (1.11)
the inverse of which is Ψ−1(x, t)=(P(x, t),t). By using these mappings any
spatial ﬁeld can be transformed into a material ﬁeld and vice versa. Let φbe a
spatial ﬁeld. Its material description is deﬁned by
φm:=φ◦Ψ. (1.12)
Similarly, let ϕbe a material ﬁeld. Its material description is deﬁned by
ϕs:=ϕ◦Ψ−1. (1.13)
In particular, the spatial description of the velocity is
v(x, t): =˙X(P(x, t),t). (1.14)

4 Chapter 1. General Deﬁnitions, Conservation Laws
Throughout this book, the following notations for diﬀerential operators are
used:
•Material ﬁelds.
˙φ=∂φ
∂t,∇φ=g r a d i e n tpφ, (1.15)
Divφ=d i v e r g e n c epφ,Curlφ=C u r l pφ. (1.16)
•Spatial ﬁelds.
ϕ/prime=∂ϕ
∂t,gradφ=g r a d i e n txϕ, (1.17)
divϕ=d i v e r g e n c exϕ,curlϕ=c u r l xϕ, (1.18)
˙ϕ= material derivative of ϕ. (1.19)
1.2 Conservation of Mass
We assume a mass distribution forBdeﬁned by a reference density ,ρ0:B→R+,
in the reference conﬁguration. The mass conservation law implies that the density
in the motion X,ρ(x, t),must satisfy
ρ(x, t)d et(F(p,t)) =ρ0(p),x=X(p,t). (1.20)
From (1.20), the following local equations of the mass conservation law can be
deduced:
ρ/prime+d i v ( ρv)=0, (1.21)
˙ρ+ρdivv=0, (1.22)
which are called the conservative and the non-conservative forms, respectively.
Weak Formulation
It is very easy to obtain a so-called weak formulation of the mass conservation law.
Lettbe any time and z:Bt→Rbe any square integrable scalar ﬁeld.
Multiplying (1.22) by zand integrating in Bt,w eg e t
/integraldisplay
Bt˙ρzdVx+/integraldisplay
BtρdivvzdVx=0. (1.23)
The above weak formulation can be used for ﬁnite element approximation.
However, for ﬁnite volume methods the following equation is used:
d
dt/integraldisplay
RρdVx+/integraldisplay
∂Rρv·ndAx=0. (1.24)

1.3. Balance of Linear and Angular Momentum 5
It can be easily obtained by integrating the conservative form (1.21) in a control
volume Rand then applying Gauss’ Theorem .
The following useful result is a straightforward consequence of the Reynolds’
transport Theorem and the mass conservation principle.
Lemma 1.2.1. LetΦbe a smooth spatial ﬁeld deﬁned in the trajectory of a motion
X. Let us assume that Φis either scalar or vector-valued. Then for any part P
and time t
d
dt/integraldisplay
PtρΦd V x=/integraldisplay
Ptρ˙Φd V x. (1.25)
1.3 Balance of Linear and Angular Momentum
Let us denote by Nthe set of unit vectors of the linear space V.
Deﬁnition 1.3.1. Asystem of forces for a body Bduring the motion Xis a pair
(s,b) of vector ﬁelds
s:N×T→V ,b:T→ V , (1.26)
with
1.s(n,x ,t), for each n∈Nand t, a smooth function of xinBt,
2.b(x, t), for each t, a continuous function of xinBt.
Vector ﬁeld srepresents the density of surface force exerted across an oriented
surface Swith positive unit normal vector nat point xby the material on the
positive side of Supon the material on the negative side ( Cauchy’s hypothesis ).
Vector ﬁeld bis called the body force and gives the force, per unit volume, exerted
by the environment of the body on x.
Thebalance of linear and angular momentum laws state that, for every part
Pand time t, the following equalities hold:
d
dt/integraldisplay
PtvρdVx=/integraldisplay
∂Pts(n)d A x+/integraldisplay
PtbdVx, (1.27)
d
dt/integraldisplay
Ptr×vρdVx=/integraldisplay
∂Ptr×s(n)d A x+/integraldisplay
Ptr×bdVx. (1.28)
According to the Cauchy Theorem , there exists a symmetric second order
tensor ﬁeld T:T→Sym(called the Cauchy stress tensor ) such that s(n,x ,t)=
T(x, t)n. Furthermore, the following equations hold:
(ρv)/prime+d i v ( ρv⊗v)=d i v T+b, (1.29)
ρ˙v=d i v T+b, (1.30)
which are the conservative and the non-conservative motion equation, respectively.

6 Chapter 1. General Deﬁnitions, Conservation Laws
In (1.30) ˙vdenotes the material time derivative which is deﬁned, for a vector
ﬁeldw,b y
˙w(x, t): =∂
∂tw(X(p,t),t)|p=P(x,t)=∂w
∂t(x, t) + grad w(x, t)v(x, t).(1.31)
The following result is known as the Theorem of Power Expended :
Theorem 1.3.2. For every part Pand time twe have
d
dt/integraldisplay
Ptρ|v|2
2dVx+/integraldisplay
PtT·DdVx=/integraldisplay
∂Pts(n)·vdAx+/integraldisplay
Ptb·vdVx,(1.32)
where
D=L+Lt
2(1.33)
andL:= grad v.
Proof. Firstly, by using a Green’s formula we get
/integraldisplay
PtdivT·vdVx=/integraldisplay
∂PtTn·vdAx−/integraldisplay
PtT·gradvdVx
=/integraldisplay
∂Pts(n)·vdAx−/integraldisplay
PtT·DdVx, (1.34)
because Tis symmetric.
Moreover, from Lemma 1.2.1,
/integraldisplay
Ptρ˙v·vdVx=d
dt/integraldisplay
Ptρ|v|2
2dVx. (1.35)
Let us make the scalar product of the motion equation (1.30) by vand
integrate in Pt.W eo b t a i n
/integraldisplay
Ptρ˙v·vdVx=/integraldisplay
PtdivT·vdVx+/integraldisplay
Ptb·vdVx. (1.36)
Now this equality easily yields (1.32) by using (1.34) and (1.35). /square
Term/integraltext
PtT·DdVxin equation (1.32) is called the power stress .
Weak Formulation. Boundary and Initial Conditions
As for the mass conservation, we can obtain a weak formulation for the balance
of momentum .L e ttbe any time and w:Bt→Vbe any smooth vector ﬁeld. By
making the scalar product of (1.30) with wand using a Green’s formula we get
/integraldisplay
Btρ˙v·wdVx=−/integraldisplay
BtT·gradwdVx+/integraldisplay
∂Bts(n)·wdAx+/integraldisplay
Btb·wdVx.
(1.37)

1.4. Balance of Energy. First Principle of Thermodynamics 7
We have to give an initial condition, the velocity ﬁeld at initial time, and
boundary conditions. The latter may have diﬀerent forms depending on the par-
ticular material. At this stage we consider two general ones. Let us divide the
boundary of Btinto two parts: ∂Bv
tand∂Bs
t. Let us suppose the velocity is given
on∂Bv
twhile the surface force density is prescribed on ∂Bs
t. More precisely,
v=¯von∂Bv
t, (1.38)
s(n)=¯son∂Bs
t, (1.39)
where ¯vands(n)=¯sare given boundary ﬁelds. In order to incorporate these
boundary conditions into the weak formulation (1.37), ﬁrstly, we only take test
functions ,w, vanishing on ∂Bv
t. Then the surface integral in (1.37) becomes
/integraldisplay
∂Bts(n)·wdAx=/integraldisplay
∂Bs
ts(n)·wdAx=/integraldisplay
∂Bs
t¯s·wdAx. (1.40)
Finally, the weak formulation consists of the following equalities:
/integraldisplay
Btρ˙v·wdVx+/integraldisplay
BtT·gradwdVx=/integraldisplay
∂Bs
t¯s·wdAx+/integraldisplay
Btb·wdVx,
∀wnull on ∂Bv
t,(1.41)
v=¯von∂Bv
t, (1.42)
v=v0inB0att=0. (1.43)
Moreover, we can also obtain a formulation of the motion equation to be used for
ﬁnite volume approximation. For this purpose we integrate the conservative form
(1.29) in a control volume Rand use Gauss’ Theorem .W eg e t
d
dt/integraldisplay
RρvdVx+/integraldisplay
∂Rρvv·ndAx=/integraldisplay
∂RTndAx+/integraldisplay
RbdVx. (1.44)
1.4 Balance of Energy. First Principle of
Thermodynamics
We begin by stating Cauchy’s hypothesis concerning the surface heat which is sim-
ilar to that for surface forces. It consists in assuming the existence of a surface heat
density g(n,x ,t) deﬁned for each unit vector nand every ( x, t) in the trajectory
of the motion, having the following property: let Sbe an oriented surface in Bt
with positive unit normal natx.T h e n g(n,x ,t) is the heat, per unit area and
time, ﬂowing from the negative side of Sto the positive side of S.
Besides this surface heat, we suppose there is another one supplied at interior
points of Bt. Such a heat is determined by a scalar ﬁeld fdeﬁned on T:f(x, t)
gives the heat, per unit volume and time, supplied by the environment at point x
at time t.

8 Chapter 1. General Deﬁnitions, Conservation Laws
The previous discussion motivates the following:
Deﬁnition 1.4.1. Asystem of heat for a body Bduring a motion (with trajectory
T)i sap a i r( g,f) of functions g:N×T→ R,f:T→Rwhere
1.g(n,x ,t), for each n∈Nandt, is a smooth function of xonBt,
2.f(x, t), for each t, is a continuous function of xonBt.
We call gthesurface heat andfthebody heat .
Deﬁnition 1.4.2. Theheat rate supplied into the part Pat time tis given by
Q(P,t)=−/integraldisplay
∂Ptg(n)d A x+/integraldisplay
PtfdVx. (1.45)
Let us consider a system of forces ( s,b)a n das y s t e mo fh e a t( g,f) during
am o t i o n Xof a body B.T h e energy conservation law (which is also called First
Principle of Thermodynamics ) asserts that there exists a scalar ﬁeld E, the speciﬁc
total energy , such that for every part Pand time t,
d
dt/integraldisplay
PtρEdVx=/integraldisplay
∂Pts(n)·vdAx+/integraldisplay
Ptb·vdVx
−/integraldisplay
∂Ptg(n)d A x+/integraldisplay
PtfdVx. (1.46)
We have the following fundamental result.
Theorem 1.4.3. (Cauchy). Suppose that the momentum balance laws hold. Then a
necessary and suﬃcient condition that the energy conservation law be satisﬁed is
the existence of a spatial vector ﬁeld q(called heat ﬂux vector) such that
1.For each unit vector n,
g(n,x ,t)=q(x, t)·n, (1.47)
2. ρ˙E=d i v ( Tv)+b·v−divq+f. (1.48)
Proof. We only prove that assuming (1.47) the local form (1.48) holds. To see that
we ﬁrst replace sbyTnandgbyq·nin (1.46). Next we use Lemma 1.2.1 to
transform the left-hand side of (1.46). We get
/integraldisplay
Ptρ˙EdVx=/integraldisplay
∂PtTn·vdAx+/integraldisplay
Ptb·vdVx−/integraldisplay
∂Ptq·ndAx+/integraldisplay
PtfdVx.
(1.49)
Then by using Gauss’ Theorem we deduce
/integraldisplay
Ptρ˙EdVx=/integraldisplay
Ptdiv(Tv)d V x+/integraldisplay
Ptb·vdVx
−/integraldisplay
PtdivqdVx+/integraldisplay
PtfdVx. (1.50)

1.4. Balance of Energy. First Principle of Thermodynamics 9
Now (1.48) follows by applying the Localization Theorem . /square
Corollary 1.4.4. Let us assume the energy conservation law holds. Then, for each
t∈Rand any smooth scalar ﬁeld z:Bt→R, we have
/integraldisplay
Ptρ˙EzdVx+/integraldisplay
PtTv·gradzdVx=/integraldisplay
∂Pts(n)·vzdAx+/integraldisplay
Ptb·vzdVx
−/integraldisplay
∂Ptg(n)zdAx+/integraldisplay
PtfzdVx, (1.51)
∀Ppart of B.
Deﬁnition 1.4.5. We denote by speciﬁc internal energy the scalar ﬁeld
e=E−|v|2
2. (1.52)
The next Proposition provides both integral and local forms of the energy
conservation law in terms of the speciﬁc internal energy.
Proposition 1.4.6. We have
1.d
dt/integraldisplay
PtρedVx=/integraldisplay
PtT·DdVx−/integraldisplay
∂Ptg(n)d A x+/integraldisplay
PtfdVx,(1.53)
for all parts Pand times t,w h e r e Dis deﬁned by (1.33).
2. ρ˙e=T·D−divq+f. (1.54)
Proof. By subtracting equality (1.32) from (1.46) we immediately get (1.53).
Now, in order to prove the second part of the Proposition, we replace g(n)
byq·nin (1.53) and then use Gauss’ Theorem .W eo b t a i n
d
dt/integraldisplay
PtρedVx=/integraldisplay
PtT·DdVx−/integraldisplay
PtdivqdVx+/integraldisplay
PtfdVx. (1.55)
Now (1.54) follows from the Localization Theorem . /square
Remark 1.4.7.Equations (1.48) and (1.54) are local non-conservative versions
of the energy conservation law . We can also write conservative forms which are
obtained by adding the mass conservation equation (1.21) multiplied by Eande
to (1.48) and (1.54), respectively. We get
(ρE)/prime+d i v ( ρEv)=d i v ( Tv)+b·v−divq+f, (1.56)
(ρe)/prime+d i v ( ρev)=T·D−divq+f. (1.57)

10 Chapter 1. General Deﬁnitions, Conservation Laws
Weak Formulation. Boundary and Initial Conditions
Let us obtain a weak formulation for the energy balance in terms of the internal
energy .L e ttbe any time and y:Bt→Rbe any smooth scalar ﬁeld. By making
the product of (1.54) with y,i n t e g r a t i n gi n Bt, and using a Green’s formula in the
boundary integral including the heat ﬂux ,w eg e t
/integraldisplay
Btρ˙eydVx=/integraldisplay
BtT·DydVx+/integraldisplay
Btq·gradydVx−/integraldisplay
∂Btg(n)ydAx
+/integraldisplay
BtfydVx.(1.58)
In order to state a well-posed mathematical problem, we have to prescribe
an initial condition, the internal energy at initial time, and boundary conditions.
We consider two types of boundary conditions. Let us divide the boundary of Bt
into two parts: ∂Be
tand∂Bg
t. Let us suppose the internal energy is given on ∂Be
t
while the heat ﬂux is prescribed on ∂Bg
t. That is, we impose
e=¯eon∂Be
t, (1.59)
g(n)=¯gon∂Bg
t. (1.60)
In order to include these boundary conditions in the weak formulation (1.58),
we only take test functions ynull on ∂Be
t. Then the boundary integral transforms
as follows:
/integraldisplay
∂Btg(n)ydAx=/integraldisplay
∂Bg
tg(n)ydAx=/integraldisplay
∂Bg
t¯gydAx. (1.61)
Thus the weak formulation of the energy equation is
/integraldisplay
Btρ˙eydVx=/integraldisplay
BtT·DydVx+/integraldisplay
Btq·gradydVx−/integraldisplay
∂Bg
t¯gydAx+/integraldisplay
BtfydVx,
∀ynull on ∂Be
t,(1.62)
e=¯eon∂Be
t, (1.63)
e=e0inB0att=0. (1.64)
For ﬁnite volume discretization we have to use the conservation forms (1.56) or
(1.57). They are ﬁrst integrated in a control volume Rand then transformed by

1.5. Second Principle of Thermodynamics. The Clausius-Duhem Inequality 11
usingGauss’ Theorem .W eg e t
d
dt/integraldisplay
∂RρEdVx+/integraldisplay
∂RρEv·ndAx=/integraldisplay
∂RTn·vdAx+/integraldisplay
Rb·vdVx
−/integraldisplay
∂Rg(n)d A x+/integraldisplay
RfdVx,(1.65)
d
dt/integraldisplay
RρedVx+/integraldisplay
∂Rρev·ndAx=/integraldisplay
RT·DdVx
−/integraldisplay
∂Rg(n)d A x+/integraldisplay
RfdVx,
(1.66)
respectively.
1.5 Second Principle of Thermodynamics.
The Clausius-Duhem Inequality
In this section we introduce the Second Principle of Thermodynamics . It asserts
that there exist a scalar ﬁeld s, the speciﬁc entropy , and a strictly positive scalar
ﬁeldθ,t h eabsolute temperature , such that
d
dt/integraldisplay
PtρsdVx≥−/integraldisplay
∂Ptq·n
θdAx+/integraldisplay
Ptf
θdVx (1.67)
for all parts Pand times t.
The quantity
S(P,t)=/integraldisplay
PtρsdVx (1.68)
is called entropy of the part Pat time t, while the sum
−/integraldisplay
∂Ptq·n
θdAx+/integraldisplay
Ptf
θdVx (1.69)
is the entropy rate supplied to the part Pat time t.
Thus the inequality (1.67) states that entropy can not be conserved because
the entropy growth rate of a part can be greater than the rate of supplied entropy
from its environment.
The following local form, called the Clausius-Duhem inequality can be eas-
ily obtained from (1.67) by using the Reynolds ,t h eGauss and the Localization
Theorems :
ρ˙s+d i v/parenleftBigq
θ/parenrightBig
−f
θ≥0. (1.70)

12 Chapter 1. General Deﬁnitions, Conservation Laws
Moreover, by multiplying (1.70) by θand then subtracting (1.54) we obtain
ρθ˙s−ρ˙e+T·D−1
θq·gradθ≥0. (1.71)
Finally, we write the second principle of thermodynamics in terms of the free
energy .
Deﬁnition 1.5.1. We denote by speciﬁc Helmholtz free energy the scalar ﬁeld ψ
deﬁned by
ψ=e−sθ. (1.72)
Since we have
˙ψ=˙e−˙sθ−s˙θ, (1.73)
by using this expression in (1.71) we get another local form of the second principle
of thermodynamics, namely,
ρs˙θ+ρ˙ψ−T·D+1
θq·gradθ≤0. (1.74)

Chapter 2
Lagrangian Coordinates
Sometimes it is convenient to write the conservation principles in material (also
called Lagrangian) coordinates rather than in spatial (also called Eulerian) ones.
This reveals some new tensor ﬁelds related with the Cauchy stress tensor.
2.1 The Piola-Kirchhoﬀ Stress Tensors
Let us introduce the First Piola-Kirchhoﬀ stress tensor (also called Boussinesq
stress tensor) which is deﬁned, in the reference conﬁguration, by
S(p,t): =d e t ( F(p,t))T(x, t)F−t(p,t),x=X(p,t). (2.1)
Similarly, the Second Piola-Kirchhoﬀ stress tensor (also called Piola-Lagrange
stress tensor) is deﬁned by
P(p,t): =d e t ( F(p,t))F−1(p,t)T(x, t)F−t(p,t),x=X(p,t). (2.2)
We notice that Pis symmetric but Sis not.
We can write the stress power representing the rate of mechanical internal
energy (see (1.53)) in two Lagrangian alternative forms.
Proposition 2.1.1. We have
/integraldisplay
PtT·DdVx=/integraldisplay
PS·∇˙udVp=/integraldisplay
PP·˙GdVp. (2.3)
Proof. We need the following Lemmas which can be found in [5].
Lemma 2.1.2. LetA,BandCbe three tensors. Then
A·BC=ACt·B=BtA·C. (2.4)

14 Chapter 2. Lagrangian Coordinates
Lemma 2.1.3. We have
˙F=LmF. (2.5)
Now, let us prove the Proposition. Since Tis symmetric we deduce
/integraldisplay
PtT·DdVx=/integraldisplay
PtT·LdVx (2.6)
and, by making the change of variable x=X(p,t),
/integraldisplay
PtT·LdVx=/integraldisplay
PTm·Lmdet(F)d V p=/integraldisplay
PTm·˙FF−1det(F)d V p
=/integraldisplay
PTmF−t·˙Fdet(F)d V p=/integraldisplay
PS·∇˙udVp. (2.7)
Moreover, we can compute the time derivative of the Green-Lagrange strain tensor,
˙G=1
2(FtF).=1
2(˙FtF+Ft˙F)=1
2/bracketleftbig
(LmF)tF+FtLmF/bracketrightbig
=1
2/bracketleftbig
FtLt
mF+FtLmF/bracketrightbig
=1
2Ft(Lm+Lt
m)F=FtDmF. (2.8)
Then we have
/integraldisplay
PtT·DdVx=/integraldisplay
PTm·Dmdet(F)d V p=/integraldisplay
PTm·F−t˙GF−1det(F)d V p
=/integraldisplay
PF−1TmF−t·˙Gdet(F)d V p=/integraldisplay
PP·˙GdVp. (2.9)
/square
2.2 The Conservation Equations in Lagrangian
Coordinates
By using the change of variable x=X(p,t), we obtain the equalities (see Lemma
1.2.1 and Appendix B)
d
dt/integraldisplay
PtρvdVx=/integraldisplay
Ptρ˙vdVx=/integraldisplay
Pρ(X(p,t),t)¨u(p,t)det(F(p,t)) dV p,(2.10)
/integraldisplay
PtbdVx=/integraldisplay
Pb(X(p,t),t)det(F(p,t)) dV p,(2.11)
/integraldisplay
∂PtT(x, t)n(x)dA(x)=/integraldisplay
∂Pdet(F(p,t))T(X(p,t),t)F−t(p,t)m(p)d A p,(2.12)
where n(x) (respectively m(p)) denotes the outward unit normal vector to ∂Pt
(respectively to ∂P)a tp o i n t x(respectively p). Then we replace these equalities
in (1.27) and apply the Gauss and the Localization Theorems .W eg e t
ρ0¨u=D i v S+b∗, (2.13)

2.2. The Conservation Equations in Lagrangian Coordinates 15
where b∗is given by
b∗(p,t)=b(X(p,t),t)d et(F(p,t)). (2.14)
Analogous computations allow us to obtain, from (1.49), the following form of the
energy equation in Lagrangian coordinates:
ρ0˙Em=D i v ( S˙u)+b∗·˙u−Divq∗+f∗, (2.15)
where
q∗(p,t)=d e t ( F(p,t))F−1(p,t)q(X(p,t),t), (2.16)
f∗(p,t)=f(X(p,t),t)d et (F(p,t)). (2.17)
Similarly, in terms of the internal energy, the energy equation can be written in
Lagrangian coordinates as
ρ0˙em=S·∇˙u−Divq∗+f∗, (2.18)
by using (2.3).

Chapter 3
Constitutive Laws
In what follows we use simultaneously material and spatial ﬁelds. As we did in
the previous chapter, sometimes subscript m(respectively, s)i su s e dt od e n o t e
the material (respectively, spatial) description of a spatial (respectively, material)
ﬁeld. But, for the sake of simplicity in notation, they will be omitted in whatfollows if no ambiguity arises.
3.1 Thermodynamic Process. Material Body
Firstly, we extend the concept of material body given in reference [5]. We begin
with the following:
Deﬁnition 3.1.1. Athermodynamic process for a body with a mass distribution
ρ0is a set of eight mappings:
X:B×R→E, (3.1)
T:T→Sym, (3.2)
b:T→ V , (3.3)
e:T→R, (3.4)
θ:T→R+, (3.5)
q:T→ V , (3.6)
f:T→R, (3.7)
s:T→R, (3.8)
such that Xis a motion, Tits trajectory, T∈C1(T;Sym),b∈C0(T;V),
e∈C1(T;R),θ∈C1(T;R),q∈C1(T;V),f∈C0(T;R),s∈C1(T;R) and,
furthermore, the following equations hold:
ρ˙v=d i v T+b, (3.9)
ρ˙e=T·D−divq+f, (3.10)

18 Chapter 3. Constitutive Laws
where
ρ(x, t)d et(F(p,t)) =ρ0(p),withx=X(p,t). (3.11)
Remark 3.1.2.We notice that, in order to deﬁne a thermodynamic process, it is
enough to prescribe the six mappings X,T,e,θ, qands, because the two remaining
onesbandfcan be deduced from (3.9) and (3.10), respectively.
With the deﬁnition above we can introduce an extended concept of material
body.
Deﬁnition 3.1.3. Amaterial body is a triple ( B,ρ0,C) consisting of a body B,
a mass distribution ρ0and a family Cof thermodynamic processes called the
constitutive class of the body.
3.2 Coleman-Noll Materials
Let us denote by Lin+the subset of second order tensors with positive determi-
nant, i.e.,
Lin+={G∈Lin:d e t ( G)>0}. (3.12)
Now we introduce a particular material body.Deﬁnition 3.2.1. Ahyperelastic material with heat conduction and viscosity is
a material body the constitutive class of which consists of all thermodynamic
processes satisfying
T(x, t)=ˆT(F(p,t),s(x, t),p)+ˆl(F(p,t),s(x, t),p)(L(x, t)), (3.13)
e(x, t)=ˆe(F(p,t),s(x, t),p), (3.14)
θ(x, t)=ˆθ(F(p,t),s(x, t),p), (3.15)
q(x, t)=ˆq(F(
p,t),s(x, t),gradθ(x, t),p), (3.16)
withx=X(p,t),for some “smooth enough” mappings
ˆT:Lin+×R×B→ Sym, (3.17)
ˆl:Lin+×R×B→L (Lin,Sym ), (3.18)
ˆe:Lin+×R×B→ R, (3.19)
ˆθ:Lin+×R×B→ R+, (3.20)
ˆq:Lin+×R×V×B→V , (3.21)
which are called response mappings of the body.
In what follows a hyperelastic body with heat conduction and viscosity will
be also called a Coleman-Noll material .
In order to prove the next theorem we need the following result:

3.2. Coleman-Noll Materials 19
Lemma 3.2.2. Letϕbe deﬁned by
ϕ(A)=d e t ( A). (3.22)
Then ϕis smooth and, if Ais invertible, then
∂ϕ
∂A(A)(U)=d e t ( A)t r(UA−1), (3.23)
for each U∈Lin.
Proof. See [5]. /square
Corollary 3.2.3. LetSbe a mapping deﬁned in an open set D⊂Rand valued in
the set of invertible tensors. Then
(det(S))·=( d e t ( S)) tr˙SS−1. (3.24)
Proof. See [5]. /square
We have the following fundamental result.
Theorem 3.2.4. Let us consider a Coleman-Noll material with constitutive class
C. We make the following assumption:
(H1)There exists a “smooth enough” function, ˆs:Lin+×R+×B→ R, such that
ifs∈R,F∈Lin+,θ∈R+andp∈B,t h e n
s=ˆs(F,θ,p )if and only if θ=ˆθ(F,s,p ). (3.25)
Then all elements in Csatisfy the second law of thermodynamics if and only if
ˆθ(F,s,p )=∂ˆe
∂s(F,s,p ), (3.26)
ˆT(F,s,p )=ρ0(p)
det(F)∂ˆe
∂F(F,s,p )Ft, (3.27)
ˆl(F,s,p )(L)·L≥0(dissipation inequality), (3.28)
ˆq(F,s,w,p)·w≤0, (3.29)
for all F∈Lin+,s∈R,p∈B,L∈Linandw∈V.
Remark 3.2.5.Notice that assumption ( H1) implies that∂ˆs
∂θ(F,θ,p )/negationslash=0∀F∈
Lin+,θ∈R+andp∈B. Indeed, from equality
s=ˆs(F,ˆθ(F,s,p ),p) (3.30)
and the chain rule, we deduce
1=∂ˆs
∂θ(F,θ,p )∂ˆθ
∂s(F,s,p ), (3.31)
withθ=ˆθ(F,s,p ), which implies the result.

20 Chapter 3. Constitutive Laws
The following Lemmas will be used in the proof of Theorem 3.2.4.
Lemma 3.2.6. LetF∈Lin+andL∈Lin. Then there exists ˜F:R→Lin+such
that˜F(0) = Fand˙˜F(t)˜F(t)−1=L. Moreover
det(˜F(t)) = det( F)etr(L)t>0. (3.32)
Proof. Let us consider the Cauchy problem:
˙˜F(t)=L˜F(t), (3.33)
˜F(0) = F. (3.34)
Existence and uniqueness of solution to this problem, deﬁned for all t∈R,i sa n
immediate consequence of the Picard-Lipschitz Theorem .S i n c e ˜Fis continuous
and det( ˜F(0)) = det( F)>0, the set
A={λ∈R:d e t ( ˜F(t))>0,0≤t≤λ} (3.35)
is nonempty. Let τ=s u p {A}. We are going to prove that τ=∞.I n d e e d ,o t h –
erwise det( ˜F(τ)) = 0 because of continuity. Moreover, det( ˜F(t))>0,∀t∈[0,τ).
Hence there exists ˜F(t)−1and, from (3.33), L=˙˜F(t)˜F(t)−1.F u r t h e r m o r e ,
(det(˜F))·(t)=d e t ( ˜F(t)) tr(˙˜F(t)˜F(t)−1)=d e t ( ˜F(t)) tr(L)∀t∈[0,τ),(3.36)
from which (3.32) follows for t∈[0,τ). By using continuity arguments, (3.32)
yields det( ˜F(τ)) = det( F)etr(L)τ>0, which is a contradiction. Then τ=∞and
so˜F(t)∈Lin+and (3.32) holds ∀t∈R. /square
Following Gurtin[5], we denote by Skwthe set of skew second order tensors
and by Orth+the set of rotations.
Lemma 3.2.7. LetW∈SkwandQ∈Orth+.L e t ˜Qbe the solution of the Cauchy
problem
˙˜Q(t)=W˜Q(t), (3.37)
˜Q(0) = Q. (3.38)
Then ˜Q(t)∈Orth+∀t∈R.
Proof. LetZ(t)=˜Q(t)˜Q(t)t.W eh a v e
˙Z(t)=˙˜Q(t)˜Q(t)t+˜Q(t)˙˜Q(t)t=W˜Q(t)˜Q(t)t−˜Q(t)˜Q(t)tW
=WZ(t)−Z(t)W. (3.39)
Since Z(0) = ˜Q(0)˜Q(0)t=QQt=I,Zis a solution of the Cauchy problem
˙Z=WZ−ZW, (3.40)
Z(0) = I, (3.41)

3.2. Coleman-Noll Materials 21
which has unique solution. But it is obvious that the identity Iis also a solution.
Hence we deduce Z(t)=I∀t∈Rand then ˜Q(t)∈Orth. On the other hand, from
the previous Lemma we deduce
det(˜Q(t)) = det( Q)etr(W)t=1, (3.42)
which completes the proof. /square
The next Lemma allows us to build up particular thermodynamic processes
in the constitutive class of a Coleman-Noll material.
Lemma 3.2.8. Let us consider a Coleman-Noll material with constitutive class
C.Suppose (H1)holds. Then, for given F∗∈Lin+andL∗∈Lin,t h e r ee x i s t s
am o t i o n Xsuch that F(p,0) = F∗andgradv=L∗. Let us denote by Tits
trajectory. Moreover, let s∗∈R,p∗∈Bandθ∈C1(T;R+)satisfying
θ(x∗,0) =θ∗:=ˆθ(F∗,s∗,p∗),x∗=X(p∗,0). (3.43)
Then there exists a thermodynamic process (X,T,b,e,θ,q,f,s)∈Csuch that
F(p,0) =F∗,∀p∈B, (3.44)
L(x, t)=L∗,∀(x, t)∈T, (3.45)
s(x∗,0) =s∗. (3.46)
Proof. Let˜F(t) be given as in Lemma 3.2.6, for F=F∗andL=L∗. Let us deﬁne
Xby
X(p,t)=o+˜F(t)(p−o), (3.47)
where ois any point in E.W eh a v e
F(p,t)=˜F(t) (3.48)
and, furthermore,
L(x, t)=˙˜F(t)˜F(t)−1=L∗. (3.49)
Now let us consider the given θand deﬁne sby
s(x, t)=ˆs(˜F(t),θ(x, t),p), (3.50)
withx=X(p,t). Then, since θ∗=ˆθ(F∗,s∗,p∗), from assumption ( H1)w eh a v e
s(x∗,0) = ˆs(F∗,θ(x∗,0),p∗)=ˆs(F∗,θ∗,p∗)=s∗. (3.51)
Finally, let
T(x, t)=ˆT(˜F(t),s(x, t),p)+ˆl(˜F(t),s(x, t),p)(L∗), (3.52)
e(x, t)=ˆe(˜F(t),s(x, t),p), (3.53)
q(x, t)=ˆq(˜F(t),s(x, t),gradθ(x, t),p), (3.54)

22 Chapter 3. Constitutive Laws
and deﬁne bandfby
b=ρ˙v−divT, (3.55)
f=ρ˙e−T·D+d i vq, (3.56)
withρ(x, t)=ρ0(p)
det(˜F(t))andx=X(p,t).
Then the thermodynamic process ( X,T,b,e,θ,q,f,s) belongs to the constitutive
class of the material and satisﬁes the requirements (3.44)–(3.46). /square
Now we are ready to prove Theorem 3.2.4.
Proof. We start with the following form of the second principle (see (1.71)):
ρθ˙s−ρ˙e+T·D−1
θq·gradθ≥0. (3.57)
By using (3.14) we deduce
˙e(x, t)=∂ˆe
∂s(F(p,t),s(x, t),p)˙s(x, t)+∂ˆe
∂F(F(p,t),s(x, t),p)·˙F(p,t),(3.58)
withx=X(p,t), which can be used in (3.57) to get
ρ/bracketleftbigg
θ−∂ˆe
∂s/bracketrightbigg
˙s+/bracketleftbigg
T·D−ρ∂ˆe
∂F·˙F/bracketrightbigg
−1
θq·gradθ≥0. (3.59)
Since Tis symmetric we can replace T·DbyT·L.M o r e o v e r , ˙F=LF(see Lemma
2.1.3). Then (3.59) becomes
ρ/bracketleftbigg
θ−∂ˆe
∂s/bracketrightbigg
˙s+/bracketleftbigg
T−ρ∂ˆe
∂FFt/bracketrightbigg
·L−1
θq·gradθ≥0, (3.60)
where we have used the Lemma 2.1.2.
Now we choose a particular thermodynamic process. For this we apply Lem-
ma 3.2.8 for L∗=0 ,a n yg i v e n F∗,s∗,p∗,a n d θsatisfying (3.43), grad θ≡0as
well as θ/prime(x∗,0) =a,abeing any real number. Then (3.60) becomes
ρ/bracketleftbigg
θ−∂ˆe
∂s/bracketrightbigg
˙s≥0. (3.61)
But ˙s=∂ˆs
∂θ˙θbecause ˙F≡0 and hence
ρ/bracketleftbigg
θ−∂ˆe
∂s/bracketrightbigg∂ˆs
∂θ˙θ≥0. (3.62)
Then we take p=p∗andt= 0 in (3.62). We get
ρ/bracketleftbigg
ˆθ(F∗,s∗,p∗)−∂ˆe
∂s(F∗,s∗,p∗)/bracketrightbigg∂ˆs
∂θ(F∗,θ∗,p∗)a≥0, (3.63)

3.2. Coleman-Noll Materials 23
from which (3.26) follows just by taking, successively, a=1a n d a=−1, and
using the fact that∂ˆs
∂θ(F∗,θ∗,p∗)/negationslash= 0 (see Remark 3.2.5).
Now we apply Lemma 3.2.8 with the following choices: L∗=αL,αbeing
any real number and Lbeing any second order tensor, any given F∗,s∗,p∗andθ
satisfying (3.43) with grad θ=0. Then (3.60) becomes
/bracketleftbigg
ˆT(˜F(t),s(x, t),p)−ρ∂ˆe
∂F(˜F(t),s(x, t),p)˜Ft(t)/bracketrightbigg
·(αL)
+ˆl(˜F(t),s(x, t),p)(αL)·(αL)≥0 (3.64)
and, by taking t=0a n d p=p∗,w eh a v e s(x∗,0) =s∗and then
α/bracketleftbigg
ˆT(F∗,s∗,p∗)−ρ0(p∗)
det(F∗)∂ˆe
∂F(F∗,s∗,p∗)(F∗)t/bracketrightbigg
·L
+α2ˆl(F∗,s∗,p∗)(L)·L≥0. (3.65)
In order for the previous inequality to hold for all αinR,it is necessary and
suﬃcient that (3.27) and (3.28) be satisﬁed.
Finally, we consider the thermodynamic process provided by Lemma 3.2.8
forL∗=0 ,a n yg i v e n F∗,s∗,p∗andθ≥0 satisfying (3.43) and grad θ(x∗,0) =w,
withw∈V. Then, from (3.60), we deduce
−ˆq(˜F(t),s(x, t),gradθ(x, t),p)·gradθ(x, t)≥0, (3.66)
because θ>0. By taking t=0a n d p=p∗we obtain (3.29).
To prove suﬃciency, we notice that (3.57) is equivalent to (3.60) for Coleman-
Noll materials. But it is obvious that if (3.26)–(3.29) are satisﬁed then the latter
holds. /square
Remark 3.2.9.In what follows we assume that ( H1)a n dt h e Second Law of Ther-
modynamics hold.
Corollary 3.2.10. For a Coleman-Noll material we have the following form of the
energy equation:
ρθ˙s=l(L)·D−divq+f. (3.67)
Proof. Firstly,
˙e(x, t)=∂ˆe
∂s(F(p,t),s(x, t),p)˙s(x, t)+∂ˆe
∂F(F(p,t),s(x, t),p)·˙F(p,t)
=θ(x, t)˙s(x, t)+1
ρ(x, t)ˆT(F(p,t),s(x, t),p)F−t(p,t)·˙F(p,t)
=θ(x, t)˙s(x, t)+1
ρ(x, t)ˆT(F(p,t),s(x, t),p)·L(x, t). (3.68)
By replacing this expression for ˙ ein (3.10) we get (3.67). /square

24 Chapter 3. Constitutive Laws
Remark 3.2.11.We have
divq=d i v/parenleftBig
θq
θ/parenrightBig
=g r a d θ·q
θ+θdiv/parenleftBigq
θ/parenrightBig
. (3.69)
Hence (3.67) can be written as
ρ˙s=1
θl(L)·D−1
θ2gradθ·q−div/parenleftBigq
θ/parenrightBig
+f
θ, (3.70)
which yields
ρ˙s+d i v/parenleftBigq
θ/parenrightBig
−f
θ=1
θ/parenleftbigg
l(L)·D−1
θgradθ·q/parenrightbigg
. (3.71)
Thus, the scalar ﬁeld
Φ: =l(L)·D−1
θgradθ·q, (3.72)
which is called dissipation rate , is responsible for production of non-reversible
entropy. From the Clausius-Duhem inequality we easily deduce that Φ ≥0i nT.
Remark 3.2.12.Property (3.29) means that heat goes from hot to cold regions in
the following sense: let ( x, t)∈Tandnbe a unit normal vector. Let Sbe any
surface containing xand having nas a normal vector at this point. Then we recall
that the heat per unit surface and time ﬂowing across Sfrom the negative side of
Sto its positive side is given by q(x, t)·n.L e tu st a k e n=gradθ(x,t)
|gradθ(x,t)|.T h e nw e
have
q(x, t)·n=ˆq(F(p,t),s(x, t),gradθ(x, t),p)·gradθ(x, t)
|gradθ(x, t)|≤0, (3.73)
by using (3.29) for w=g r a d θ(x, t). Now we recall that grad θ(x, t) is the direction
of highest growth of θ(·,t).
Deﬁnition 3.2.13. A thermodynamic process is called isentropic if ˙s≡0.
Remark 3.2.14.This means that material points conserve entropy during motion,
i.e., that there exists a function s0:B→Rsuch that
s(X(p,t),t)=s0(p). (3.74)
In particular, if s0is constant then s(x, t)=s0∀(x, t)∈T. Some authors say that
an isentropic process is a thermodynamic process such that s(x, t) is constant in
T.
The next result is a straightforward consequence of the above deﬁnition and
(3.67).

3.2. Coleman-Noll Materials 25
Proposition 3.2.15. A thermodynamic process of a Coleman-Noll material is isen-
tropic if and only if the following equation holds:
ˆl(F(p,t),s0(p),p)(L)·D−divq+f=0. (3.75)
Moreover, the motion equation becomes
ρ˙v−divˆT(F(p,t),s0(p),p)−div/bracketleftBig
ˆl(F(p,t),s0(p),p)(L)/bracketrightBig
=b. (3.76)
Remark 3.2.16.We notice that for an isentropic process the mass and the motion
equations can be solved independently of the energy equation.
Isentropic thermodynamic processes are of great importance in continuum
mechanics. An important ﬁeld where they arise is in standard (i.e., non-dissipative)
linear acoustics. This subject will be studied in Chapter 11.
Deﬁnition 3.2.17. A thermodynamic process is called adiabatic ifq≡0andf≡0.
Deﬁnition 3.2.18. A thermodynamic process is called Eulerian ifl≡0.
Proposition 3.2.19. Any Eulerian adiabatic thermodynamic process is isentropic.
Proof. It is an immediate consequence of (3.67) and the above Deﬁnitions. /square

Chapter 4
The Principle of Material
Frame-Indiﬀerence
In this chapter we state the principle of material frame-indiﬀerence and prove
that it implies that the response functions of Coleman-Noll material bodies have
particular forms.
4.1 Change in the Observer. The Indiﬀerence Principle
We start by recalling some deﬁnitions.
Deﬁnition 4.1.1. Two thermodynamic processes ( X,T,b,e,θ,q,f,s)a n d
(X∗,T∗,b∗,e∗,θ∗,q∗,f∗,s∗)a r erelated by a change in the observer if there exist
two C3functions q:R→EandQ:R→Orth+such that
X∗(p,t)=q(t)+Q(t)(X(p,t)−o), (4.1)
T∗(x∗,t)=Q(t)T(x, t)Q(t)t, (4.2)
e∗(x∗,t)=e(x, t), (4.3)
θ∗(x∗,t)=θ(x, t), (4.4)
q∗(x∗,t)=Q(t)q(x, t), (4.5)
s∗(x∗,t)=s(x, t), (4.6)
for some o∈E,w i t h x=X(p,t)a n d x∗=X∗(p,t).
Deﬁnition 4.1.2. We say that the response of a material body is independent of
the observer if its constitutive class, C, has the following property:
If (X,T,b,e,θ,q,f,s)∈Cand (X∗,T∗,b∗,e∗,θ∗,q∗,f∗,s∗) is another ther-
modynamic process which is related to the previous one by a change in the ob-
server, then ( X∗,T∗,b∗,e∗,θ∗,q∗,f∗,s∗)∈C.

28 Chapter 4. The Principle of Material Frame-Indiﬀerence
Deﬁnition 4.1.3. We say that a material body satisﬁes the material frame-indif-
ference principle if its response is independent of the observer, in the sense of the
previous deﬁnition.
4.2 Consequences for Coleman-Noll Materials
Firstly we can prove the following:
Proposition 4.2.1. Let us consider a Coleman-Noll material. If it satisﬁes the
material frame-indiﬀerence principle, then the following conditions hold:
ˆe(F,s,p )=ˆe(QF,s ,p ), (4.7)
ˆθ(F,s,p )=ˆθ(QF,s ,p ), (4.8)
QˆT(F,s,p )Qt=ˆT(QF,s ,p ), (4.9)
Qˆl(F,s,p )(L)Qt=ˆl(QF,s ,p )(QLQt+W), (4.10)
Qˆq(F,s,w,p)=ˆq(QF,s ,Q w,p), (4.11)
for all F∈Lin+,L∈Lin,s∈R,p∈B,Q∈Orth+andW∈Skw. The converse
is also true.
Proof. Let (X,T,b,e,θ,q,f,s)∈C. For arbitrarily given C3functions q:R→E
andQ:R→Orth+, we introduce another thermodynamic process as follows,
X∗(p,t)=q(t)+Q(t)(X(p,t)−o), (4.12)
T∗(x∗,t)=Q(t)T(x, t)Q(t)t, (4.13)
e∗(x∗,t)=e(x, t), (4.14)
θ∗(x∗,t)=θ(x, t), (4.15)
q∗(x∗,t)=Q(t)q(x, t), (4.16)
s∗(x∗,t)=s(x, t), (4.17)
withb∗andf∗deﬁned so as to satisfy the motion and the energy equations.
According to Deﬁnition 4.1.1, this process is related to ( X,T,b,e,θ,q,f,s)∈C
by a change in the observer. Hence it must belong to C,w h i c hm e a n s
T∗(x∗,t)=ˆT(F∗(p,t),s∗(x∗,t),p)+ˆl(F∗(p,t),s∗(x∗,t),p)(L∗(x∗,t)),(4.18)
e∗(x∗,t)=ˆe(F∗(p,t),s∗(x∗,t),p),(4.19)
θ∗(x∗,t)=ˆθ(F∗(p,t),s∗(x∗,t),p),(4.20)
q∗(x∗,t)=ˆq(F∗(p,t),s∗(x∗,t),grad∗θ∗(x∗,t),p),(4.21)

4.2. Consequences for Coleman-Noll Materials 29
and therefore
ˆT(Q(t)F(p,t),s(x, t),p)
+ˆl(Q(t)F(p,t),s(x, t),p)/parenleftBig
Q(t)L(x, t)Q(t)t+˙Q(t)Q(t)t/parenrightBig
=Q(t)ˆT(F(p,t),s(x, t),p)Q(t)t+Q(t)ˆl(F(p,t),s(x, t),p)(L(x, t))Q(t)t,(4.22)
ˆe(Q(t)F(p,t),s(x, t),p)=ˆe(F(p,t),s(x, t),p), (4.23)
ˆθ(Q(t)F(p,t),s(x, t),p)=ˆθ(F(p,t),s(x, t),p), (4.24)
ˆq(Q(t)F(p,t),s(x, t),Q(t)grad θ(x, t),p)
=Q(t)ˆq(F(p,t),s(x, t),gradθ(x, t),p), (4.25)
w h e r ew eh a v eu s e dt h a t( s e e[ 5 ] )
F∗(p∗,t)=Q(t)F(p,t), (4.26)
L∗(x∗,t)=Q(t)L(x, t)Q(t)t+˙Q(t)Q(t)t, (4.27)
and the fact that grad∗θ∗=Qgradθ.
Now we use Lemma 3.2.8 to build a particular thermodynamic process in C.
Let us take L∗=0a n da n y F∗,s∗,p∗,a n dθsatisfying (3.43) with grad θ(x∗,0) =
w∈V.M o r e o v e r ,l e t Q(t)=QwithQ∈Orth+. Then, from (4.22)–(4.25) we
have
ˆT(QF∗,s(x, t),p)=QˆT(F∗,s(x, t),p)Qt, (4.28)
ˆe(QF∗,s(x, t),p)=ˆe(F∗,s(x, t),p), (4.29)
ˆθ(QF∗,s(x, t),p)=ˆθ(F∗,s(x, t),p), (4.30)
ˆq(QF∗,s(x, t),Qgradθ(x, t),p)=Qˆq(F∗,s(x, t),gradθ(x, t),p), (4.31)
and taking t=0a n d p=p∗,
ˆT(QF∗,s∗,p∗)=QˆT(F∗,s∗,p∗)Qt, (4.32)
ˆe(QF∗,s∗,p∗)=ˆe(F∗,s∗,p∗), (4.33)
ˆθ(QF∗,s∗,p∗)=ˆθ(F∗,s∗,p∗), (4.34)
ˆq(QF∗,s∗,Qw,p∗)=Qˆq(F∗,s∗,w,p∗). (4.35)
Now, by using (4.32) in (4.22) we obtain
ˆl(Q(t)F(p,t),s(x, t),p)/parenleftBig
Q(t)L(x, t)Q(t)t+˙Q(t)Q(t)t/parenrightBig
=Q(t)ˆl(F(p,t),s(x, t),p)(L(x, t))Q(t)t. (4.36)
Let us consider the thermodynamic process given by Lemma 3.2.8 for the following
choices: any F∗∈Lin+,L∗∈Lin, s∗∈R,p∗∈Bandθsatisfying (3.43). This

30 Chapter 4. The Principle of Material Frame-Indiﬀerence
process satisﬁes
F(p,0) =F∗, (4.37)
L(x, t)=L∗, (4.38)
s(x∗,0) =s∗, (4.39)
θ(x∗,0) =θ∗, (4.40)
withx∗=X(p∗,0).
Moreover, let ˜Q(t) be deﬁned by Lemma 3.2.7 for any given Q∈Orth+and
W∈Skw. Then (4.36) yields
ˆl(˜Q(t)F(p,t),s(x, t),p)/parenleftBig
˜Q(t)L∗˜Q(t)t+˙˜Q(t)˜Q(t)t/parenrightBig
=˜Q(t)ˆl(F(p,t),s(x, t),p)(L∗)˜Q(t)t. (4.41)
Finally, by taking t=0a n d p=p∗we get (4.10). /square
As a consequence of this important Proposition we obtain very simple ex-
pressions for l.
Corollary 4.2.2. We have
ˆl(F,s,p )(L)=ˆl(F,s,p )(D), (4.42)
where Ddenotes the symmetric part of L, i.e., D=(L+Lt)
2.
Proof. It is enough to take Q=IandW=−(L−Lt)
2in (4.10). /square
The next Corollary is also a straightforward consequence of (4.10).
Corollary 4.2.3. The following equality holds for any Coleman-Noll material sat-
isfying the material-frame indiﬀerence principle:
ˆl(QF,s ,p )(QDQt)=Qˆl(F,s,p )(D)Qt∀Q∈Orth+. (4.43)
Proof. It is enough to use the previous Corollary and to take into account in (4.10)
that the symmetric part of QLQt+WisQDQt. /square
Proposition 4.2.4. For any Coleman-Noll material satisfying the material-frame
indiﬀerence principle, if furthermore
ˆl(QF,s ,p )=ˆl(F,s,p )∀Q∈Orth+, (4.44)
then there exist two functions
ˆη:Lin+×R×B→ R, (4.45)
ˆξ:Lin+×R×B→ R, (4.46)

4.2. Consequences for Coleman-Noll Materials 31
such that
ˆl(F,s,p )(D)=2 ˆη(F,s,p )D+ˆξ(F,s,p )t r(D)I. (4.47)
Moreover, ˆη(F,s,p )=ˆη(QF,s ,p )andˆξ(F,s,p )=ˆξ(QF,s ,p ),for all Q∈Orth+.
Proof. It follows immediately from Corollary 4.2.3 and the representation theorem
for isotropic linear tensor functions (see Appendix A). This result would be applied,
for any ﬁxed F,s,p , to the mapping G=ˆl(F,s,p ). /square
Proposition 4.2.5. Suppose that
ˆl(F,s,p )(D)=2 ˆη(F,s,p )D+ˆξ(F,s,p )t r(D)I. (4.48)
Then the dissipation inequality holds if and only if
ˆη≥0,2ˆη+3ˆξ≥0. (4.49)
Proof. We have tr( D)=I·Dand then
|tr(D)|2≤|I|2|D|2=3|D|2. (4.50)
Moreover
l(L)·L=l(D)·D=2 ˆη|D|2+ˆξ(t r(D))2. (4.51)
Let us assume that (4.49) holds. Then
l(L)·L=l(D)·D≥2ˆη
3(t r(D))2+ˆξ(t r (D))2=2ˆη+3ˆξ
3(t r(D))2≥0.
(4.52)
Conversely, let D/negationslash= 0 with tr( D)=0 .W eh a v e
0≤l(D)·D=2 ˆη|D|2(4.53)
and then ˆ η≥0.
Now we take D=I.W eo b t a i n
0≤l(I)·I=2 ˆη|I|2+ˆξ(t r (I))2=( 2 ˆη+3ˆξ)3, (4.54)
from which (4.49) follows. /square

Chapter 5
Replacing Entropy with
Temperature
The equations we have deduced from the conservation and the constitutive laws
involve unknowns ρ,X,sandθ. While θcan be eliminated in terms of Xand
sthrough the response function ˆθ(F,s,p ), in most cases it is more interesting to
retain this ﬁeld instead of entropy. In this chapter we use the Helmholtz free energy
deﬁned in (1.72) and assumption ( H1) to eliminate entropy in the motion and the
energy equations.
5.1 The Conservation Equations in Terms
of Temperature
F i r s t l yw eh a v et h ef o l l o w i n g :
Proposition 5.1.1. Let us write the free energy ψas a function of Fandθ,n a m e l y ,
ψ=ˆψ(F,θ,p ).T h e n
∂ˆψ
∂θ(F,θ,p )=−s, (5.1)
∂ˆψ
∂F(F,θ,p )=∂ˆe
∂F(F,s,p )=det(F)
ρ0(p)ˆT(F,s,p )F−t, (5.2)
withs=ˆs(F,θ,p ).
Proof. We apply the chain rule to the mapping
ˆe(F,s,p )=ˆψ(F,ˆθ(F,s,p ),p)+ˆθ(F,s,p )s (5.3)

34 Chapter 5. Replacing Entropy with Temperature
and use (3.26) to deduce
ˆθ(F,s,p )=∂ˆe
∂s(F,s,p )=∂ˆψ
∂θ(F,θ,p )∂ˆθ
∂s(F,s,p )+∂ˆθ
∂s(F,s,p )s
+ˆθ(F,s,p ), (5.4)
from which it follows that
/bracketleftBigg
∂ˆψ
∂θ(F,θ,p )+s/bracketrightBigg
∂ˆθ
∂s(F,s,p ) = 0 (5.5)
and then (5.1).
Moreover, by taking the derivative with respect to Fin (5.3) and using (5.1)
we obtain
∂ˆe
∂F(F,s,p )=∂ˆψ
∂F(F,θ,p )+∂ˆψ
∂θ(F,θ,p )∂ˆθ
∂F(F,s,p )+∂ˆθ
∂F(F,s,p )s
=∂ˆψ
∂F(F,θ,p ). (5.6)
/square
From (5.2) we obtain the following:
Corollary 5.1.2. We have
ˆT(F,θ,p )=ρ0(p)
det(F)∂ˆψ
∂F(F,θ,p )Ft, (5.7)
where x=X(p,t).
If we know the free energy response function ˆψor, more precisely, its partial
derivatives with respect to Fandθ, then we can state the equations to describe any
thermodynamic process of a Coleman-Noll material. Indeed, according to (5.7) and
using equation (3.67), we can write the mass, momentum and energy conservation
equations as
˙ρ+ρdivv=0, (5.8)
ρ˙v=d i v/bracketleftBigg
ρ∂ˆψ
∂F(F,θ,p )Ft/bracketrightBigg
+d i v ( l(D)) +b, (5.9)
−ρθ/parenleftBigg
∂ˆψ
∂θ(F,θ,p )/parenrightBigg·
=l(D)·D−divq+f. (5.10)
Deﬁnition 5.1.3. The speciﬁc heat at constant deformation is the scalar ﬁeld de-
ﬁned by
cF(x, t)=ˆcF(F(p,t),θ(x, t),p), (5.11)

5.1. The Conservation Equations in Terms of Temperature 35
with
ˆcF(F,θ,p ): =∂ˆe
∂θ(F,θ,p ). (5.12)
Proposition 5.1.4. We have
ˆcF(F,θ,p )=θ∂ˆs
∂θ(F,θ,p ). (5.13)
Proof. Indeed,
∂ˆe
∂θ(F,θ,p )=∂ˆe
∂s(F,ˆs(F,θ,p ),p)∂ˆs
∂θ(F,θ,p )=θ∂ˆs
∂θ(F,θ,p ). (5.14)
/square
Now we transform equation (5.10) by using the speciﬁc heat. Assuming
enough regularity we have
/parenleftBigg
∂ˆψ
∂θ(F,θ,p )/parenrightBigg·
=∂2ˆψ
∂θ2(F,θ,p )˙θ+∂2ˆψ
∂F∂θ(F,θ,p )·˙F. (5.15)
Moreover, from (5.1) we obtain
∂2ˆψ
∂θ2(F,θ,p )=−∂ˆs
∂θ(F,θ,p ), (5.16)
and, on the other hand,
∂2ˆψ
∂F∂θ(F,θ,p )=∂2ˆψ
∂θ∂F(F,θ,p )=det(F)
ρ0(p)∂ˆT
∂θ(F,θ,p )F−t, (5.17)
where we have used (5.2). Then, by using (5.13) we obtain
/parenleftBigg
∂ˆψ
∂θ(F,θ,p )/parenrightBigg·
=−1
θcF˙θ+det(F)
ρ0(p)∂ˆT
∂θ(F,θ,p )F−t(p,t)·˙F
=−1
θcF˙θ+det(F)
ρ0(p)∂ˆT
∂θ(F,θ,p )·L. (5.18)
By replacing this equality in the energy equation (5.10), we deduce
ρcF˙θ=θ∂ˆT
∂θ(F,θ,p )·L+l(D)·D−divq+f, (5.19)
or, equivalently,
ρcF˙θ=θ∂ˆT
∂θ(F,θ,p )·D+l(D)·D−divq+f, (5.20)
because ˆTisSym-valued and then∂ˆT
∂θ(F,θ,p )∈Sym.

Chapter 6
Isotropy
Isotropy is a property which leads to som e interesting simpliﬁcations in the con-
stitutive laws. It is satisﬁed by many Coleman-Noll materials, in particular, bythe ﬂuids to be considered in Chapter 10.
6.1 The Extended Symmetry Group
Deﬁnition 6.1.1. Let us consider a Coleman-Noll material. A symmetry transfor-
mation atp∈Bis a tensor H∈Lin+such that
ˆT(F,s,p )=ˆT(FH,s ,p ), (6.1)
ˆe(F,s,p )=ˆe(FH,s ,p ), (6.2)
ˆθ(F,s,p )=ˆθ(FH,s ,p ), (6.3)
ˆl(F,s,p )=ˆl(FH,s ,p ), (6.4)
ˆq(F,s,w,p)=ˆq(FH,s, w,p), (6.5)
∀F∈Lin+,s∈R,w∈V.
Roughly speaking, a symmetry transformation is an orientation-preserving
change in the reference conﬁguration which leaves the response of the materialunaltered.
We denote by H
pthe set of all symmetry transformations at p.I ti sv e r ye a s y
to see that Hpis a subgroup of Lin+called the extended symmetry group atp.
Remark 6.1.2.We notice that, if the material satisﬁes the Second Principle ,t h e n
(6.2) implies (6.3). It also implies (6.1) if det( H)=1 .
We are also interested in replacing sbyθas an independent variable. For
this purpose we need the next Lemmas.

38 Chapter 6. Isotropy
Lemma 6.1.3. Let us assume the material indiﬀerence principle holds. Then
QˆT(F,θ,p )Qt=ˆT(QF,θ,p ), (6.6)
Qˆq(F,θ,w,p)=ˆq(QF,θ,Q w,p), (6.7)
for all Q∈Orth+.
Proof. We have
ˆT(QF,θ,p )=ˆT(QF,ˆs(QF,θ,p ),p)=QˆT(F,ˆs(QF,θ,p ),p)Qt
=QˆT(F,ˆs(F,θ,p ),p)Qt=QˆT(F,θ,p )Qt, (6.8)
where we have used (4.9) and (4.8) and assumption ( H1).
Similarly, from (4.11), (4.8) and assumption ( H1), we have
ˆq(QF,θ,Q w,p)= ˆq(QF,ˆs(QF,θ,p ),Qw,p)=Qˆq(F,ˆs(QF,θ,p ),w,p)
=Qˆq(F,ˆs(F,θ,p ),w,p)=Qˆq(F,θ,w,p), (6.9)
which completes the proof. /square
Lemma 6.1.4. The following equalities hold for all H∈H p:
ˆT(F,θ,p )=ˆT(FH,θ,p ), (6.10)
ˆq(F,θ,w,p)=ˆq(FH,θ, w,p). (6.11)
Proof. We have
ˆT(FH,θ,p )=ˆT(FH,ˆs(FH,θ,p ),p)=ˆT(F,ˆs(FH,θ,p ),p)
=ˆT(F,ˆs(F,θ,p ),p)=ˆT(F,θ,p ), (6.12)
where we have used (6.1) and (6.3), and assumption ( H1).
Moreover,
ˆq(FH,θ, w,p)=ˆq(FH,ˆs(FH,θ,p ),w,p)=ˆq(F,ˆs(FH,θ,p ),w,p)
=ˆq(F,ˆs(F,θ,p ),w,p)=ˆq(F,θ,w,p).(6.13)
/square
As a consequence of these two last Lemmas the results given below concerning
the response functions ˆTand ˆqare true for yreplaced by either speciﬁc entropy
or temperature.
Firstly we are able to prove the following:
Proposition 6.1.5. The Functions ˆT(F,y,p )andˆq(F,y,w,p)are invariant under
Hp∩Orth+, i.e.,
ˆT(QFQt,y,p)=QˆT(F,y,p )Qt, (6.14)
ˆq(QFQt,y,Qw,p)=Qˆq(F,y,w,p), (6.15)
∀Q∈H p∩Orth+.

6.2. Isotropic Bodies 39
Proof. By using successively Lemma 6.1.3 and Lemma 6.1.4 we deduce
QˆT(F,y,p )Qt=ˆT(QF,y,p )=ˆT(QFQt,y,p), (6.16)
because if Q∈H p∩Orth+,t h e n Qt=Q−1∈H p∩Orth+.
Similarly,
Qˆq(F,y,w,p)=ˆq(QF,y,Q w,p)=ˆq(QFQt,y,Qw,p). (6.17)
/square
6.2 Isotropic Bodies
Deﬁnition 6.2.1. A material body is said to be isotropic atpif its symmetry group
at this point contains all rotations, i.e.,
Orth+⊂H p. (6.18)
Corollary 6.2.2. Let us assume that the material is isotropic at p. Then, for each
(y,p), the mappings ˆT(·,y,p)andq(·,y,·,p)are isotropic, that is,
ˆT(QFQt,y,p)=QˆT(F,y,p )Qt, (6.19)
ˆq(QFQt,y,Qw,p)=Qˆq(F,y,w,p), (6.20)
∀Q∈Orth+.
Now we recall the Theorem of Representation of isotropic tensor functions
(see for instance [5]).
Theorem 6.2.3. A mapping
G:A⊂Sym→Sym (6.21)
is isotropic if and only if there exist three scalar functions,
ϕi:I(A)→Ri=0,1,2, (6.22)
such that
G(A)=ϕ0(IA)I+ϕ1(IA)A+ϕ2(IA)A2(6.23)
for every A∈A,w h e r e IAdenotes the set of the three principal invariants of
tensor A.
We notice that this Theorem cannot be directly applied to ˆT(F,y,p )b e c a u s e ,
in general, Fis not symmetric. However, with this result and Corollary 6.2.2 we
easily prove the following:

40 Chapter 6. Isotropy
Proposition 6.2.4. Let us assume the material at pis isotropic. Then the response
function ˆT(F,y,p )c a nb ew r i t t e ni nt h ef o r m
ˆT(F,y,p )=β0(IB,y,p)I+β1(IB,y,p)B+β2(IB,y,p)B2, (6.24)
w h e r ew er e c a l lt h a t B:=FFtis the left Cauchy-Green strain tensor.
Proof. Let us consider the polar decomposition of F,n a m e l y ,
F=RU=VR , R ∈Orth+,U , V ∈Psym. (6.25)
We recall that V=B1/2=RURt.Since the material at pis isotropic we have
ˆT(F,y,p )=ˆT(VR,y,p )=ˆT(V,y,p )=ˆT(B1/2,y,p). (6.26)
Let us introduce the mapping
˜T(B,y,p ): =ˆT(B1/2,y,p). (6.27)
We are going to prove that ˜T(B,y,p ) is isotropic. Indeed, for every Q∈Orth+,
˜T(QBQt,y,p)=˜T(QB1/2QtQB1/2Qt,y,p)=ˆT(QB1/2Qt,y,p)
=QˆT(B1/2,y,p)Qt=Q˜T(B,y,p )Qt. (6.28)
Then, for y∈Rwe can apply Theorem 6.2.3 to the tensor mapping
G(B): =˜T(B,y,p ), (6.29)
which yields the result. /square
Since the eigenvalues of BandCare the same, their corresponding principal
invariants are also the same and then we can easily prove the following:
Corollary 6.2.5. Let us assume the material is isotropic at p. Then the response
function ˆP(F,y,p )for the Second Piola-Kirchhoﬀ stress tensor can be written in
the form
ˆP(F,y,p )=˜P(C,y,p )=γ0(IC,y,p)I+γ1(IC,y,p)C+γ2(IC,y,p)C2.(6.30)
Proof.
ˆP(F,y,p )=d e t ( F)F−1ˆT(F,y,p )F−t(6.31)
=d e t ( F)F−1[β0(IB,y,p)I+β1(IB,y,p)B+β2(IB,y,p)B2]F−t
(6.32)
=i3(C)1/2[β0(IC,y,p)C−1+β1(IC,y,p)I+β2(IC,y,p)C].(6.33)
The result follows by using the Cayley-Hamilton Theorem , allowing us to write
C−1in terms of I,CandC2. /square

6.2. Isotropic Bodies 41
Deﬁnition 6.2.6. A Coleman-Noll material is called a Saint Venant-Kirchhoﬀ ma-
terial if its response function for the second Piola-Kirchhoﬀ stress is of the form
ˆP(F,y,p )=λtr(G)I+2µG (6.34)
where G=1
2(FtF−I)i st h e Green-Saint Venant strain tensor andλandµare
two constants.
For further results in nonlinear elasticity we refer to [2].
In order to give a simpler form of ˆqin the case of isotropy we recall another
Theorem of Representation for vector valued functions (see [8])
Theorem 6.2.7. Af u n c t i o n
g:A×C⊂ Sym×V→V (6.35)
is isotropic if and only if there exist three scalar functions,
φi:A×C⊂ Sym×V→ Ri=0,1,2 (6.36)
of the form
φi(A,w)=ϕi(IA,w·w,w·Aw,w·A2w),i=0,1,2, (6.37)
such that
g(A,w)=φ0(A,w)w+φ1(A,w)Aw+φ2(A,w)A2w (6.38)
for every A∈A,w∈C.
Again, this theorem cannot be directly applied to ˆq(F,y,w,p) but we have
a similar result to Proposition 6.2.4:
Proposition 6.2.8. Let us assume that the material at pis isotropic. Then the
response function ˆq(F,y,w,p)can be written as
ˆq(F,y,w,p)=α0(B,w,y,p)w+α1(B,w,y,p)Bw+α2(B,w,y,p)B2w,
(6.39)
where αi,i=0,1,2are functions of the following form:
αi(B,w,y,p)=γi(IB,w·w,w·Bw,w·B2w,y,p). (6.40)
Proof. Since the material at pis isotropic,
ˆq(F,y,w,p)=ˆq(VR ,y , w,p)=ˆq(V,y,w,p)
=ˆq(B1/2,y,w,p). (6.41)

42 Chapter 6. Isotropy
Let us introduce the mapping
˜q(B,y,w,p): =ˆq(B1/2,y,w,p). (6.42)
Now we prove that ˜q(·,y,w,p) is isotropic. Indeed, for every Q∈Orth+,
˜q(QBQt,y,Qw,p)= ˜q(QB1/2QtQB1/2Qt,y,Qw,p)
=ˆq(QB1/2Qt,y,Qw,p)= Qˆq(B1/2,y,w,p)=Q˜q(B,y,w,p).(6.43)
Then, for every y∈Rwe can apply Theorem 6.2.7 to the vector mapping
g(B,w): =˜q(B,y,w,p), (6.44)
which yields the result. /square
Corollary 6.2.9. Let us assume that the material is isotropic at pand that ˆqdoes
not depend on F. Then there exists a mapping ˆk:R×R+×B→ Rsuch that
ˆq(F,y,w,p)=−ˆk(y,|w|2,p)w. (6.45)
Proof. It is enough to take F=Iin (6.39). Function ˆkis given by
ˆk(y,|w|2,p)=−/bracketleftbig
γ0(|w|2,y,p)+γ1(|w|2,y,p)+γ2(|w|2,y,p)/bracketrightbig
. (6.46)
/square
Remark 6.2.10.We notice that, under the assumption of the above Corollary, the
heat ﬂux qis given by
q=−ˆk(y,|gradθ|2,p)gradθ, (6.47)
which is a generalized version of the Fourier law . Moreover, from (3.29) we deduce
q·gradθ=−ˆk(y,|gradθ|2,p)|gradθ|2≤0 (6.48)
and then
ˆk(y,|gradθ|2,p)≥0, (6.49)
if grad θ/negationslash=0.
The scalar ﬁeld k(x, t)=ˆk(y(x, t),|gradθ(x, t)|2,p) is called thermal conduc-
tivity.

Chapter 7
Equations in Lagrangian
Coordinates
In previous chapters we have introduced some partial diﬀerential equations which
should be solved in order to determine the thermodynamic process followed by a
material body. The diﬃculty of this system is that it takes place in the deformedconﬁguration along time, i.e., in the trajectory Twhich, in many cases, is not
known in advance. This is generally true for solids. For ﬂuids, since they usually
move in domains bounded by solids which can be considered as rigid, the set Tis
a data of the problem. Of course, we must exclude free boundary ﬂows from this
situation (see Appendix D).
From Eulerian to Lagrangian Coordinates
In this section we show how one can change from spatial (Eulerian) to material
(Lagrangian) coordinates, in order to obtain a new system of partial diﬀerential
equations which holds in the (known) reference conﬁguration B.
We can introduce a response function for the ﬁrst Piola-Kirchhoﬀ stress ten-
sor (see (2.1)) from the response function for the Cauchy stress tensor, namely
ˆS(F,H,θ,p ): =ˆSelas(F,θ,p )+ˆSvisc(F,H,θ,p ), (7.1)
with
ˆSelas(F,θ,p ): =d e t ( F)ˆT(F,θ,p )F−t(7.2)
and
ˆSvisc(F,H,θ,p ): =d e t ( F)ˆl(F,θ,p )(HF−1)F−t. (7.3)
We notice that, in order to compute the viscous part of S, we have to replace H
with ˙Fin the above formula. Then HF−1=˙FF−1=L(see Lemma 2.1.3).

44 Chapter 7. Equations in Lagrangian Coordinates
Then, by performing the change of variable x=X(p,t) in (1.30) and (5.20),
the following equations, stated in the reference conﬁguration, can be obtained:
Theorem 7.1. Any thermodynamic process of a Coleman-Noll material satisﬁes
ρ0¨u=D i v ˆS(F,˙F,θ,p )+b∗, (7.4)
ρ0cF˙θ=θ∂ˆSelas
∂θ(F,θ,p )·∇˙u+ˆSvisc(F,˙F,θ,p )·∇˙u
−Divˆq∗(F,θ,∇θ,p)+f∗, (7.5)
whereb∗,ˆq∗andf∗are given by
b∗(p,t)=b(X(p,t),t)d et(F(p,t)), (7.6)
ˆq∗(F,θ,w,p)=d e t ( F)F−1ˆq(F,θ,F−tw,p), (7.7)
f∗(p,t)=f(X(p,t),t)d et (F(p,t)). (7.8)
The proof of this theorem relies upon the following well-known results on
change of variable in integrals (see Appendix B):
Lemma 7.2. Letw:T→ V be a continuous vector ﬁeld and R:T→ Lina
continuous tensor ﬁeld. For any part PofBwe have
/integraldisplay
Ptw(x, t)d V x=/integraldisplay
Pw(X(p,t),t)d et(F(p,t)) dV p, (7.9)
/integraldisplay
∂Ptw(x, t)·n(x)dA(x)=/integraldisplay
∂Pdet(F(p,t))F−1(p,t)w(X(p,t),t)·m(p)d A p,
(7.10)
/integraldisplay
∂PtR(x, t)n(x)dA(x)=/integraldisplay
∂Pdet(F(p,t))R(X(p,t),t)F−t(p,t)m(p)d A p,(7.11)
where n(x)(respectively m(p)) denotes the outward unit normal vector to ∂Pt
(respectively to ∂P)a tp o i n t x(respectively p).
Proof of Theorem 7.1. Firstly we recall that equation (7.4) has been already ob-
tained in (2.13) by making the change of variable x=X(p,t) in (1.27). In order
to prove (7.5) we begin with the energy equation written in the form (5.20). By
integrating in Pta n dt h e nu s i n gt h e Gauss Theorem we get
/integraldisplay
PtρcF˙θdVx=/integraldisplay
Ptθ∂ˆT
∂θ(F,θ,p )·DdVx+/integraldisplay
Ptl(L)·DdVx
−/integraldisplay
∂Ptq·ndA(x)+/integraldisplay
PtfdVx. (7.12)

45
Now we perform the change of variable x=X(p,t). We get
/integraldisplay
PtρcF˙θdVx=/integraldisplay
Pρ0cFm˙θmdVp, (7.13)
/integraldisplay
Ptθ∂ˆT
∂θ(F,θ,p )·DdVx=/integraldisplay
Ptθ∂ˆT
∂θ(F,θ,p )·LdVx
=/integraldisplay
Pθ/parenleftBigg
∂ˆT
∂θ/parenrightBigg
m·(˙FF−1)d et (F)d V p=/integraldisplay
Pdet(F)θ/parenleftBigg
∂ˆT
∂θ/parenrightBigg
mF−t·˙FdVp
=/integraldisplay
Pθ∂ˆSelas
∂θ·∇˙udVp, (7.14)
/integraldisplay
Ptl(D)·DdVx=/integraldisplay
Ptl(L)·LdVx=/integraldisplay
Pl(˙FF−1)·(˙FF−1)d et(F)d V p
=/integraldisplay
Pdet(F)l(∇˙uF−1)F−t·∇˙udVp, (7.15)
/integraldisplay
∂Ptq·ndA(x)=/integraldisplay
∂Pdet(F)F−1qm·mdAp=/integraldisplay
PDiv(det( F)F−1qm)d V p,
(7.16)/integraldisplay
PtfdVx=/integraldisplay
Pfmdet(F)d V p. (7.17)
From these equalities and the deﬁnitions (7.1) and (7.6)–(7.8) we easily get (7.5),
by using the Gauss and the Localization Theorems . /square
The following important result is a consequence of (5.7).
Corollary 7.3. We have
ˆSelas(F,θ,p )=ρ0(p)∂ˆψ
∂F(F,θ,p ). (7.18)
Remark 7.4.From (7.18) we see that, if l=0 ,t h e nt h e response function ˆSis the
partial derivative with respect to Fof the Helmholtz free energy density function
ˆσ(F,θ,p )= :ρ0(p)ˆψ(F,θ,p ). In this sense, any inviscid Coleman-Noll material is
hyperelastic (see for instance [5, Section 28]).

Chapter 8
Linearized Models
Firstly we emphasize that this chapter applies to any Coleman-Noll material, no
matter whether it is a solid or a ﬂuid. A particularization to ﬂuids of the results
we obtain below will be done in Chapter 11.
Equations (7.4), (7.5) constitute a nonlinear evolutionary system of partial
diﬀerential equations, the unknowns of which are the ﬁelds Xandθ. Hence its
analysis and numerical solution are diﬃcult tasks.
However, in many practical situations, we may assume that only small de-
formations and changes of temperature arise and then this system can be approx-
imated by a linear one. Throughout this chapter we suppose that the gradient of
displacement from an initial equilibrium state, ∇u, is small and that variation of
temperature from this state is also small.
8.1 Linear Approximation of the Motion Equation
We choose as reference conﬁguration the initial equilibrium position of the body,
i.e.,B=B0.L e tb0,f0,ρ0,T0,θ0andq0be the initial values of body sources, den-
sity, Cauchy stress tensor, temperature and heat ﬂux. Assuming initial equilibrium
we have
DivT0+b0=0, (8.1)
−Divq0+f0=0, (8.2)
where both equations hold in B.
Now we consider a “small” thermodynamic process from this initial state.
This means, in particular, that X(p,0) =psoF(p,0) =Iand furthermore
ˆSelas(I,θ0(p),p)=ˆT(I,θ0(p),p)=T0(p), (8.3)
ˆq∗(I,θ0(p),∇θ0(p),p)=ˆq(I,θ0(p),∇θ0(p),p)=q0(p). (8.4)
We need the following technical results:

48 Chapter 8. Linearized Models
Lemma 8.1.1. LetˆKbe the mapping deﬁned in the set of invertible tensors by
ˆK(A)=A−1.T h e n ˆKis diﬀerentiable and, furthermore,
∂ˆK
∂A(A)(H)=−A−1HA−1. (8.5)
Proof. We assume the diﬀerentiability and prove (8.5). Let us consider the auxil-
iary mapping ˆMdeﬁned in LinbyˆM(A)=A.T h e n ˆM(A)ˆK(A)=I.B yu s i n g
the product rule of diﬀerential calculus we get
∂ˆM
∂A(A)(H)ˆK(A)+ˆM(A)∂ˆK
∂A(A)(H)=0.
Since ˆM(A) is linear,∂ˆM
∂A(A)(H)=ˆM(H) from which the result follows. /square
Lemma 8.1.2. The response function ˆT(F,y,p ),w h e r e yrepresents either temper-
ature or speciﬁc entropy, satisﬁes
∂ˆT
∂F(F,y,p )(WF)=WˆT(F,y,p )+ˆT(F,y,p )Wt, (8.6)
for all F∈Lin+,W∈Skwandy∈R.
Proof. Let us consider the ordinary diﬀerential equation, ˙Q=WQ,w i t hi n i t i a l
condition Q(0) = I,w h e r e W∈Skw. From Lemma 3.2.7 we know that Q(t)i sa
rotation for all tand˙Q(0) = W.
Moreover, by taking the derivative of equation (6.6) with respect to twe get
∂ˆT
∂F(QF,y,p )/parenleftBig
˙QF/parenrightBig
=˙Q(t)ˆT(F,y,p )Qt(t)+Q(t)ˆT(F,y,p )˙Qt(t). (8.7)
Finally, by choosing t= 0 in the above equation we obtain (8.6). /square
IfF=I, the previous Lemma yields the following:
Corollary 8.1.3.
∂ˆT
∂F(I,y,p)(W)=WˆT(I,y,p)+ˆT(I,y,p)Wt. (8.8)
From now on let us denote by EandWthe symmetric and skew parts of
tensor ∇udeﬁned by
E=∇u+∇ut
2, (8.9)
W=∇u−∇ut
2. (8.10)
We recall that E∈Symis the inﬁnitesimal strain tensor (see [5]).

8.1. Linear Approximation of the Motion Equation 49
Proposition 8.1.4. We have
1.∂ˆSelas
∂F(I,y,p)(∇u)=∂ˆT
∂F(I,y,p)(E)+D i v uˆT(I,y,p)
−ˆT(I,y,p)E+WˆT(I,y,p). (8.11)
2.∂ˆSvisc
∂F(I,H,y,p )(∇u)=∂l
∂F(I,y,p)(∇u)(H)+D i v uˆl(I,y,p)(H)
−ˆl(I,y,p)(H)∇ut−ˆl(I,y,p)(H∇u). (8.12)
3.∂ˆSelas
∂y(I,y,p)=∂ˆT
∂y(I,y,p). (8.13)
4.∂ˆSvisc
∂y(I,H,y,p )=∂l
∂y(I,y,p)(H). (8.14)
Proof. 1. We have
/hatwideSelas(F,y,p )Ft=d e t ( F)ˆT(F,y,p ) (8.15)
and, by using the product rule of the diﬀerential calculus, we get
∂ˆSelas
∂F(F,y,p )(U)Ft+ˆSelas(F,y,p )Ut
=d e t ( F)t r (UF−1)ˆT(F,y,p )+d e t ( F)∂ˆT
∂F(F,y,p )(U). (8.16)
IfF=IandU=∇u,b yt a k i n gi n t oa c c o u n tt h a t ˆSelas(I,y,p)=ˆT(I,y,p)
we obtain
∂ˆSelas
∂F(I,y,p)(∇u)=D i v uˆT(I,y,p)
+∂ˆT
∂F(I,y,p)(∇u)−ˆT(I,y,p)∇ut. (8.17)
By writing ∇u=E+Wand∇ut=E−W, we deduce
∂ˆSelas
∂F(I,y,p)(∇u)= D i v uˆT(I,y,p)
+∂ˆT
∂F(I,y,p)(E+W)−ˆT(I,y,p)(E−W). (8.18)
Since the diﬀerential is a linear mapping, and using Corollary 8.1.3, the pre-
vious equality yields
∂ˆSelas
∂F(I,y,p)(∇u)= D i v uˆT(I,y,p)+∂ˆT
∂F(I,y,p)(E)
+WˆT(I,y,p)+ˆT(I,y,p)Wt−ˆT(I,y,p)E+ˆT(I,y,p)W. (8.19)
AsWt+W= 0 because Wis skew, then equation (8.19) leads to (8.11).

50 Chapter 8. Linearized Models
2. Firstly we have
ˆSvisc(F,H,y,p )Ft=d e t ( F)ˆl(F,y,p )(HF−1). (8.20)
By applying the product rule and Lemma 8.1.1 we obtain
∂ˆSvisc
∂F(F,H,y,p )(U)Ft+ˆSvisc(F,H,y,p )Ut
=d e t ( F)t r (UF−1)ˆl(F,y,p )(HF−1)+d e t ( F)∂l
∂F(F,y,p )(U)(HF−1)
+d e t ( F)ˆl(F,y,p )(−HF−1UF−1). (8.21)
Let us take F=IandU=∇u.S i n c e ˆSvisc(I,H,y,p )=ˆl(I,y,p)(H),we
ﬁnally deduce (8.12).
3. It follows from the fact that ˆSelas(I,y,p)=ˆT(I,y,p).
4. It follows from the fact that ˆSvisc(I,H,y,p )=ˆl(I,y,p)(H). /square
The following Corollary is an immediate consequence of the diﬀerentiability
of both ˆSelasandˆSviscwith respect to FandyatF=Iandy=y0.
Corollary 8.1.5. We have
ˆS(F,H,y,p )=T0+∂ˆT
∂F(I,y0,p)(E)+D i v uT0−T0E+WT0
+∂ˆT
∂y(I,y0,p)(y−y0)+ˆl(I,y0,p)(H)+∂l
∂F(I,y0,p)(∇u)(H)
+D i vuˆl(I,y0,p)(H)−ˆl(I,y0,p)(H)∇ut−ˆl(I,y0,p)(H∇u)
+∂l
∂y(I,y0,p)(H)(y−y0)+o(∇u)+o(y−y0). (8.22)
Deﬁnition 8.1.6. The linear operator C(θ0,p)∈L(Lin,Lin ) deﬁned by
C(θ0,p): =∂ˆT
∂F(I,θ0,p) (8.23)
is called a (fourth order) elasticity tensor at point pand temperature θ0.
M o r e o v e r ,w eu s et h en o t a t i o n
Y(θ0,p): =∂ˆT
∂θ(I,θ0,p). (8.24)
The next properties follow from the fact that ˆTis symmetric tensor valued.
Proposition 8.1.7. We have
C(θ0,p)(H)∈Sym ∀H∈Lin, (8.25)
Y(θ0,p)∈Sym. (8.26)

8.1. Linear Approximation of the Motion Equation 51
Proposition 8.1.8. Let us suppose the initial stress, T0, is null. Then the elasticity
tensor C(θ0,p)is symmetric, i.e.,
H·C(θ0,p)(G)=C(θ0,p)(H)·G∀H,G∈Lin. (8.27)
Proof. Firstly, if T0=0t h e n
C(θ0,p)=∂ˆSelas
∂F(I,θ0,p) (8.28)
and hence
C(θ0,p)=ρ0(p)∂2ˆψ
∂F2(I,θ0,p). (8.29)
The result follows from the symmetry of the second derivative. /square
Let us assume that∂ˆT
∂F(I,θ0,p) is invertible when considered as a linear map-
ping from SymintoSym. Then, by the Implicit Function Theorem , the equation
T=ˆT(F,θ,p ) deﬁnes Fas a function of Tandθin a neighborhood of F=I,
θ=θ0andT=T0. Furthermore, the derivative of this function, which is called
tensor of thermal expansion at constant (elastic) stress ,i sg i v e nb y
∂ˆF
∂θ(T,θ,p )=−/parenleftBigg
∂ˆT
∂F(F,θ,p )/parenrightBigg−1
∂ˆT
∂θ(F,θ,p ) (8.30)
withT=ˆT(F,θ,p ). By evaluating this expression at F=I,θ=θ0, and hence at
T=T0,w eh a v e
A(θ0,p): =∂ˆF
∂θ(T0,θ0,p)=−C(θ0,p)−1(Y(θ0,p)). (8.31)
Now we assume that the thermodynamic process is a small perturbation of
the reference state, more precisely,
∇u=O(ε),θ−θ0=O(ε),˙F=∇˙u=O(ε), (8.32)
εbeing a small parameter. Then, from Corollary 8.1.5 we can approximate the
Piola-Kirchhoﬀ stress tensor:
S=T0+C(θ0,p)(E)+D i v uT0
−T0E+WT0+Y(θ0,p)(θ−θ0)+ˆl(I,θ0,p)(˙F)+o(ε).
(8.33)
Ifθ=θ0,ˆl≡0 and the initial state is stress free , i.e., T0= 0, then this approxi-
mation becomes
S=C(θ0,p)(E), (8.34)

52 Chapter 8. Linearized Models
which is called the generalized Hooke law .
By replacing (8.33) in (7.4) we get the linearized motion equation
ρ0¨u=D i v[ T0+D i vuT0−T0E+WT0]
+D i v[ C(θ0,p)(E−A(θ0,p)(θ−θ0))]
+D i v/bracketleftBig
ˆl(I,θ0,p)(∇˙u)/bracketrightBig
+b∗. (8.35)
IfT0= 0 then the above equation simpliﬁes to
ρ0¨u=D i v[ C(θ0,p)(E−A(θ0,p)(θ−θ0))]
+D i v/bracketleftBig
ˆl(I,θ0,p)(∇˙u)/bracketrightBig
+b∗. (8.36)
We recall that, under assumption (4.44), lis given by (4.47) and then
ˆl(I,θ0,p)(∇˙u)=2 ˆη(I,θ0,p)˙E+ˆξ(I,θ0,p)D iv˙uI. (8.37)
Moreover, if the reference state is spatially homogeneous ,
ρ0¨u=D i v[ C(θ0,p)(E−A(θ0,p)(θ−θ0))]
+η0∆˙u+(η0+ξ0)∇Div˙u+b∗. (8.38)
Remark 8.1.9.Usually, body force bis gravity force. In this situation, by using
the mass conservation equation (1.20), we get
b∗(p,t)=d e t F(p,t)b(X(p,t),t)=d e t ( F(p,t))ρ(X(p,t),t)g(X(p,t))
=ρ0(p)g(X(p,t)) =ρ0(p)g(p+u(p,t)). (8.39)
Thusb∗can be approximated by neglecting the o(u(p)) term in the following
equality:
b∗(p,t)=ρ0(p)[g(p)+∇g(p)u(p,t)+o(u(p,t))]. (8.40)
8.2 Linear Approximation of the Energy Equation
Let us go to the energy equation. Firstly we linearize the heat ﬂux.

8.2. Linear Approximation of the Energy Equation 53
Proposition 8.2.1. We have
1. ∂ˆq∗
∂F(I,y,w,p)(∇u)=∂ˆq
∂F(I,y,w,p)(∇u)+D i v uˆq(I,y,w,p)
−∇uˆq(I,y,w,p)−∂ˆq
∂w(I,y,w,p)∇utw. (8.41)
2. ∂ˆq∗
∂y(I,y,w,p)=∂ˆq
∂y(I,y,w,p). (8.42)
3. ∂ˆq∗
∂w(I,y,w,p)=∂ˆq
∂w(I,y,w,p). (8.43)
Proof. From (7.7) we have
Fˆq∗(F,y,w,p)=d e t ( F)ˆq(F,y,F−tw,p) (8.44)
and then
Uˆq∗(F,y,w,p)+F∂ˆq∗
∂F(F,y,w,p)(U)=d e t ( F)t r(UF−1)ˆq(F,y,F−tw,p)
+d e t ( F)∂ˆq
∂F(F,y,F−tw,p)(U)+d e t ( F)∂ˆq
∂w(F,y,F−tw,p)(−F−tUtF−tw).
(8.45)
Then (8.41) follows by taking F=IandU=∇u.
Equalities (8.42) and (8.43) follow from the fact that
ˆq∗(I,y,w,p)=ˆq(I,y,w,p). (8.46)
/square
Corollary 8.2.2. We have
ˆq∗(F,y,w,p)=ˆ q(I,y0,w0,p)+∂ˆq
∂F(I,y0,w0,p)(∇u)+D i v uq0
−∇uq0−∂ˆq
∂w(I,y0,w0,p)∇utw0
+∂ˆq
∂y(I,y0,w0,p)(y−y0)+∂ˆq
∂w(I,y0,w0,p)(w−w0)
+o(∇u)+o(y−y0)+o(w−w0). (8.47)
Assuming that ∇u=O(ε),˙F=∇˙u=O(ε),y−y0=O(ε)a n dw−w0=
O(ε), we can approximate ˆq∗(F,y,w,p) by neglecting the three last terms in the
right hand side of the above equality.
Moreover, in order to linearize th e energy equation (7.5), we take y=θ,
w=∇θand approximate terms∂ˆSelas
∂y(F,θ,p ),cF(F,θ,p ),θ∂ˆSelas
∂θ(F,θ,p )a n d

54 Chapter 8. Linearized Models
ˆSvisc(F,˙F,θ,p ) by their values at the reference conﬁguration, i.e., F=I,˙F=0
andθ=θ0.T h u sw eg e t
ρ0ˆcF(I,θ0,p)˙θ=θ0Y(θ0,p)·∇˙u
−Div/bracketleftbigg
q0+∂ˆq
∂F(I,θ0,∇θ0,p)(∇u)+D i v uq0
−∇uq0−∂ˆq
∂w(I,θ0,∇θ0,p)∇ut∇θ0
+∂ˆq
∂θ(I,θ0,∇θ0,p)(θ−θ0)
+∂ˆq
∂w(I,θ0,∇θ0,p)(∇θ−∇θ0)/bracketrightbigg
+f∗. (8.48)
We notice that the viscous dissipation term has been neglected in this equation
because its main part, ˆl(I,θ0,p)(∇˙u)·∇˙u, is second order in ∇˙u.M o r e o v e r ,t h e
heat source f∗can be approximated by f∗=( 1+D i v u)faccording to the Lemma
below.
Lemma 8.2.3. We have
det(F)=1+D i v u+o(∇u). (8.49)
Proof. From Lemma 3.2.2, since det( F) is diﬀerentiable at F=I,w eg e t
det(F)=d e t ( I)+d e t ( I)t r(F−I)+o(F−I), (8.50)
from which the result follows because F−I=∇uand tr( ∇u)=D i v u. /square
Let us assume that ˆqis given by the generalized Fourier law (see (6.47)):
ˆq(F,θ,w,p)=−ˆk(θ,|w|2,p)w (8.51)
and that the reference temperature θ0is spatially homogeneous. Then q0=0and
it is straightforward to see that equation (8.48) becomes
ρ0ˆcF(I,θ0,p)˙θ=θ0Y(θ0,p)·∇˙u+D i v/bracketleftBig
ˆk(θ0,0,p)∇θ/bracketrightBig
+f∗, (8.52)
which is the standard linearized energy equation for thermoelasticity .
8.3 Isotropic Linear Thermoviscoelasticity
If the body is isotropic, operators CandAbecome much simpler as a consequence
of the following.

8.3. Isotropic Linear Thermoviscoelasticity 55
Proposition 8.3.1. LetHpbe the extended symmetry group at p.T h e n
1.Cis invariant under Hp∩Orth+, that is,
QC(θ0,p)(U)Qt=C(θ0,p)(QUQt). (8.53)
2.Y(θ0,p)commutes with each Q∈H p∩Orth+.
Proof. T h eﬁ r s tp a r th o l d sb e c a u s e ˆTis invariant under Hp∩Orth+(see (6.14))
and the derivative of an invariant mapping is also invariant (see Gurtin[5]).
For the second part we take the derivative of (6.14) with respect to θ.W e
get
Q∂ˆT
∂θ(F,θ,p )Qt=∂ˆT
∂θ(QFQt,θ,p). (8.54)
By taking F=Iandθ=θ0we obtain
QY(θ0,p)Qt=Y(θ0,p). (8.55)
/square
A sac o n s e q u e n c ew eh a v et h ei m p o r t a n t
Proposition 8.3.2. Let us assume the body at pis isotropic. Then there exist func-
tions ˆλ(θ0,p),ˆµ(θ0,p),ˆy(θ0,p)andˆχ(θ0,p)deﬁned in R+×Bsuch that
C(θ0,p)(E)=2 ˆµ(θ0,p)E+ˆλ(θ0,p)t r (E)I, (8.56)
Y(θ0,p)=ˆy(θ0,p)I, (8.57)
A(θ0,p)=ˆχ(θ0,p)I. (8.58)
Moreover,
ˆχ(θ0,p)=−ˆy(θ0,p)
2ˆµ(θ0,p)+3ˆλ(θ0,p). (8.59)
Proof. The ﬁrst assertion follows from the Representation Theorem for Linear
Isotropic Tensor Functions (see Appendix A).
The second one is a consequence of the fact that symmetric tensors which
commute with any rotation must be isotropic, i.e., proportional to the identity
tensor (see also Appendix A).
Finally, the expression for A(θ0,p) and equality (8.59) can be obtained from
(8.31). /square
Deﬁnition 8.3.3. Functions ˆλ(θ0,p)a n dˆ µ(θ0,p) are called Lam´e’s coeﬃcients of
the body at point pat temperature θ0, while ˆ χ(θ0,p) is called the coeﬃcient of
linear thermal expansion at constant stress of the body at point pat temperature
θ0.

56 Chapter 8. Linearized Models
Deﬁnition 8.3.4. We say that a fourth order tensor D∈L(Lin,Sym ) is positive
deﬁnite if
E·D(E)>0 (8.60)
for all symmetric tensors E/negationslash=0 .
Proposition 8.3.5. Let us assume that the material at pis isotropic. Then C(θ0,p)
is positive deﬁnite if and only if the Lam´ e coeﬃcients satisfy the inequalities
ˆµ(θ0,p)>0,2ˆµ(θ0,p)+3ˆλ(θ0,p)>0. (8.61)
Proof. See [5]. /square
We end this section by writing the equations for linearized thermoviscoelas-
ticity under the following assumptions:
•The body is an isotropic Coleman-Noll material.
•The heat ﬂux is given by the generalized Fourier law.
•The initial state is spatially homogeneous, i.e., initial ﬁelds are constant func-
tions.
ρ0¨u=µ0∆u+(µ0+λ0)∇Divu−(2µ0+3λ0)χ0∇θ
+η0∆˙u+(η0+ξ0)∇Div˙u+b∗, (8.62)
ρ0cF0˙θ=−θ0(2µ0+3λ0)χ0Div˙u+k0∆θ+f∗. (8.63)

Chapter 9
Quasi-static Thermoelasticity
In this chapter we state the equations for quasi-static thermoelasticity and give
an approximated method to solve them.
9.1 Statement of the Equations
In the case of quasi-static thermoelasticity terms ρ0¨u,ˆSviscand
θ∂ˆS
∂θ(F,θ,p )·∇˙u (9.1)
may be neglected and the n (7.4) and (7.5) yield
DivˆSelas(F,θ,p )+b∗=0, (9.2)
ρ0cF˙θ+D i vˆ q(F,θ,∇θ,p)=f∗. (9.3)
9.2 Time Discretization
In order to write a discretization in time we use the idea underlying incremental
methods in nonlinear elasticity. For the sake of simplicity, we restrict ourselves
to the case of small displacements . However, we do not make any assumption on
temperature.
Let us take the derivative of equation (9.2) with respect to time. We get
Div/parenleftBigg
∂ˆSelas
∂F(F,θ,p )(˙F)+∂ˆSelas
∂θ(F,θ,p )˙θ/parenrightBigg
+˙b∗=0. (9.4)
We assume small deformations ,n a m e l y ,
F−I=∇u=O(ε). (9.5)

58 Chapter 9. Quasi-static Thermoelasticity
Then we have
∂ˆSelas
∂F(F,θ,p )=∂ˆSelas
∂F(I,θ,p)+O(ε) (9.6)
and similarly
∂ˆSelas
∂θ(F,θ,p )=∂ˆSelas
∂θ(I,θ,p)+O(ε). (9.7)
Thus (9.4) can be approximated by
Div/parenleftBigg
∂ˆSelas
∂F(I,θ,p)(˙F)+∂ˆSelas
∂θ(I,θ,p)˙θ/parenrightBigg
+˙b∗=0. (9.8)
Let us assume that the heat ﬂux obeys the generalized Fourier law. The previ-
ous equation can be discretized by using an implicit Euler method : we compute
sequences u1,…,uNandθ1,…,θNby solving
Div/bracketleftBigg
∂ˆSelas
∂F(I,θn+1,p)/parenleftbiggFn+1−Fn
δt/parenrightbigg
+∂ˆSelas
∂θ(I,θn+1,p)/parenleftbiggθn+1−θn
δt/parenrightbigg/bracketrightBigg
+bn+1
∗−bn
∗
δt=0,(9.9)
ρ0ˆcF(Fn,θn,p)θn+1−θn
δt−Div/parenleftBig
ˆk(θn,|∇θn|,p)∇θn+1/parenrightBig
=fn+1
∗,
(9.10)
where δtis atime step ,N=/bracketleftbigT−t0
δt/bracketrightbig
,Fn=I+∇un,bn
∗=b∗(tn),fn
∗=f∗(tn)
andtn=t0+nδt.
By using (8.11), (8.23) and (8.24), and introducing the notations
δun:=un+1−un, (9.11)
δθn:=θn+1−θn, (9.12)
δbn
∗:=bn+1
∗−bn
∗, (9.13)
discretized equation (9.9) becomes
Div/bracketleftBig
C(θn+1,p)(E(δun)) + Div δunˆT(I,θn+1,p)−ˆT(I,θn+1,p)E(δun)
+W(δun)ˆT(I,θn+1,p)+Y(θn+1,p)δθn/bracketrightBig
+δbn
∗=0.
(9.14)
In terms of the tensor of thermal expansion at constant stress this equation
can be written as follows:
Div/bracketleftBig
C(θn+1,p)/parenleftbig
E(δun)−A(θn+1,p)δθn/parenrightbig
+D i v δunˆT(I,θn+1,p)
−ˆT(I,θn+1,p)E(δun)+W(δun)ˆT(I,θn+1,p)/bracketrightBig
+δbn
∗=0. (9.15)

9.3. A Particular Case 59
We notice that if the body is isotropic then (8.56) and (8.57) hold and they
can be replaced in (9.15).
Moreover, after solving (9.15) and (9.10) for n=0,…,N −1, the displace-
ments can be computed by
un+1=u0+n/summationdisplay
k=0δuk, (9.16)
while stresses are given by
Sn+1=ˆT(I,θ0,p)+n/summationdisplay
k=0/bracketleftbig
C(θk+1,p)/parenleftbig
E(δuk)−A(θk+1,p)δθk/parenrightbig
+DivδukˆT(I,θk+1,p)−ˆT(I,θk+1,p)E(δuk)+W(δuk)ˆT(I,θk+1,p)/bracketrightBig
.
(9.17)
9.3 A Particular Case
Sometimes, instead of the tensor of thermal expansion A, we know the tensor of
theaccumulated thermal expansion between two temperatures θ0andθ,t ob e
denoted by Υ( θ0,θ,p). Parameters Aand Υ are related by
A(θ,p)=∂Υ
∂θ(θ0,θ,p). (9.18)
Thus Ais the inverse of a temperature while Υ is non-dimensional.
A discrete version of (9.18) is
A(θn+1,p)≈Υ(θ0,θn+1,p)−Υ(θ0,θn,p)
θn+1−θn(9.19)
and therefore
n−1/summationdisplay
k=0A(θk+1,p)(θk+1−θk)≈n−1/summationdisplay
k=0/bracketleftbig
Υ(θ0,θk+1,p)−Υ(θ0,θk,p)/bracketrightbig
=Υ (θ0,θn,p)−Υ(θ0,θ0,p)=Υ ( θ0,θn,p). (9.20)
Now let us assume that tensor Cisindependent of temperature and that the terms
due to the initial stresses can be neglected. Then, by summing up equations (9.15)
from 0 to n−1w eg e t
Div/bracketleftbig
C(p)/parenleftbig
∇un−Υ(θ0,θn)I/parenrightbig/bracketrightbig
+bn
∗=0. (9.21)
This means that, under the above assumption, we can compute the state at time
tnby making only one step from the initial solution.

Chapter 10
Fluids
In this chapter we study a subclass of Coleman-Noll materials called ﬂuids.
10.1 The Concept of Fluid, First Properties
Deﬁnition 10.1.1. Aﬂuidis a Coleman-Noll material having the unimodular group
Unim ={H∈Lin+:d e t ( H)=1} (10.1)
as extended symmetry group.
Remark 10.1.2.Since Orth+⊂Unim , any ﬂuid is isotropic in the sense of Deﬁ-
nition 6.1.1.
The following Lemma characterizes the functions which are invariant by uni-
modular transformations.
Lemma 10.1.3. LetWbe a ﬁnite dimensional normed space and ˆG:Lin+→W
a mapping such that
ˆG(FH)=ˆG(F)∀H∈Unim ∀F∈Lin+. (10.2)
Then, for any given constant ρ0, there exists a mapping ˆg:R→W such that
ˆG(F)=ˆg(ν), (10.3)
where ν:=det(F)
ρ0and
dˆG
dF(F)(U)=νF−t·Uˆg/prime(ν). (10.4)
Furthermore, if W=Rthen
dˆG
dF(F)(U)=νˆg/prime(ν)F−t·U. (10.5)

62 Chapter 10. Fluids
Proof. For any given F∈Lin+,l e tu sc h o o s e H=( d e t ( F))1/3F−1.T h e n H∈
Unim and
ˆG(F)=ˆG(FH)=ˆG(F(det(F))1/3F−1)=ˆG((det( F))1/3I). (10.6)
Let ˆg:R→W be deﬁned by
ˆg(r)=ˆG/parenleftBig
(ρ0r)1/3I/parenrightBig
. (10.7)
Then (10.3) holds. Indeed,
ˆg(ν)=ˆg(det(F)
ρ0)=ˆG((det( F))1/3I)=ˆG(F). (10.8)
Now the second part of the Lemma is a consequence of the chain rule:
dˆG
dF(F)(U)=ˆg/prime/parenleftbiggdet(F)
ρ0/parenrightbigg/parenleftbigg1
ρ0det(F)t r(UF−1)/parenrightbigg
=νF−t·Uˆg/prime(ν). (10.9)
/square
In what follows we deal with a Coleman-Noll material which is a ﬂuid in the
sense of Deﬁnition 10.1.1. According to Deﬁnition 3.1.3 and the previous Lemma
there must exist functions still denoted with “hat”:
ˆT:R+×R×B→ Sym, (10.10)
ˆl:R+×R×B→L (Lin,Sym ), (10.11)
ˆe:R+×R×B→ R, (10.12)
ˆθ:R+×R×B→ R+, (10.13)
ˆq:R+×R×V×B→V , (10.14)
“smooth enough” and satisfying
T(x, t)=ˆT(ν(x, t),s(x, t),p)+ˆl(ν(x, t),s(x, t),p)(L), (10.15)
e(x, t)=ˆe(ν(x, t),s(x, t),p), (10.16)
θ(x, t)=ˆθ(ν(x, t),s(x, t),p), (10.17)
q(x, t)=ˆq(ν(x, t),s(x, t),gradθ(x, t),p), (10.18)
withx=X(p,t).
Now we recall assumption ( H1) introduced in Theorem 3.2.4. In the case of
a ﬂuid it becomes:
There exists a function ˆs:R+×R+×B→ Rsuch that
s=ˆs(ν,θ,p)⇐⇒θ=ˆθ(ν,s,p)∀ν∈R+∀θ∈R+∀p∈B. (10.19)
As a consequence of Theorem 3.2.4 and equality (10.4) we have the following:

10.2. Motion Equation. Thermodynamic Pressure 63
Theorem 10.1.4. Assume (H1)holds. Then all thermodynamic processes in the
constitutive class Cof a ﬂuid satisfy the second law of thermodynamics if and only
if
ˆθ(ν,s,p)=∂ˆe
∂s(ν,s,p), (10.20)
ˆT(ν,s,p)=∂ˆe
∂ν(ν,s,p)I, (10.21)
ˆl(ν,s,p)(L)·L≥0 (dissipation inequality), (10.22)
ˆq(ν,s,w,p)·w≤0, (10.23)
for all ν∈R+,s∈R,p∈B,L∈Linandw∈V.
10.2 Motion Equation. Thermodynamic Pressure
Deﬁnition 10.2.1. We denote by thermodynamic pressure the scalar ﬁeld π(x, t)=
ˆπ(ν(x, t),s(x, t),p), with ˆ πgiven by
ˆπ(ν,s,p)=−∂ˆe
∂ν(ν,s,p). (10.24)
Remark 10.2.2.Since ﬂuids are isotropic Coleman-Noll materials, we deduce from
the above theorem that functions in (6.24) satisfy
β0(IB,s ,p)=β0(det(B),s ,p)=−ˆπ(ν,s,p), (10.25)
β1(IB,s ,p)=0,β2(IB,s ,p)=0. (10.26)
In what follows we assume that the Principle of Material Frame-Indiﬀerence
holds. As an immediate consequence of Proposition 4.2.4 we have the following
Proposition 10.2.3. For any ﬂuid there exist two functions
ˆη:R+×R×B→ R, (10.27)
ˆξ:R+×R×B→ R, (10.28)
such that
ˆl(ν,s,p)(L)=ˆl(ν,s,p)(D)=2 ˆη(ν,s,p)D+ˆξ(ν,s,p)t r(D)I. (10.29)
Proof. It follows from Proposition 4.2.4 and the fact that property (4.44) holds
for any ﬂuid. Indeed, let us suppose F∈Lin+andQ∈Orth+.T h e n ν(QF)=
det(QF)
ρ0=det(F)
ρ0=ν(F). /square

64 Chapter 10. Fluids
Deﬁnition 10.2.4. Scalar ﬁeld η(x, t)=ˆη(ν(x, t),s(x, t),p) is called dynamic vis-
cosity while the scalar ﬁeld ξ(x, t)=ˆξ(ν(x, t),s(x, t),p) is called second viscosity
coeﬃcient . Moreover, the scalar ﬁeld
ζ=ξ+2
3η, (10.30)
is called bulk viscosity . From the dissipation inequality and Proposition 4.2.5 we
deduce
η≥0,ζ≥0. (10.31)
Corollary 10.2.5. For a ﬂuid, the motion equation becomes
ρ˙v+g r a d π=d i v ( l(D)) +b, (10.32)
withl(D)given by (10.29).
10.3 Energy Equation, Enthalpy
Proposition 10.3.1. For a ﬂuid the response function ˆqsatisﬁes
ˆq(ν,y,w,p)=−ˆk(ν,y,|w|2,p)w (10.33)
where ycan be replaced by sorθ,
Proof. It follows from Proposition 6.2.8. Indeed, since ˆqonly depends on det( F),
equality (6.39) yields
ˆq(ν,y,w,p)=α0(ν,w,y,p)w=γ0(ν,w·w,y,p)w, (10.34)
which is (10.33) for ˆk(ν,y,|w|2,p)=−γ0(ν,w·w,y,p).
Deﬁnition 10.3.2. We denote by speciﬁc enthalpy the scalar spatial ﬁeld deﬁned
by
h(x, t): =e(x, t)+π(x, t)
ρ(x, t). (10.35)
The next Proposition provides a local form of the energy conservation law in
terms of the speciﬁc enthalpy.
Proposition 10.3.3. The following equality holds:
ρ˙h−˙π=l(D)·D−divq+f. (10.36)
Proof. We have
˙e=˙h−(πν)·=˙h−˙πν−π˙ν=˙h−˙πν−πνdivv, (10.37)
where we have used the mass conservation eq uation (1.22) to get the last equality.
Then ρ˙e=ρ˙h−˙π−πdivvand, by replacing this equality in (3.10), we easily
obtain (10.36). /square

10.3. Energy Equation, Enthalpy 65
The following Proposition gives the so-called Crocco form of the motion equa-
tion.
Proposition 10.3.4. Under the assumptions
•ˆeis independent of p,
•the thermodynamic process is Eulerian,
•the body force is conservative, i.e.,b
ρ=−gradβ, for some scalar ﬁeld β,
the following equation holds:
v/prime+c u r lv×v+g r a d ( h+|v|2
2+β)−θgrads=0. (10.38)
Proof. We have
grade=∂ˆe
∂νgradν+∂ˆe
∂sgrads=π
ρ2gradρ+θgrads. (10.39)
On the other hand
gradh=g r a d e+1
ρgradπ−π
ρ2gradρ=π
ρ2gradρ+θgrads
+1
ρgradπ−π
ρ2gradρ=θgrads+1
ρgradπ, (10.40)
which implies
1
ρgradπ=g r a d h−θgrads. (10.41)
By replacing (10.41) in the motion equation
˙v+1
ρgradπ=b
ρ, (10.42)
and using the equality
˙v=v/prime+g r a d|v|2
2+c u r lv×v, (10.43)
we deduce (10.38). /square
Remark 10.3.5.From the motion equation, ˙v=−1
ρgradπ−gradβ, and (10.41)
we deduce that, if the speciﬁc entropy is spatially homogeneous, then the acceler-
ation is the gradient of a potential ,n a m e l y , −(h+β). In this case, the important
Theorems of Lagrange-Cauchy, Kelvin and Transport of Vorticity apply (see [5],
Section 11).
Deﬁnition 10.3.6. The scalar ﬁeld H=h+|v|2
2is called speciﬁc stagnation en-
thalpy .

66 Chapter 10. Fluids
Deﬁnition 10.3.7. Suppose the body force is conservative. A thermodynamic pro-
cess is called homentropic if (H+β)˙ = 0.
Deﬁnition 10.3.8. A thermodynamic process ( X,T,b,e,θ,q,f,s)∈Cis said to be
steady ifXis asteady motion (see [5]) and all ﬁelds T,b,e,θ,q,f,sare indepen-
dent of time.
We have the following:
Proposition 10.3.9. Let us consider a steady thermodynamic process satisfying the
assumptions of Proposition 10.3.4. Then it is isentropic if and only if it is homen-
tropic.
Proof. We make the scalar product of (10.38) with v.S i n c e v/prime=0we obtain
v·grad(H+β)−θgrads·v=0 (10.44)
and then
(H+β)˙−θ˙s=0. (10.45)
The conclusion follows because θ>0. /square
Theorem 10.3.10. (Crocco). Let us consider an Eulerian homentropic steady ther-
modynamic process under a conservative force, with H+βconstant in T.T h e n ,
ifsis not constant in T, the process cannot be irrotational.
Proof. Since H+βis constant, equation (10.38) becomes
curlv×v=θgrads. (10.46)
Then curl v=0implies grad s=0. /square
Remark 10.3.11 .We notice that the converse is not always true. Indeed, there
may exist steady isentropic thermodynamic processes which are not irrotational.
For this it is enough that curl vandvbe parallel.
10.4 Thermodynamic Coeﬃcients and Equalities
In what follows we use standard notation in books on thermodynamics: let us
suppose that a thermodynamic variable zis a function of any two other called x
andy,t h a ti s , z=ˆz(x, y). Then the partial derivatives of ˆ zwith respect to xand
ywill be denoted by/parenleftbig∂z
∂x/parenrightbig
yand/parenleftBig
∂z
∂y/parenrightBig
x. Accordingly, we will also write zinstead
of ˆz.

10.4. Thermodynamic Coeﬃcients and Equalities 67
Proposition 10.4.1. There exists a mapping ˆs:R+×R×B→ Rsuch that s=
ˆs(ν,e,p). Moreover,
/parenleftbigg∂s
∂e/parenrightbigg
ν=1
θ, (10.47)
/parenleftbigg∂s
∂ν/parenrightbigg
e=π
θ. (10.48)
Proof. Since/parenleftbig∂e
∂s/parenrightbig
ν=θ>0, the mapping s∈R→ˆe(ν,s,p) is strictly increasing
so it is injective for any given ν∈R+andp∈B. Then it has an inverse deﬁned
in the image set. The inverse mapping theorem immediately gives (10.47) from
(10.20).
To prove (10.48) it is enough to apply the chain rule to the identity
e=ˆe(ν,ˆs(ν,e,p),p). (10.49)
Indeed, by taking the derivative with respect to ν,w eo b t a i n
0=/parenleftbigg∂e
∂ν/parenrightbigg
s+/parenleftbigg∂e
∂s/parenrightbigg
ν/parenleftbigg∂s
∂ν/parenrightbigg
e, (10.50)
which yields (10.48). /square
Now we notice that, under assumption ( H1), we can write the speciﬁc internal
energy as a function of the speciﬁc volume and the temperature. Next, we can
replace νby1
ρto get e=ˆe(ρ, θ, p) which gives sense to the following:
Deﬁnition 10.4.2. We denote by speciﬁc heat at constant volume the scalar ﬁeld
cvdeﬁned by cv(x, t)=ˆcv(ρ(x, t),θ(x, t),p)w i t hˆ cvgiven by
ˆcv(ρ, θ, p): =∂ˆe
∂θ(ρ, θ, p). (10.51)
We notice that, for ﬂuids, the speciﬁc heat at constant volume coincides with
thespeciﬁc heat at constant deformation deﬁned in (5.11).
Now we make the following assumption:
(H2)For any given p, the mapping Φp:R+×R→R+×Rdeﬁned by
(θ,π)=Φ p(ν,s): =/parenleftbigg∂ˆe
∂s(ν,s,p),−∂ˆe
∂ν(ν,s,p)/parenrightbigg
(10.52)
has an inverse allowing us to write νandsas functions of θandπ.
Deﬁnition 10.4.3. We denote by coeﬃcient of volumetric thermal expansion at
constant pressure the scalar ﬁeld αdeﬁned by α(x, t)=ˆα(θ(x, t),π(x, t),p)w i t h
ˆαgiven by
ˆα(θ,π,p): =1
ˆν(θ,π,p)∂ˆν
∂θ(θ,π,p). (10.53)

68 Chapter 10. Fluids
This coeﬃcient is related with the tensor of thermal expansion at constant
elastic stress given in (8.31). Indeed, since
ˆν(θ,π,p)=detˆF(θ,π,p)
ρ0, (10.54)
by using the chain rule and Lemma 3.2.2 we get
∂ˆν
∂θ(θ,π,p)=detˆF(θ,π,p)
ρ0tr/parenleftBigg
∂ˆF
∂θ(θ,π,p)(ˆF(θ,π,p))−1/parenrightBigg
(10.55)
and then
ˆα(θ,π,p)=t r/parenleftBigg
∂ˆF
∂θ(θ,π,p)(ˆF(θ,π,p))−1/parenrightBigg
. (10.56)
By replacing θwithθ0andπwithπ0, and using that ˆF(θ0,π0,p)=I(see
Chapter 8) and that ﬂuids are isotropic materials we deduce
∂ˆν
∂θ(θ0,π0,p)=ν03ˆχ(θ0,p) (10.57)
and hence
ˆα(θ0,π0,p)=3ˆχ(θ0,p). (10.58)
Deﬁnition 10.4.4. We denote by coeﬃcient of isothermal compressibility the scalar
ﬁeldκdeﬁned by κ(x, t)=ˆκ(θ(x, t),π(x, t),p)w i t hˆ κgiven by
ˆκ(θ,π,p): =−1
ˆν(θ,π,p)∂ˆν
∂π(θ,π,p). (10.59)
Remark 10.4.5.We notice that, since ν=1
ρ,w eh a v e
α=−1
ρ/parenleftbigg∂ρ
∂θ/parenrightbigg
π, (10.60)
κ=1
ρ/parenleftbigg∂ρ
∂π/parenrightbigg
θ. (10.61)
By using αandκwe can write the mass conservation equation as follows:
Corollary 10.4.6. We have
−α˙θ+κ˙π+d i vv=0. (10.62)
Proof. From (10.60) and (10.61) we deduce
˙ρ=/parenleftbigg∂ρ
∂θ/parenrightbigg
π˙θ+/parenleftbigg∂ρ
∂π/parenrightbigg
θ˙π=−ρα˙θ+ρκ˙π. (10.63)
Then the result can be obtained by replacing this expression for ˙ ρin the mass
conservation equation (1.22). /square

10.4. Thermodynamic Coeﬃcients and Equalities 69
Under assumption ( H2), the speciﬁc enthalpy can be written as a function
ofθandπ. Indeed
h=e+π
ρ=ˆe(ˆν(θ,π,p),ˆs(θ,π,p),p)+ˆν(θ,π,p)π. (10.64)
Hence the following deﬁnition makes sense.
Deﬁnition 10.4.7. We denote by speciﬁc heat at constant pressure the scalar ﬁeld
cπdeﬁned by cπ(x, t)=ˆcπ(θ(x, t),π(x, t),p)w i t hˆ cπgiven by
ˆcπ(θ,π,p): =∂ˆh
∂θ(θ,π,p). (10.65)
The next Proposition gives the partial derivatives of swhen it is considered
as a function of θandπ.
Proposition 10.4.8. We have
/parenleftbigg∂s
∂θ/parenrightbigg
π=cπ
θ, (10.66)
/parenleftbigg∂s
∂π/parenrightbigg
θ=1
ρ2/parenleftbigg∂ρ
∂θ/parenrightbigg
π=−α
ρ, (10.67)
/parenleftbigg∂h
∂π/parenrightbigg
θ=1
ρ+θ
ρ2/parenleftbigg∂ρ
∂θ/parenrightbigg
π. (10.68)
Proof. Firstly we make use of Proposition 10.4.1 and assumption ( H2)t ow r i t e s
as a function of θandπ:
ˆs(θ,π,p)=ˆs(ˆν(θ,π,p),ˆe(θ,π,p),p)
=ˆs/parenleftBig
ˆν(θ,π,p),ˆh(θ,π,p)−πˆν(θ,π,p),p/parenrightBig
. (10.69)
Then,
/parenleftbigg∂s
∂θ/parenrightbigg
π=/parenleftbigg∂s
∂e/parenrightbigg
ν/bracketleftbigg/parenleftbigg∂h
∂θ/parenrightbigg
π−π/parenleftbigg∂ν
∂θ/parenrightbigg
π/bracketrightbigg
+/parenleftbigg∂s
∂ν/parenrightbigg
e/parenleftbigg∂ν
∂θ/parenrightbigg
π
=/parenleftbigg∂s
∂e/parenrightbigg
ν/parenleftbigg∂h
∂θ/parenrightbigg
π+/bracketleftbigg
−π/parenleftbigg∂s
∂e/parenrightbigg
ν+/parenleftbigg∂s
∂ν/parenrightbigg
e/bracketrightbigg/parenleftbigg∂ν
∂θ/parenrightbigg
π
=1
θcπ+/bracketleftBig
−π
θ+π
θ/bracketrightBig/parenleftbigg∂ν
∂θ/parenrightbigg
π=1
θcπ. (10.70)
Similarly,
/parenleftbigg∂s
∂π/parenrightbigg
θ=/parenleftbigg∂s
∂e/parenrightbigg
ν/bracketleftbigg/parenleftbigg∂h
∂π/parenrightbigg
θ−ν−π/parenleftbigg∂ν
∂π/parenrightbigg
θ/bracketrightbigg
+/parenleftbigg∂s
∂ν/parenrightbigg
e/parenleftbigg∂ν
∂π/parenrightbigg
θ
=/parenleftbigg∂s
∂e/parenrightbigg
ν/bracketleftbigg/parenleftbigg∂h
∂π/parenrightbigg
θ−ˆν(θ,π,p)/bracketrightbigg
+/bracketleftbigg
−/parenleftbigg∂s
∂e/parenrightbigg
νπ+/parenleftbigg∂s
∂ν/parenrightbigg
e/bracketrightbigg/parenleftbigg∂ν
∂π/parenrightbigg
θ
=1
θ/bracketleftbigg/parenleftbigg∂h
∂π/parenrightbigg
θ−1
ρ/bracketrightbigg
.(10.71)

70 Chapter 10. Fluids
Now we compute the partial derivatives of (10.70) and (10.71) with respect to π
andθ, respectively. We obtain
/parenleftbigg∂
∂π/parenleftbigg∂s
∂θ/parenrightbigg
π/parenrightbigg
θ=1
θ/parenleftbigg∂
∂π/parenleftbigg∂h
∂θ/parenrightbigg
π/parenrightbigg
θ, (10.72)
/parenleftbigg∂
∂θ/parenleftbigg∂s
∂π/parenrightbigg
θ/parenrightbigg
π=∂
∂θ/parenleftbigg1
θ/bracketleftbigg/parenleftbigg∂h
∂π/parenrightbigg
θ−1
ρ/bracketrightbigg/parenrightbigg
π
=−1
θ2/parenleftbigg∂h
∂π/parenrightbigg
θ+1
ρθ2+1
θ/parenleftbigg∂
∂θ/parenleftbigg∂h
∂π/parenrightbigg
θ/parenrightbigg
π+1
θρ2/parenleftbigg∂ρ
∂θ/parenrightbigg
π. (10.73)
BySchwarz’s Theorem these two second derivatives are equal. Hence
1
θ/parenleftbigg∂
∂π/parenleftbigg∂h
∂θ/parenrightbigg
π/parenrightbigg
θ=−1
θ2/parenleftbigg∂h
∂π/parenrightbigg
θ+1
ρθ2+1
θ/parenleftbigg∂
∂θ/parenleftbigg∂h
∂π/parenrightbigg
θ/parenrightbigg
π+1
θρ2/parenleftbigg∂ρ
∂θ/parenrightbigg
π
(10.74)
and using again Schwarz’s Theorem
/parenleftbigg∂
∂π/parenleftbigg∂h
∂θ/parenrightbigg
π/parenrightbigg
θ=/parenleftbigg∂
∂θ/parenleftbigg∂h
∂π/parenrightbigg
θ/parenrightbigg
π(10.75)
we can deduce (10.68).
Finally, by replacing (10.68) in (10.71) we obtain (10.67), which completes
the proof. /square
Corollary 10.4.9. Let us assume that ˆcπ(θ,π,p)/negationslash=0. Then we have
/parenleftbigg∂θ
∂π/parenrightbigg
s=−θ
ρ2cπ/parenleftbigg∂ρ
∂θ/parenrightbigg
π=θα
ρcπ. (10.76)
Proof. Firstly, from assumption ( H2) we can write the speciﬁc entropy as a func-
tion of temperature and pressure:
s=ˆs(θ,π,p). (10.77)
As a consequence of the Implicit Function Theorem , for any given sandp∈Bwe
can obtain θas a function of π, because∂ˆs
∂θ(θ,π,p)/negationslash= 0 according to (10.66) and
the assumption of the Corollary.
Finally, by using (10.66) and (10.67), we deduce
/parenleftbigg∂θ
∂π/parenrightbigg
s=−/parenleftbig∂s
∂π/parenrightbig
θ
/parenleftbig∂s
∂θ/parenrightbig
π=−1
ρ2/parenleftBig
∂ρ
∂θ/parenrightBig
π
cπ
θ=−θ
ρ2cπ/parenleftbigg∂ρ
∂θ/parenrightbigg
π. (10.78)
/square
Corollary 10.4.10. The following form of the energy equation holds:
ρcπ˙θ−αθ˙π=l(D)·D−divq+f. (10.79)

10.4. Thermodynamic Coeﬃcients and Equalities 71
Proof. From (10.66) and (10.67) we deduce
˙s=/parenleftbigg∂s
∂θ/parenrightbigg
π˙θ+/parenleftbigg∂s
∂π/parenrightbigg
θ˙π=cπ
θ˙θ−α
ρ˙π. (10.80)
By replacing this equality in (3.67) we get the result. /square
The next Proposition gives express ions for the partial derivatives of sas a
function of ρandθ.
Proposition 10.4.11. We have
1./parenleftbigg∂s
∂θ/parenrightbigg
ρ=1
θcv. (10.81)
2./parenleftbigg∂s
∂ρ/parenrightbigg
θ=−1
ρ2/parenleftbigg∂π
∂θ/parenrightbigg
ρ. (10.82)
3./parenleftbigg∂e
∂ρ/parenrightbigg
θ=1
ρ2/bracketleftBigg
π−θ/parenleftbigg∂π
∂θ/parenrightbigg
ρ/bracketrightBigg
. (10.83)
Proof. We ﬁrst write sas a function of ρandθ:
ˆs(ρ, θ, p)=ˆs(ˆν(ρ),ˆe(ρ, θ, p),p). (10.84)
Then we use the chain rule to compute the partial derivative:
/parenleftbigg∂s
∂θ/parenrightbigg
ρ=/parenleftbigg∂s
∂e/parenrightbigg
ν/parenleftbigg∂e
∂θ/parenrightbigg
ρ=1
θcv, (10.85)
which proves (10.81). Similarly,
/parenleftbigg∂s
∂ρ/parenrightbigg
θ=/parenleftbigg∂s
∂e/parenrightbigg
ν/parenleftbigg∂e
∂ρ/parenrightbigg
θ+/parenleftbigg∂s
∂ν/parenrightbigg
e/parenleftbigg∂ν
∂ρ/parenrightbigg
θ
=1
θ/parenleftbigg∂e
∂ρ/parenrightbigg
θ+π
θ/parenleftbigg
−1
ρ2/parenrightbigg
. (10.86)
By calculating the derivatives of (10.85) and (10.86) with respect to ρandθwe
get
/parenleftBigg
∂
∂ρ/parenleftbigg∂s
∂θ/parenrightbigg
ρ/parenrightBigg
θ=1
θ/parenleftbigg∂cv
∂ρ/parenrightbigg
θ=1
θ/parenleftBigg
∂
∂ρ/parenleftbigg∂e
∂θ/parenrightbigg
ρ/parenrightBigg
θ,(10.87)
/parenleftbigg∂
∂θ/parenleftbigg∂s
∂ρ/parenrightbigg
θ/parenrightbigg
ρ=−1
θ2/parenleftbigg∂e
∂ρ/parenrightbigg
θ+1
θ/parenleftbigg∂
∂θ/parenleftbigg∂e
∂ρ/parenrightbigg
θ/parenrightbigg
ρ−1
ρ2/bracketleftBigg/parenleftbig∂π
∂θ/parenrightbig
ρθ−π
θ2/bracketrightBigg
.(10.88)

72 Chapter 10. Fluids
BySchwarz’s Theorem , these two derivatives must be equal. Hence,
1
θ/parenleftBigg
∂
∂ρ/parenleftbigg∂e
∂θ/parenrightbigg
ρ/parenrightBigg
θ=−1
θ2/parenleftbigg∂e
∂ρ/parenrightbigg
θ+1
θ/parenleftbigg∂
∂θ/parenleftbigg∂e
∂ρ/parenrightbigg
θ/parenrightbigg
ρ
−1
ρ2/bracketleftBigg/parenleftbig∂π
∂θ/parenrightbig
ρθ−π
θ2/bracketrightBigg
, (10.89)
and using the fact that
/parenleftBigg
∂
∂ρ/parenleftbigg∂e
∂θ/parenrightbigg
ρ/parenrightBigg
θ=/parenleftbigg∂
∂θ/parenleftbigg∂e
∂ρ/parenrightbigg
θ/parenrightbigg
ρ, (10.90)
we immediately deduce (10.83) from (10.89).
Finally, (10.82) can be easily obtained by replacing (10.83) in (10.86). This
completes the proof. /square
Corollary 10.4.12. Let us assume that ˆcv(ρ, θ, p)/negationslash=0. Then we have
/parenleftbigg∂θ
∂ρ/parenrightbigg
s=θ
ρ2cv/parenleftbigg∂π
∂θ/parenrightbigg
ρ. (10.91)
Proof. Firstly, we write the speciﬁc entropy as a function of density and temper-
ature:
s=ˆs(ρ, θ, p). (10.92)
Now, for any given sandp∈B, we can apply the Implicit Function Theorem to
obtain θas a function of ρ, because∂ˆs
∂θ(ρ, θ, p)/negationslash= 0 according to (10.81) and the
assumption of the Corollary. Finally, by using (10.81) and (10.82) we obtain
/parenleftbigg∂θ
∂ρ/parenrightbigg
s=−/parenleftBig
∂s
∂ρ/parenrightBig
θ
/parenleftbig∂s
∂θ/parenrightbig
ρ=−−1
ρ2/parenleftbig∂π
∂θ/parenrightbig
ρ
cv
θ=θ
ρ2cv/parenleftbigg∂π
∂θ/parenrightbigg
ρ. (10.93)
/square
From now on let us assume that cvandcπare always strictly positive.
The next deﬁnition makes sense if∂ˆπ
∂ρ(ρ, s, p)≥0.
Deﬁnition 10.4.13. We denote by sound speed the scalar ﬁeld cdeﬁned by c(x, t): =
ˆc(ρ(x, t),s(x, t),p)w i t hˆ cdeﬁned by
ˆc(ρ, s, p)=/radicalBigg
∂ˆπ
∂ρ(ρ, s, p). (10.94)
Proposition 10.4.14. The following equality holds:
c2=cπ
cv/parenleftbigg∂π
∂ρ/parenrightbigg
θ. (10.95)

10.4. Thermodynamic Coeﬃcients and Equalities 73
Proof. As in the proof of Corollary 10.4.9, we can write θas a function of πand
s. More precisely,
θ=ˆθ(ρ, s, p)=ˆθ(ˆπ(ρ, s, p),s ,p)). (10.96)
Hence, by the chain rule,
/parenleftbigg∂θ
∂ρ/parenrightbigg
s=/parenleftbigg∂θ
∂π/parenrightbigg
s/parenleftbigg∂π
∂ρ/parenrightbigg
s, (10.97)
from which it follows that
c2=/parenleftbigg∂π
∂ρ/parenrightbigg
s=/parenleftBig
∂θ
∂ρ/parenrightBig
s
/parenleftbig∂θ
∂π/parenrightbig
s=θ
ρ2cv/parenleftbig∂π
∂θ/parenrightbig
ρ
−θ
ρ2cπ/parenleftBig
∂ρ
∂θ/parenrightBig
π, (10.98)
by taking (10.76) and (10.91) into account.
Now, we use the fact that ρ=ˆρ(π,θ,p)t ow r i t e
π=ˆπ(ˆρ(π,θ,p),θ,p). (10.99)
By taking the derivative of this equality with respect to θwe get
0=/parenleftbigg∂π
∂ρ/parenrightbigg
θ/parenleftbigg∂ρ
∂θ/parenrightbigg
π+/parenleftbigg∂π
∂θ/parenrightbigg
ρ(10.100)
and then/parenleftbigg∂π
∂ρ/parenrightbigg
θ=−/parenleftbig∂π
∂θ/parenrightbig
ρ
/parenleftBig
∂ρ
∂θ/parenrightBig
π. (10.101)
By replacing this equality in (10.98) we immediately get (10.95). /square
Proposition 10.4.15. The following equality holds:
/parenleftbigg∂π
∂s/parenrightbigg
ρ=θ
cv/parenleftbigg∂π
∂θ/parenrightbigg
ρ. (10.102)
Proof. Firstly, we prove that
/parenleftbigg∂π
∂s/parenrightbigg
ρ=−/parenleftbigg∂θ
∂ν/parenrightbigg
s. (10.103)
Indeed, let us recall that
π=−/parenleftbigg∂e
∂ν/parenrightbigg
s,θ=/parenleftbigg∂e
∂s/parenrightbigg
ν. (10.104)

74 Chapter 10. Fluids
Moreover, by Schwarz’s Theorem
/parenleftbigg∂
∂s/parenleftbigg∂e
∂ν/parenrightbigg
s/parenrightbigg
ν=/parenleftbigg∂
∂ν/parenleftbigg∂e
∂s/parenrightbigg
ν/parenrightbigg
s. (10.105)
Then we immediately deduce (10.103) from (10.104).
Now, by using (10.91) we obtain
/parenleftbigg∂θ
∂ν/parenrightbigg
s=/parenleftbigg∂θ
∂ρ/parenrightbigg
sdρ
dν=−θ
ρ2cv/parenleftbigg∂π
∂θ/parenrightbigg
ρρ2
=−θ
cv/parenleftbigg∂π
∂θ/parenrightbigg
ρ. (10.106)
/square
From the previous Proposition we can easily prove the following:
Proposition 10.4.16. We have
/parenleftbigg∂π
∂s/parenrightbigg
ρ=c2ραθ
cπ. (10.107)
Proof. From (10.95) and (10.53) we deduce
c2ραθ
cπ=1
cv/parenleftbigg∂π
∂ρ/parenrightbigg
θρ1
ν/parenleftbigg∂ν
∂θ/parenrightbigg
πθ=−1
cv/parenleftbigg∂π
∂ν/parenrightbigg
θ/parenleftbigg∂ν
∂θ/parenrightbigg
πθ. (10.108)
Now, by taking the derivative with respect to θin the equality
π=ˆπ(ˆν(θ,π,p),θ,p) (10.109)
we get
0=/parenleftbigg∂π
∂ν/parenrightbigg
θ/parenleftbigg∂ν
∂θ/parenrightbigg
π+/parenleftbigg∂π
∂θ/parenrightbigg
ν, (10.110)
and using this in (10.108)
c2ραθ
cπ=θ
cv/parenleftbigg∂π
∂θ/parenrightbigg
ν=θ
cv/parenleftbigg∂π
∂θ/parenrightbigg
ρ. (10.111)
Now the results follows from (10.107). /square
Corollary 10.4.17. We have
/parenleftbigg∂ρ
∂s/parenrightbigg
π=−ραθ
cπ. (10.112)

10.4. Thermodynamic Coeﬃcients and Equalities 75
Proof. We apply the Implicit Function Theorem to get ρas a function of πfrom
equality π=ˆπ(ρ, s, p). We have
/parenleftbigg∂ρ
∂s/parenrightbigg
π=−/parenleftbig∂π
∂s/parenrightbig
ρ
/parenleftBig
∂π
∂ρ/parenrightBig
s. (10.113)
Now the result follows by using (10.107) and (10.94). /square
Corollary 10.4.18. We have
/parenleftbigg∂π
∂θ/parenrightbigg
ρ=c2ρα
γ, (10.114)
/parenleftbigg∂π
∂ρ/parenrightbigg
θ=c2
γ, (10.115)
where γ:=cπ
cv.
Proof. Equality (10.114) follows immediately from (10.111) while (10.115) is a
consequence of (10.95). /square
Now, we can obtain new forms of the motion and energy equations which are
useful, for instance, in dissipative acoustics.
Proposition 10.4.19. Let us assume that function ˆe(ρ, θ, p)does not depend on p.
Then we have
ρ˙v+c2
γgradρ+c2ρα
γgradθ=d i v l(D)+b. (10.116)
Proof. It is enough to write πas a function of ρandθ. From the previous Corollary
we deduce
gradπ=/parenleftbigg∂π
∂ρ/parenrightbigg
θgradρ+/parenleftbigg∂π
∂θ/parenrightbigg
ρgradθ
=c2
γgradρ+c2ρα
γgradθ. (10.117)
By replacing this equality in the motion equation (10.32) we deduce (10.116). /square
Proposition 10.4.20. We have
ρcv˙θ−αθc2
γ˙ρ=l(D)·D−divq+f. (10.118)

76 Chapter 10. Fluids
Proof. Let us consider sas a function of ρandθ. From (10.81), (10.82) and (10.114)
we deduce
˙s=/parenleftbigg∂s
∂ρ/parenrightbigg
θ˙ρ+/parenleftbigg∂s
∂θ/parenrightbigg
ρ˙θ=−αc2
ργ˙ρ+cv
θ˙θ. (10.119)
Now the result follows by replacing this equality in the energy equation (3.67). /square
Remark 10.4.21 .As we said above, for ﬂuids, the cFdeﬁned by (5.11) coincides
withcv. Actually, equation (10.118) is nothing but (5.20) for ﬂuids.
The next Proposition gives two useful relations between some of the thermo-
dynamic ﬁelds we have introduced in this section.
Proposition 10.4.22. The following equalities hold:
1. γ=ρκc2. (10.120)
2. cπ−cv=c2α2θ
γ. (10.121)
Proof. Equality (10.120) is a straightforward consequence of (10.61), (10.115) and
the fact that/parenleftbigg∂π
∂ρ/parenrightbigg
θ=/parenleftbigg∂ρ
∂π/parenrightbigg−1
θ. (10.122)
To prove (10.121) we ﬁrst write the speciﬁc internal energy as a function of ρand
θ. From (10.35) we deduce
e=ˆe(ρ, θ, p)=ˆh(ˆπ(ρ, θ, p),θ,p)−ˆπ(ρ, θ, p)
ρ. (10.123)
Now we take the partial derivative with respect to θ,
/parenleftbigg∂e
∂θ/parenrightbigg
ρ=/parenleftbigg∂h
∂π/parenrightbigg
θ/parenleftbigg∂π
∂θ/parenrightbigg
ρ+/parenleftbigg∂h
∂θ/parenrightbigg
π−1
ρ/parenleftbigg∂π
∂θ/parenrightbigg
ρ
=/parenleftbigg∂h
∂θ/parenrightbigg
π+/parenleftbigg∂π
∂θ/parenrightbigg
ρ/bracketleftbigg/parenleftbigg∂h
∂π/parenrightbigg
θ−1
ρ/bracketrightbigg
. (10.124)
From the deﬁnitions of the speciﬁc heats and (10.68) we deduce
cv=cπ+/parenleftbigg∂π
∂θ/parenrightbigg
ρθ
ρ2/parenleftbigg∂ρ
∂θ/parenrightbigg
π. (10.125)
Then, from (10.114) and (10.60) we get
cv=cπ+c2ρα
γθ
ρ2(−ρα)=cπ−c2α2θ
γ, (10.126)
which ﬁnishes the proof. /square

10.5. Gibbs Free Energy 77
10.5 Gibbs Free Energy
Deﬁnition 10.5.1. We denote by speciﬁc Gibbs free energy the scalar spatial ﬁeld
deﬁned by
G(x, t): =h(x, t)−θ(x, t)s(x, t). (10.127)
From (1.72) we deduce the expression
G(x, t)=ψ(x, t)+π(x, t)
ρ(x, t)(10.128)
relating the Gibbs free energy to the Helmholtz free energy .
Proposition 10.5.2. We have
/parenleftBigg
∂G
∂π/parenrightBigg
θ=ν, (10.129)
/parenleftBigg
∂G
∂θ/parenrightBigg
π=−s. (10.130)
Proof. By using Proposition 10.4.8 and formula (10.127) we get
/parenleftBigg
∂G
∂π/parenrightBigg
θ=/parenleftBigg
∂h
∂π/parenrightBigg
θ−θ/parenleftBigg
∂s
∂π/parenrightBigg
θ=1
ρ+θ
ρ2/parenleftBigg
∂ρ
∂θ/parenrightBigg
π−θ
ρ2/parenleftBigg
∂ρ
∂θ/parenrightBigg
π=1
ρ=ν.(10.131)
Similarly, by taking into account the deﬁnition of cπ(see Deﬁnition 10.4.7) and
equality (10.66) we have
/parenleftBigg
∂G
∂θ/parenrightBigg
π=/parenleftBigg
∂h
∂θ/parenrightBigg
π−s+θ/parenleftBigg
∂s
∂θ/parenrightBigg
π=cπ−s+θcπ
θ=−s. (10.132)
/square
10.6 Statics of Fluids
In this section we study the ﬂuids at rest in a gravitational ﬁeld.
Mechanical Equilibrium
F i r s t l yw eh a v et h ef o l l o w i n g :
Deﬁnition 10.6.1. A steady thermodynamic process is a mechanical equilibrium if
v≡0.

78 Chapter 10. Fluids
For a ﬂuid in mechanical equilibrium, the motion equation becomes
gradπ=b. (10.133)
Thus, a necessary condition for a thermodynamics process to be a mechanical
equilibrium is that the body force be a gradient.
Let us consider a mechanical equilibrium in a gravitational ﬁeld, i.e., b=ρg
withg=−g(x3)e3. Then the equilibrium equation (10.133) implies
∂π
∂x1=∂π
∂x2= 0 (10.134)
and, furthermore,
ρ=ρ(x3)=−1
g(x3)∂π
∂x3. (10.135)
Therefore, if a ﬂuid is in mechanical equilibrium in a gravitational ﬁeld, then pres-
sure and density are functions of the altitude only. Furthermore, since temperature
can be obtained from pressure and density through a constitutive law, it will de-
pend only on x3. Conversely, if temperature is diﬀerent at points with the same
altitude x3, then the thermodynamic process cannot be a mechanical equilibrium,
in other words, the ﬂuid cannot be at rest.
Deﬁnition 10.6.2. A steady thermodynamic process is a thermomechanical equi-
librium ifv≡0and the temperature is a constant ﬁeld.
It is obvious that any thermomechanical equilibrium is also a mechanical
equilibrium. Moreover, since the heat ﬂux qmust be identically null (see (10.33)),
we deduce from energy equation (10.118) that for any thermomechanical equilib-
rium the body heat, f, must be identically null.
10.7 The Boussinesq Approximation, Natural
Convection
In this section we obtain an approximate model for thermodynamic processes close
to a mechanical equilibrium under gravity. For the sake of simplicity, we omit the
explicit dependency on pof the response functions.
Linearization
Let us denote the ﬁelds corresponding to the mechanical equilibrium with a 0
subscript. We have
gradπ0=ρ0g, (10.136)
divq0=f0, (10.137)
ρ0=ˆρ(π0,θ0). (10.138)

10.7. The Boussinesq Approximation, Natural Convection 79
Let us consider a thermodynamic process close to the previous one in the following
sense:
divv=O(ε),α0(θ−θ0)=O(ε),κ0(π−π0)=O(ε), (10.139)
where εis a small parameter and α0andκ0denote, respectively, the coeﬃcients of
thermal expansion andisothermal compressibility at the initial equilibrium state.
From (10.60) and (10.61) we obtain
ρ=ρ0−α0ρ0(θ−θ0)+κ0ρ0(π−π0)+o(θ−θ0)+o(π−π0).(10.140)
Let us denote by πEtheEulerian ﬂuctuation of pressure deﬁned by
πE(x, t)=π(x, t)−π0(x). (10.141)
Then
gradπ=g r a d π0+g r a d πE=ρ0g+g r a d πE. (10.142)
By using this equality and (10.140) the motion equation can be approximated by
ρ0˙v+g r a d πE−div (2η0D)=ρ0[−α0(θ−θ0)+κ0(π−π0)]g
=ρ0[α0(θ−θ0)−κ0(π−π0)]g(x3)e3, (10.143)
where η0=ˆη(π0,θ0). Moreover, according to assumptions (10.139), we can ap-
proximate the mass conservation equation by the incompressibility condition,
divv=0. (10.144)
Equations (10.143) and (10.144) are similar to incompressible Navier-Stokes equa-
tions (13.18) and (13.2). Since (10.143) involves temperature not only on the right-hand side but also in the viscosity coeﬃcient, we also need to solve the energy
equation. We use the form (10.79) which can be approximated in our case by
ρ
0cπ0˙θ−α0θ0˙π=2η0|D|2−div(k0gradθ)+f. (10.145)
Usually, the terms involving the pressure and the viscous dissipation in this equa-
tion are small compared to the other terms, so they are neglected. Equations
(10.143), (10.144) and (10.145) constitute the Boussinesq model fornatural con-
vection .

Chapter 11
Linearized Models for Fluids,
Acoustics
In this chapter we specialize the linearized thermoviscoelastic equations obtained
in Chapter 8 to the case of ﬂuids. These equations will be used, in particular,
to model acoustic waves in ﬂuids so they are the fundamental equations of sound
propagation theory. Moreover, they also constitute a suitable model in cases where
a linear approximation of the general models can be used in the neighbourhood of
an initial state. Thus, their applicability is not merely to sound propagation. For
instance they can deal with internal gravity waves in ﬂuids.
As in Chapter 8, we assume the ﬂuid is at rest at the initial state. A similar
linearization procedure could be done to obtain the approximate equations for
small perturbed motion from a ﬂuid ﬂow (see [1]).
11.1 General Equations, Dissipative Acoustics
In this section we obtain linearized models for small perturbations from an initial
state in which the ﬂuid is at rest (i.e., in mechanical equilibrium). We recall that,
for ﬂuids, ˆT(F,θ,p )=−ˆπ(ν,θ,p)Iwithν=1/ρ=detF
ρ0.
Proposition 11.1.1. The following equalities hold for a ﬂuid:
1. ∂ˆT
∂F(I,θ0,p)(U)=C(θ0,p)(U)=ρ0c2
0
γ0tr(U)I, (11.1)
2. ∂ˆT
∂θ(I,θ0,p)=Y(θ0,p)=−ρ0α0c2
0
γ0I, (11.2)
3. ∂ˆT
∂F(I,s0,p)(U)=ρ0c2
0tr(U)I, (11.3)

82 Chapter 11. Linearized Models for Fluids, Acoustics
4. ∂ˆT
∂s(I,s0,p)=−ρ0α0θ0c2
0
cπ0I, (11.4)
5.∂ˆq∗
∂F(I,y,w,p)(∇u)=−ρ0∂ˆq
∂ρ(ρ0,y,w,p)D ivu+D i vuˆq(ρ0,y,w,p)
−∇uˆq(ρ0,y,w,p)−∂ˆq
∂w(ρ0,y,w,p)∇utw, (11.5)
where ydenotes either temperature or speciﬁc entropy.
Proof. 1. By using the Lemma 10.1.3 we have
∂ˆT
∂F(F,θ,p )(U)=−ν∂ˆπ
∂ν(ν,θ,p)(F−t·U)I
=1
ν∂ˆπ
∂ρ(ρ, θ, p)(F−t·U)I. (11.6)
By taking F=Iandθ=θ0,i nw h i c hc a s e ν=ν0=1/ρ0, and using (10.115)
we deduce that the elasticity tensor Cis given by
C(θ0,p)(U)=∂ˆT
∂F(I,θ0,p)(U)=ρ0∂ˆπ
∂ρ(ρ0,θ0,p)t r(U)I
=ρ0c2
0
γ0tr(U)I. (11.7)
2. From (10.114) we deduce
∂ˆT
∂θ(F,θ,p )=−∂ˆπ
∂θ(ν,θ,p)I=−ραc2
γI. (11.8)
The result follows by taking F=Iandθ=θ0.
3. From Lemma 10.1.3 we have
∂ˆT
∂F(F,s,p )(U)=−ν∂ˆπ
∂ν(ν,s,p)(F−t·U)I
=1
ν∂ˆπ
∂ρ(ρ, s, p)(F−t·U)I. (11.9)
By taking F=I,s=s0and using Deﬁnition 10.4.13 we deduce
∂ˆT
∂F(I,s0,p)(U)=ρ0∂ˆπ
∂ρ(ρ0,s0,p)t r(U)I
=ρ0c2
0tr(U)I. (11.10)

11.1. General Equations, Dissipative Acoustics 83
4. From (10.107) we get
∂ˆT
∂s(F,s,p )=−∂ˆπ
∂s(ν,s,p)I=−ραθc2
cπI. (11.11)
The result follows by taking F=Iands=s0.
5. By using again Lemma 10.1.3 we have
∂ˆq
∂F(F,y,w,p)(U)=ν∂ˆq
∂ν(ν,y,w,p)F−t·U=−ρ∂ˆq
∂ρ(ρ, y,w,p)F−t·U.
(11.12)
Then equality (11.5) follows from Proposition 8.2.1 by using the previous
equality for F=IandU=∇u. /square
Remark 11.1.2.We recall that ﬂuids are isotropic Coleman-Noll materials so
(8.56)–(8.59) hold. In particular, (11.1) shows that the Lam´ec o e ﬃ c i e n t s for a
ﬂuid are
µ(θ0,p)=0,λ (θ0,p)=ρ0(p)c2
0(p)
γ0(p). (11.13)
Moreover, by using the previous equ alities, (8.59) and (11.2) yield
χ(θ0,p)=α0(p)
3. (11.14)
Now we can obtain approximations for the Piola-Kirchhoﬀ stress tensor and
theheat ﬂux vector in a ﬂuid in terms of either absolute temperature or speciﬁc
entropy.
Let us assume that the thermodynamic process is a small perturbation of the
reference state, more precisely,
∇u=O(ε),˙F=∇˙u=O(ε),θ−θ0=O(ε),∇θ−∇θ0=O(ε),(11.15)
εbeing a small parameter.
As an inmediate consequence of Corollaries 8.1.5 and 8.2.2 and of the above Propo-
sition, we can get approximations of the Piola-Kirchhoﬀ stress tensor of the ﬁrst
kind:
Corollary 11.1.3. We have
1.ˆS(F,˙F,θ,p )=−π0I−Divuπ0I+π0∇ut+ρ0c2
0
γ0DivuI
−ρ0α0c2
0
γ0(θ−θ0)I+ˆl(ρ0,θ0,p)(∇˙u)+o(ε).
(11.16)

84 Chapter 11. Linearized Models for Fluids, Acoustics
2. ˆS(F,˙F,s,p )=−π0I−Divuπ0I+π0∇ut+ρ0c2
0DivuI
−ρ0α0θ0c2
0
cπ0(s−s0)I+ˆl(ρ0,s0,p)(∇˙u)+o(ε).
(11.17)
/square
Corollary 11.1.4. For a ﬂuid we have
ˆq∗(F,y,w,p)=ˆq(ρ0,y0,w0,p)−ρ0∂ˆq
∂ρ(ρ0,y0,w0,p)D ivu+D i vuˆq(ρ0,y0,w0,p)
−∇uˆq(ρ0,y0,w0,p)−∂ˆq
∂w(ρ0,y0,w0,p)∇utw0+∂ˆq
∂y(ρ0,y0,w0,p)(y−y0)
+∂ˆq
∂w(ρ0,y0,w0,p)(w−w0)+o(∇u)+o(y−y0)+o(w−w0).
(11.18)
By replacing (11.16) or (11.17) in (7.4) we get linearized motion equations
for a ﬂuid in terms of temperature or speciﬁc entropy, respectively:
ρ0¨u=D i v/bracketleftbigg
−π0I−Divuπ0I+π0∇ut+ρ0c2
0
γ0DivuI−ρ0α0c2
0
γ0(θ−θ0)I/bracketrightbigg
+D i v/bracketleftBig
ˆl(ρ0,θ0,p)(∇˙u)/bracketrightBig
+b∗, (11.19)
ρ0¨u=D i v/bracketleftbigg
−π0I−Divuπ0I+π0∇ut+ρ0c2
0DivuI−ρ0α0θ0c2
0
cπ0(s−s0)I/bracketrightbigg
+D i v/bracketleftBig
ˆl(ρ0,s0,p)(∇˙u)/bracketrightBig
+b∗.
(11.20)
By using the equalities of vector calculus
∇(ϕψ)=ϕ∇ψ+ψ∇ϕ, (11.21)
Div(ϕA)=A∇ϕ+ϕDivA, (11.22)
Div(a⊗c)=(∇a)c+aDivc, (11.23)
∇(a·c)=(∇a)tc+(∇c)ta, (11.24)
we easily deduce
Div(−Divuπ0I+π0∇ut)=−Divu∇π0+∇ut∇π0
=D i v ( −∇π0⊗u+∇π0·uI). (11.25)
These equalities allow us to write the linearized motion equations (11.19) and
(11.20) in another form. For instance, in terms of temperature we have
ρ0¨u=−∇π0−Divu∇π0+∇ut∇π0+∇/parenleftbiggρ0c2
0
γ0Divu/parenrightbigg
−∇/parenleftbiggρ0α0c2
0
γ0(θ−θ0)/parenrightbigg
+D i v/bracketleftBig
ˆl(ρ0,θ0,p)(∇˙u)/bracketrightBig
+b∗, (11.26)

11.2. The Isentropic Case, Non-Dissipative Acoustics 85
as well as the conservative form
ρ0¨u=−∇π0+D i v ( −∇π0⊗u+∇π0·uI)+∇/parenleftbiggρ0c2
0
γ0Divu/parenrightbigg
−∇/parenleftbiggρ0α0c2
0
γ0(θ−θ0)/parenrightbigg
+D i v/bracketleftBig
ˆl(ρ0,θ0,p)(∇˙u)/bracketrightBig
+b∗. (11.27)
If the reference state is spatially homogeneous, then the above equation becomes
ρ0¨u=ρ0c2
0
γ0∇Divu−ρ0α0c2
0
γ0∇θ+η0∆˙u+(η0+ξ0)∇Div˙u+b∗,
(11.28)
where we have used a similar result to (10. 29) by replacing speciﬁc entropy with
temperature.
Now we deal with the energy equation. By using (11.2) and Corollary 11.1.4,
equation (8.48) becomes
ρ0cv0˙θ=−θ0ρ0α0c2
0
γ0Div˙u−Div/bracketleftBig
q0−ρ0∂ˆq
∂ρ(ρ0,θ0,∇θ0,p)D ivu
+D i vuq0−∇uq0−∂ˆq
∂w(ρ0,θ0,∇θ0,p)∇ut∇θ0+
∂ˆq
∂θ(ρ0,θ0,∇θ0,p)(θ−θ0)+∂ˆq
∂w(ρ0,θ0,∇θ0,p)(∇θ−∇θ0)/bracketrightbigg
+f∗,
(11.29)
where cv0=: ˆcv(ρ0,θ0,p).
If the reference state is spatially homogeneous, this equation simpliﬁes to
ρ0cv0˙θ=−θ0ρ0α0c2
0
γ0Div˙u+k0∆θ+f∗, (11.30)
where we have used (10.33) and k0:=ˆk(ρ0,θ0,0,p).
Equations (11.28) and (11.30) constitute a suitable model for dissipative
acoustics .
11.2 The Isentropic Case, Non-Dissipative Acoustics
Sometimes it turns out that the thermodynamic process can be considered as
isentropic. This is, for instance, the usual assumption in the fundamental case of
standard (non-dissipative) acoustics. It is obvious that, in such a situation, it is
more convenient to write the motion and energy equations in terms of speciﬁc
enthalpy rather than temperature.

86 Chapter 11. Linearized Models for Fluids, Acoustics
By using (11.25), equation (11.20) can be written as
ρ0¨u=−∇π0−Divu∇π0+∇ut∇π0+∇/parenleftbig
ρ0c2
0Divu/parenrightbig
−∇/parenleftbiggρ0α0θ0c2
0
cπ0(s−s0)/parenrightbigg
+D i v/parenleftBig
ˆl(ρ0,s0,p)(∇˙u)/parenrightBig
+b∗, (11.31)
as well as in conservative form:
ρ0¨u=−∇π0+ Div ( −∇π0⊗u+∇π0·uI)+∇/parenleftbig
ρ0c2
0Divu/parenrightbig
−∇/parenleftbiggρ0α0θ0c2
0
cπ0(s−s0)/parenrightbigg
+D i v/parenleftBig
ˆl(ρ0,s0,p)(∇˙u)/parenrightBig
+b∗. (11.32)
If the thermodynamic process is isentropic, then s(p,t)=s0(p) (see (3.74))
and (11.31) becomes
ρ0¨u=−∇π0−Divu∇π0+∇ut∇π0+∇/parenleftbig
ρ0c2
0Divu/parenrightbig
+D i v/parenleftBig
ˆl(ρ0,s0,p)(∇˙u)/parenrightBig
+b∗. (11.33)
Remark 11.2.1.From Proposition 3.2.19 we easily deduce that any Eulerian and
adiabatic thermodynamic process is isentropic. However, these conditions are not
strictly necessary for a thermodynamic process to be isentropic as far as equality
(3.75) holds.
When b∗can be approximated by b0=∇π0(see (8.1)) and the viscous
eﬀects can be neglected, (11.33) becomes
ρ0¨u=−Divu∇π0+∇ut∇π0+∇/parenleftbig
ρ0c2
0Divu/parenrightbig
. (11.34)
This is the Galbrun equation . In standard acoustics, the ﬁrst two terms on the
right-hand side are also neglected and then (11.34) becomes
ρ0¨u=∇/parenleftbig
ρ0c2
0Divu/parenrightbig
. (11.35)
Let us deﬁne the Lagrangian ﬂuctuation of pressure by
πL(p,t)=π(X(p,t),t)−π0(p). (11.36)
Then πL(p,t) can be approximated by −ρ0c2
0Divu(p,t).Indeed,
π(X(p,t),t)=ˆπ(F(p,t),s0(p),p)=π0(p)+∂ˆπ
∂F(I,s0(p),p)(∇u(p,t))
+o(∇u(p,t)) =π0(p)−ρ0(p)c2
0(p)D ivu(p,t)+o(∇u(p,t)). (11.37)
(see the Proof of point 3 in Proposition 11.1.1).
By using this equality and (11.36), equation (11.35) can be written as
ρ0¨u=−∇πL. (11.38)

11.3. Linearized Models under Gravity 87
Now we apply the divergence operator after dividing by ρ0.W eg e t
Div¨u=−Div(1
ρ0∇πL) (11.39)
and using (11.37)
¨πL−ρ0c2
0Div(1
ρ0∇πL)=0. (11.40)
The above equation is known as the Pekeris equation .
Finally, if the initial state is spatially homogeneous, ρ0andc2
0are constant
and this equation becomes the standard wave equation
¨πL−c2
0∆πL=0. (11.41)
11.3 Linearized Models under Gravity
Let us go back to equation (11.33) and consider the case where the body force is
the gravity force, that is b(x, t)=ρ(x, t)g(x),gbeing the intensity of the gravity
ﬁeld.T h e n b0(p)=ρ0(p)g(p) and, according to Remark 8.1.9, b∗can be written
as
b∗(p,t)=b0(p)+ρ0(p)∇g(p)u(p,t)+o(u(p,t)). (11.42)
Moreover, the initial state satisﬁes the equations
∇π0=ρ0g, (11.43)
−Divˆq(ρ0,θ0,∇θ0,p)+f0=0, (11.44)
π0=ˆπ(ρ0,θ0,p), (11.45)
which can be solved to get ρ0,θ0andπ0. In this case, equation (11.27) becomes
ρ0¨u=−ρ0g−Div(ρ0g⊗u)+∇(ρ0g·u)+∇/parenleftbiggρ0c2
0
γ0Divu/parenrightbigg
−∇/parenleftbiggρ0α0c2
0
γ0(θ−θ0)/parenrightbigg
+D i v/parenleftBig
ˆl(ρ0,θ0,p)(∇˙u)/parenrightBig
+b∗. (11.46)
By using the vector calculus equalities (11.23) and
∇(ϕa)=ϕ∇a+a⊗∇ϕ, (11.47)
we deduce
Div(ρ0g⊗u)= ∇(ρ0g)u+ρ0gDivu
=ρ0∇gu+(g⊗∇ρ0)u+ρ0gDivu
=ρ0∇gu+u·∇ρ0g+ρ0Divug. (11.48)

88 Chapter 11. Linearized Models for Fluids, Acoustics
Let us assume u(p,t)=O(ε). By using (11.42) and (11.48) in (11.46), we get
ρ0¨u=−(u·∇ρ0+ρ0Divu)g+∇/parenleftbigg
ρ0g·u+ρ0c2
0
γ0Divu/parenrightbigg
−∇/parenleftbiggρ0α0c2
0
γ0(θ−θ0)/parenrightbigg
+D i v/parenleftBig
ˆl(ρ0,θ0,p)(∇˙u)/parenrightBig
. (11.49)
Moreover
(u·∇ρ0+ρ0Divu)g=(u·∇ρ0−ρ0
c2
0u·g)g+(ρ0Divu+ρ0
c2
0g·u)g
=/parenleftbigg
g⊗(∇ρ0−ρ0
c2
0g)/parenrightbigg
u+(ρ0Divu+ρ0
c2
0g·u)g. (11.50)
The tensor
B=1
ρ0/parenleftbigg
g⊗(∇ρ0−ρ0
c2
0g)/parenrightbigg
, (11.51)
has dimension inverse of a square of time. It is called the V¨ais¨al¨a-Brunt tensor .
By replacing (11.50) in (11.49) we obtain
ρ0¨u=−ρ0Bu−(ρ0Divu+ρ0
c2
0g·u)g+∇/parenleftbigg
ρ0g·u+ρ0c2
0
γ0Divu/parenrightbigg
−∇/parenleftbiggρ0α0c2
0
γ0(θ−θ0)/parenrightbigg
+D i v/parenleftBig
ˆl(ρ0,θ0,p)(∇˙u)/parenrightBig
. (11.52)
Let us now look at some considerations about the V¨ais¨al¨a-Brunt tensor. From the
state equation, π=ˆπ(ρ, s), we have
∇π0=∂ˆπ
∂ρ(ρ0,s0)∇ρ0+∂ˆπ
∂s(ρ0,s0)∇s0, (11.53)
and then
ρ0g=c2
0∇ρ0+c2
0ρ0α0θ0
cπ0∇s0, (11.54)
where we have used (10.94) and (10.107).
From (11.54) we deduce
∇ρ0−ρ0
c2
0g=−ρ0α0θ0
cπ0∇s0, (11.55)
so the V¨ais¨al¨a-Brunt tensor is also given by
B=−α0θ0
cπ0(g⊗∇s0). (11.56)

11.3. Linearized Models under Gravity 89
This shows that, if the speciﬁc entropy at the reference state is constant, then the
V¨ais¨al¨a-Brunt tensor is null.
Let us suppose g(p)=−g(p3)e3. From (11.43) we deduce that π0only de-
pends on the third coordinate p3. Moreover, this equality yields
ρ0(p)=−1
g(p3)∂π0
∂p3(p) (11.57)
and hence ρ0only depends on p3.C o n s e q u e n t l y
∇ρ0=∂ρ0
∂p3e3. (11.58)
From this result and (11.54) we get a similar property for the speciﬁc entropy,
namely,
∇s0=∂s0
∂p3e3. (11.59)
Then (11.51) and (11.56) become
B=−1
ρ0g/parenleftbigg∂ρ0
∂p3+ρ0
c2
0g/parenrightbigg
e3⊗e3=α0θ0g
cπ0∂s0
∂p3e3⊗e3. (11.60)
The number Ndeﬁned by
N2=−1
ρ0g/parenleftbigg∂ρ0
∂p3+ρ0
c2
0g/parenrightbigg
=α0θ0g
cπ0∂s0
∂p3(11.61)
is called V¨ais¨al¨a-Brunt frequency . We notice that it is a function of p3only. More-
over,Nwill be real or imaginary depending on whether the speciﬁc entropy, s0,
is increasing or decreasing with altitude, respectively. Now the ﬁrst term on the
right-hand side of (11.52), ρ0Bu, can also be written as
ρ0Bu=ρ0N2u·e3e3=ρ0N2u3e3. (11.62)
The Isentropic Case
Let us consider the particular case where the process is isentropic. Firstly, similar
computations to those made above allow us to transform equation (11.32) into the
following one:
ρ0¨u=−ρ0Bu−(ρ0Divu+ρ0
c2
0g·u)g+∇/parenleftbig
ρ0g·u+ρ0c2
0Divu/parenrightbig
−∇/parenleftbiggρ0α0θ0c2
0
cπ0(s−s0)/parenrightbigg
+D i v/parenleftBig
ˆl(ρ0,s0,p)(∇˙u)/parenrightBig
. (11.63)

90 Chapter 11. Linearized Models for Fluids, Acoustics
If the thermodynamic process is isentropic, then term ∇/parenleftBig
ρ0α0θ0c2
0
cπ0(s−s0)/parenrightBig
disap-
pears. Let us recall that the Eulerian ﬂuctuation of pressure is
πE(p,t)=π(p,t)−π0(p). (11.64)
We have
πE(p,t)=π(p,t)−π(X(p,t),t)+π(X(p,t),t)−π0(p)
=−∇π(p,t)·u(p,t)+o(u(p,t)+πL(p,t) (11.65)
and then a useful approximation is given by (see (11.37))
πE=−∇π0·u−ρ0c2
0Divu=−ρ0g·u−ρ0c2
0Divu. (11.66)
Using this equality in (11.63) we obtain the following mixed formulation (in terms
of displacement and pressure):
ρ0¨u=−ρ0Bu+1
c2
0πEg−∇πE+D i v/parenleftBig
ˆl(ρ0,s0,p)(∇˙u)/parenrightBig
, (11.67)
which, if g=−ge3, can also be written as
ρ0¨u=−ρ0N2u·e3e3+1
c2
0πEg−∇πE+D i v/parenleftBig
ˆl(ρ0,s0,p)(∇˙u)/parenrightBig
. (11.68)
We end this section by obtaining an energy conservation principle . For this purpose
we make the scalar product of (11.68) with ˙uand integrate in P,ap a r to ft h e
body. We get
1
2d
dt/integraldisplay
Pρ0|˙u|2dVp=−1
2d
dt/integraldisplay
Pρ0N2|u·e3|2dVp+/integraldisplay
P1
c2
0πEg·˙udVp
−/integraldisplay
P∇πE·˙udVp+/integraldisplay
PDiv/parenleftBig
ˆl(ρ0,s0,p)(∇˙u)/parenrightBig
·˙udVp. (11.69)
By replacing the expression (11.66) for πEwe get
1
2d
dt/integraldisplay
Pρ0|˙u|2dVp=−1
2d
dt/integraldisplay
Pρ0N2|u·e3|2dVp
−/integraldisplay
P1
c2
0(ρ0g·u+ρ0c2
0Divu)g·˙udVp+/integraldisplay
P∇(ρ0g·u+ρ0c2
0Divu)·˙udVp
+/integraldisplay
PDiv/parenleftBig
ˆl(ρ0,s0,p)(∇˙u)/parenrightBig
·˙udVp (11.70)

11.3. Linearized Models under Gravity 91
and using Green’s formulas in the two rightmost terms we obtain
1
2d
dt/integraldisplay
Pρ0|˙u|2dVp=−1
2d
dt/integraldisplay
Pρ0N2|u·e3|2dVp−1
2d
dt/integraldisplay
Pρ0
c2
0|g·u|2dVp
−/integraldisplay
Pρ0Divug·˙udVp+/integraldisplay
∂Pρ0g·u˙u·ndAp−/integraldisplay
Pρ0g·uDiv˙udVp
+/integraldisplay
∂Pρ0c2
0Divu˙u.ndAp−/integraldisplay
Pρ0c2
0DivuDiv˙udVp
+/integraldisplay
∂Pˆl(ρ0,s0,p)(∇˙u)n·˙udAp−/integraldisplay
Pˆl(ρ0,s0,p)(∇˙u)·∇˙udVp. (11.71)
By using again (11.66) we easily deduce
1
2d
dt/integraldisplay
P/braceleftbigg
ρ0|˙u|2+ρ0N2|u·e3|2+1
ρ0c2
0|πE|2/bracerightbigg
dVp
+/integraldisplay
Pˆl(ρ0,s0,p)(∇˙u)·∇˙udVp=−/integraldisplay
∂PπE˙u·ndAp
+/integraldisplay
∂Pˆl(ρ0,s0,p)(∇˙u)n·˙udAp. (11.72)
The sum
W=1
2/bracketleftbigg
ρ0|˙u|2+ρ0N2|u·e3|2+1
ρ0c2
0|πE|2/bracketrightbigg
(11.73)
represents the density of mechanical energy , while the vector ﬁeld I=πE˙uis called
acoustic intensity . The physical meaning of Iis the following: the inner product
I·nis the rate at which the acoustic energy is transported across a surface element
normal to n, per unit area. If viscous eﬀects are neglected ( l≡0), by using the
Localization Theorem in (11.72) we can deduce the following energy conservation
equation :
∂W
∂t=−DivI. (11.74)

Chapter 12
Perfect Gases
In this chapter we study particular Coleman-Noll ﬂuids: the so-called perfect gases.
12.1 Deﬁnition, General Properties
Deﬁnition 12.1.1. Aperfect gas is a ﬂuid for which the following state equation
holds:
π=ρRθ. (12.1)
More precisely, ˆ π(ρ, θ, p)=ρR(p)θ,w h e r e Rhas a speciﬁc value for each perfect
gas. Actually, R(p)=R
M(p),w h e r e Ris a universal constant ( R=8314 J/(kmolK))
andM(p)i st h e molecular mass of the gas at material point p.
Proposition 12.1.2. The speciﬁc internal energy of a perfect gas only depends on
temperature. Therefore, the same is true for the speciﬁc heat at constant volume.
Proof. By using (12.1) in (10.83) we get
/parenleftbigg∂e
∂ρ/parenrightbigg
θ=1
ρ2[ρRθ−θρR]=0. (12.2)
The second part follows immediately from the deﬁnition of cv. /square
A similar result is valid for the speciﬁc enthalpy:
Proposition 12.1.3. The speciﬁc enthalpy of a perfect gas only depends on temper-
ature. Therefore, the same is true for the speciﬁc heat at constant pressure.
Proof. It is a consequence of the deﬁnition of h.W eh a v e
h=e+π
ρ=ˆe(θ,p)+R(p)θ. (12.3)
/square

94 Chapter 12. Perfect Gases
Corollary 12.1.4. For a perfect gas, the following relation between the speciﬁc heats,
called Mayer equation, holds:
ˆcπ(θ,p)=ˆcv(θ,p)+R(p). (12.4)
Proof. It is enough to take the partial derivative of (12.3) with respect to θ./square
Proposition 12.1.5. The sound speed of a perfect gas only depends on temperature.
More precisely we have:
c=/radicalbig
γRθ, (12.5)
where γis the ratio of speciﬁc heats, γ=cπ
cv.
Proof. From Proposition 10.4.14 and equality (12.1) we deduce
c2=cπ
cv/parenleftbigg∂π
∂ρ/parenrightbigg
θ=cπ
cvRθ=γRθ. (12.6)
/square
The next Proposition easily follows from (12.1).
Proposition 12.1.6. For a perfect gas, the coeﬃcients of thermal expansion αand
isothermal compressibility κare given by
α=1
θ, (12.7)
κ=1
π. (12.8)
Corollary 12.1.7. For a perfect gas we have the following expressions for internal
energy and enthalpy:
1.ˆe(θ,p)=ˆe(θ0,p)+/integraldisplayθ
θ0ˆcv(r, p)dr, (12.9)
2. ˆh(θ,p)=ˆh(θ0,p)+/integraldisplayθ
θ0ˆcπ(r, p)dr. (12.10)
Moreover, ˆh(θ0,p)=ˆe(θ0,p)+R(p)θ0.
12.2 Entropy and Free Energy
Now we obtain an expression for the entropy of a perfect gas. For this we notice
that (10.66) and (10.67) yield
/parenleftbigg∂s
∂θ/parenrightbigg
π=cπ
θ, (12.11)
/parenleftbigg∂s
∂π/parenrightbigg
θ=1
θ/bracketleftbigg
−1
ρ/bracketrightbigg
=−1
ρθ=−R
π. (12.12)

12.2. Entropy and Free Energy 95
Therefore
ˆs(θ,π,p)=ˆs(θ0,π0,p)+/integraldisplayθ
θ0ˆcπ(r, p)
rdr−/integraldisplayπ
πoR(p)
rdr. (12.13)
Similarly, from (10.81) and (10.82) we have
/parenleftbigg∂s
∂θ/parenrightbigg
ρ=cv
θ, (12.14)
/parenleftbigg∂s
∂ρ/parenrightbigg
θ=−1
ρ2/parenleftbigg∂π
∂θ/parenrightbigg
ρ=−1
ρ2ρR=−R
ρ. (12.15)
Then we have
ˆs(ρ, θ, p)=ˆs(ρ0,θ0,p)+/integraldisplayθ
θ0ˆcv(r, p)
rdr−/integraldisplayρ
ρoR(p)
rdr. (12.16)
Deﬁnition 12.2.1. A perfect gas is called calorically perfect if ˆcv, and hence ˆ cπ,d o
not depend on θ.
Monoatomic gases are examples of calorically perfect gases. In this case
ˆcπ(p)=5
2R(p)a n dˆ cv(p)=3
2R(p). Then, for monoatomic gases γ=5
3.
Diatomic gases can be considered as calorically perfect at low temperature.
Actually, the speciﬁc heat at constant pressure of a diatomic gas is given by
ˆcπ(θ,p)=7
2R(p)+R(p)/bracketleftbiggθν(p)/(2θ)
sinh(θν(p)/(2θ))/bracketrightbigg2
, (12.17)
where θνis the so-called characteristic temperature of vibration .F o r O2,θν=
2230Kwhile, for N2,θν= 3340 K.
For low temperature the second term in (12.17) can be neglected. Hence cπ
can be approximated by7
2Rand, according to (12.4), cv=5
2Randγ=1.4.
In the case of calorically perfect gases, expression (12.16) becomes
ˆs(ρ, θ, p)= ˆs(ρ0,θ0,p)+ˆcv(p)ln (θ
θ0)−R(p)ln (ρ
ρ0)
=ˆs(ρ0,θ0,p)+l n/parenleftbigg(θ/θ0)ˆcv(p)
(ρ/ρ0)R(p)/parenrightbigg
=ˆs(ρ0,θ0,p)+l n/parenleftbigg((θρ)/(θ0ρ0))ˆcv(p)
(ρ/ρ0)R(p)+ˆcv(p)/parenrightbigg
=ˆs(ρ0,θ0,p)+ˆcv(p)ln/parenleftBigg
(θρ)/(θ0ρ0)
(ρ/ρ0)R(p)+ˆcv(p)
ˆcv(p)/parenrightBigg
=ˆs(ρ0,θ0,p)+ˆcv(p)ln/parenleftbigg(θρ)/(θ0ρ0)
(ρ/ρ0)γ(p)/parenrightbigg
,(12.18)

96 Chapter 12. Perfect Gases
because
R(p)+ˆcv(p)
ˆcv(p)=ˆcπ(p)−ˆcv(p)+ˆcv(p)
ˆcv(p)=ˆcπ(p)
ˆcv(p)=γ(p). (12.19)
From equation (12.18) we can obtain the pressure as a function of ρands;
indeed, we have
ˆs(ρ, θ, p)−ˆs(ρ0,θ0,p)
ˆcv(p)=l n/parenleftbigg(θρ)/(θ0ρ0)
(ρ/ρ0)γ(p)/parenrightbigg
=l n/parenleftbigg(π/π0)
(ρ/ρ0)γ(p)/parenrightbigg
.(12.20)
Then
(π/π0)
(ρ/ρ0)γ(p)=eˆs(ρ,θ,p)−ˆs(ρ0,θ0,p)
ˆcv(p) (12.21)
and ﬁnally
π=π0
ργ(p)
0ργ(p)eˆs(ρ,θ,p)−ˆs(ρ0,θ0,p)
ˆcv(p) . (12.22)
Remark 12.2.2.Any inviscid (i.e., l≡0) calorically perfect gas undergoing an
isentropic thermodynamic process behaves like an elastic ﬂuid in the sense of [5].
Indeed, since each material point conserves entropy during the process, we deduce
from (12.22)
π=π0
ργ(p)
0ργ(p)=D(p)ργ(p)(12.23)
withD(p)=π0
ργ(p)
0. Actually, in [5], elastic ﬂuids having a constitutive law as
(12.23) are called ideal gases (see exercise 19.1 of this reference).
Finally we obtain an expression of the speciﬁc Gibbs free energy for aperfect
gas. From Proposition 10.5.2 we deduce
ˆG(θ,π)=ˆG(θ0,π0)−/integraldisplayθ
θ0ˆs(r, π0)dr+/integraldisplayπ
π0ˆν(θ,r)dr. (12.24)
By using (12.13) we obtain
ˆG(θ,π)=ˆG(θ0,π0)−/integraltextθ
θ0/parenleftBigg
ˆs(θ0,π0)+/integraltextz
θ0ˆcπ(r)
rdr/parenrightBigg
dz+/integraltextπ
π0ˆν(θ,r)dr
=ˆG(θ0,π0)−ˆs(θ0,π0)(θ−θ0)−/integraltextθ
θ0/parenleftBigg
/integraltextz
θ0ˆcπ(r)
rdr/parenrightBigg
dz+/integraltextπ
π0Rθ
rdr. (12.25)
The previous expression can also be written in the form
ˆG(θ,π)=ˆG0(θ)+Rθln (π
π0), (12.26)

12.3. The Compressible Navier-Stokes Equations 97
where ˆG0(θ)=ˆG(θ,π0) is the speciﬁc Gibbs free energy of formation at tempe-
rature θand at reference pressure π0.W eh a v e
ˆG0(θ)=ˆG(θ0,π0)−ˆs(θ0,π0)(θ−θ0)−/integraldisplayθ
θ0/parenleftBigg/integraldisplayz
θ0ˆcπ(r)
rdr/parenrightBigg
dz. (12.27)
Usually π0is taken to be 1 atm in which case ˆG0(θ) is called standard speciﬁc
Gibbs free energy at temperature θ.
12.3 The Compressible Navier-Stokes Equations
Now we write the equations to be satisﬁed for any thermodynamic process of a
perfect gas. Firstly, in the energy equation, we write the internal energy and the
enthalpy in terms of the temperature. To do that, we recall that for a perfect gas
˙e(x, t)=ˆcv(θ(x, t),p)˙θ(x, t), (12.28)
˙h(x, t)=ˆcπ(θ(x, t),p)˙θ(x, t). (12.29)
We use these equalities in (1.54) and in (10.36). They yield, respectively,
ρˆcv(θ,p)˙θ=T·D−divq+f, (12.30)
ρˆcπ(θ,p)˙θ−˙π=l(D)·D−divq+f. (12.31)
By using (10.19), we can express viscosity coeﬃcients ηandξin terms of θrather
thans. Moreover, we use the expression for ˆqgiven by (10.33). Thus we get the
following equations:
˙ρ+ρdivv=0, (12.32)
ρ˙v+g r a d π−div/braceleftBig
2ˆη(ν,θ,p)D+ˆξ(ν,θ,p)d ivvI/bracerightBig
=b, (12.33)
π=ρRθ, (12.34)
and as energy equation, either
ρˆcv(θ,p)˙θ=−πdivv+2ˆη(ν,θ,p)|D|2+ˆξ(ν,θ,p)(divv)2
+d i v( ˆk(ν,θ,|gradθ|2,p)gradθ)+f (12.35)
or
ρˆcπ(θ,p)˙θ−˙π=2 ˆη(ν,θ,p)|D|2+ˆξ(ν,θ,p)(divv)2
+d i v ( ˆk(ν,θ,|gradθ|2,p)grad θ)+f. (12.36)
This system is called the compressible Navier-Stokes equations .
Moreover, it is easy to obtain the following expression for the dissipation rate
Φ=η
2|gradv+g r a d vt|2+ξ(divv)2+kθ−1|gradθ|2. (12.37)

98 Chapter 12. Perfect Gases
12.4 The Compressible Euler Equations
This system arises when we neglect the viscous and heat conduction eﬀects in the
compressible Navier-Stokes equations .W eg e t
˙ρ+ρdivv=0, (12.38)
ρ˙v+g r a d π=b, (12.39)
π=ρRθ, (12.40)
and either
ρˆcv(θ,p)˙θ=−πdivv+f (12.41)
or
ρˆcπ(θ,p)˙θ−˙π=f. (12.42)
Sometimes, for numerical purposes it is more convenient to write this system
in a diﬀerent equivalent way. Let us go back to the energy equation written in
terms of the speciﬁc total energy E. Under the above assumptions, equation (1.48)
becomes
ρ˙E+d i v ( πv)=0, (12.43)
w h e r ew eh a v es u p p o s e d b=0andf= 0 for simplicity.
Then we replace (12.41) or (12.42) by (12.43) and the algebraic equation,
E=1
2|v|2+ˆe(θ,p). (12.44)
We recall that, since we are dealing with perfect gases, function ˆ e(., p) is nothing
but a primitive of ˆ cv(·,p). Equation (12.44) allows us to obtain θonceEhas been
computed.
In order to use ﬁnite volume methods, which are the most used discretization
techniques for solving the compressible Euler equations, we must rewrite them in
a conservative form. By using (1.21), (1.29) and (1.56) we get
ρ/prime+d i v ( ρv)=0, (12.45)
(ρv)/prime+d i v ( ρv⊗v+πI)=0, (12.46)
(ρE)/prime+d i v( ( ρE+π)v)=0, (12.47)
π=ρRθ, (12.48)
E=1
2|v|2+ˆe(θ,p). (12.49)
NowρE+πin (12.47) can be written in terms of the speciﬁc stagnation enthalpy .
Indeed, by using Deﬁnitions 10.3.2 and 10.3.6 we get
ρE+π=ρ|v|2
2+ρe+π=ρ|v|2
2+ρh=ρH. (12.50)

12.4. The Compressible Euler Equations 99
If the thermodynamic process is stationary the above equations yield
div(ρv)=0, (12.51)
div(ρv⊗v+πI)=0, (12.52)
div (ρHv)=0, (12.53)
π=ρRθ, (12.54)
H=1
2|v|2+ˆe(θ,p)+π
ρ. (12.55)
Moreover
div (ρHv)=Hdiv (ρv)+ρv·gradH=ρv·gradH, (12.56)
by using (12.51). Thus equation (12.53) yields ˙H=H/prime+v·gradH= 0 and hence
the thermodynamic process is homentropic . This means that His constant along
streamlines. In particular, if all streamlines arise from the same uniform ﬂow, then
Hwill be given and constant everywhere, in which case (12.53) can be eliminated.
Moreover if the gas is calorically perfect ,t h e n e=cvθand (12.54) yields
π=ρRe
cv. (12.57)
By using the Mayer equation (12.4), R=cπ−cvis also constant and the state
law becomes
π=ρcπ−cv
cve=ρ(γ−1)e. (12.58)
Replacing (12.48) or (12.54) by (12.58) and ˆ e(θ,ρ)b yein (12.49) or (12.55) allows
us to eliminate temperature, which can be computed afterwards from e,b ys o l v i n g
the algebraic equation e=ˆe(θ,p).

Chapter 13
Incompressible Fluids
In this chapter we study material bodies which are not Coleman-Noll materials in
the sense of the Deﬁnition 3.2.1. They are called incompressible ﬂuids.
13.1 Isochoric Processes
They are volume preserving thermodynamic processes:
Deﬁnition 13.1.1. A thermodynamic process is said to be isochoric if, for each t
and part P,
d
dt/integraldisplay
PtdVx=0. (13.1)
It is easy to prove that a thermodynamic process is isochoric if and only if
divv= 0 (13.2)
(see, for instance, [5]).
13.2 Newtonian Fluids
Viscous ﬂuids are called Newtonian ﬂuids when the viscous stress depends linearly
on the velocity gradient. Let us denote by Lin0(respectively by Sym 0)t h es p a c e
of tensors (respectively, symmetric tensors) with null trace.
Deﬁnition 13.2.1. Anincompressible Newtonian ﬂuid is a material body, the con-
stitutive class of which consists of all isochoric thermodynamic processes such that

102 Chapter 13. Incompressible Fluids
there exist mappings
ˆl:R×B→L (Lin0,Sym 0), (13.3)
ˆe:R×B→ R, (13.4)
ˆθ:R×B→ R+, (13.5)
ˆq:R×V×B→V (13.6)
“smooth enough” and satisfying
Td(x, t)=ˆl(s(x, t),p)(L), (13.7)
e(x, t)=ˆe(s(x, t),p), (13.8)
θ(x, t)=ˆθ(s(x, t),p), (13.9)
q(x, t)=ˆq(s(x, t),gradθ(x, t),p), (13.10)
withx=X(p,t),where Tdis the deviatoric part of T, i.e.,
Td=T−1
3tr(T)I. (13.11)
Analogous to Theorem 3.2.4 we have the following
Theorem 13.2.2. Let us consider an incompressible ﬂuid with constitutive class C.
We make the following assumption:
(H1∗)There exists a function ˆs:R+×B→ R, “smooth enough”, such that
s=ˆs(θ,p)⇐⇒θ=ˆθ(s, p)∀θ∈R+∀s∈R∀p∈B. (13.12)
Then all elements in Csatisfy the second law of thermodynamics if and only
if
ˆθ(s, p)=∂ˆe
∂s(s, p), (13.13)
ˆl(s, p)(L)·L≥0 (dissipation inequality), (13.14)
ˆq(s,w,p)·w≤0, (13.15)
for all s∈R,p∈B,L∈Lin0andw∈V.
Proof. It is a straightforward adaptation of that of Theorem 3.2.4. /square
Deﬁnition 13.2.3. The scalar ﬁeld πdeﬁned by
π(x, t)=−1
3tr(T(x, t)) (13.16)
is called the pressure of the thermodynamic process.
Assuming the material-frame indiﬀerence principle, we can prove an analo-
gous result to Proposition 10.2.3:

13.3. Ideal Fluids 103
Proposition 13.2.4. There exists a function ˆη:R×B → R,c a l l e d dynamic viscosity
of the ﬂuid, such that
ˆl(s, p)(D)=2 ˆη(s, p)D. (13.17)
Proof. It follows from the fact that tr( D)=d i v v. /square
Now we write the equations to be satisﬁed by the thermodynamic processes
ofincompressible Newtonian ﬂuids .
First of all, we use (13.12) to replace sbyθin the constitutive laws. Then,
we notice that the mass conservation equation (1.20) yields ρ(x, t)=ρ0(p), with
x=X(p,t), because det F= 1. Furthermore we have (13.2). Hence, the motion
equation becomes
ρ0(p)˙v+g r a d π−div{2ˆη(θ,p)D}=b. (13.18)
Concerning the energy equation, we use the fact that e=ˆe(θ,p)t oo b t a i n
˙e(x, t)=ˆcv(θ,p)˙θ(x, t), (13.19)
withx=X(p,t).
Finally we assume qobeys the Fourier’s law, i.e., q=−ˆk(θ,|gradθ|2,p)grad θ
and write the energy equation as follows:
ρ0(p)ˆcv(θ,p)˙θ=2 ˆη(θ,p)|D|2+d i v ( ˆk(θ,|gradθ|2,p)grad θ)+f, (13.20)
because div v=0 .
Remark 13.2.5.Unlike general Coleman-Noll materials, equations (13.2) and
(13.18) (which are called the non-homogeneous incompressible Navier-Stokes equa-
tions) can be solved independently of the energy equation, as far as ˆ ηdoes not
depend on temperature.
13.3 Ideal Fluids
Now we introduce another material body, the constitutive class of which is a
subclass of the constitutive class of the incompressible Newtonian ﬂuid given in
Deﬁnition 13.2.1 above.
Deﬁnition 13.3.1. Anideal ﬂuid is an incompressible ﬂuid characterized by the
following additional requirements:
1. The mass distribution ρ0is a constant function.
2. All thermodynamic processes in its constitutive class are Eulerian and adia-
batic.

104 Chapter 13. Incompressible Fluids
Consequently, the partial diﬀerential equations satisﬁed by any thermody-
namic process of an ideal ﬂuid are the following:
ρ0˙v+g r a d π=b, (13.21)
divv=0, (13.22)
ρ0cv˙θ=0. (13.23)
Notice that a similar Remark to 13.2.5 applies for an ideal ﬂuid: the equations
(13.21), (13.22) which are called the incompressible Euler equations ,c a nb es o l v e d
independently of the energy equation.
Moreover, the energy equation is equivalent to ˙θ= 0. Hence material points
conserve temperature during motion. Furthermore, if the process is steady ,t h e n
temperature is constant on streamlines .

Chapter 14
Turbulent Flow of
Incompressible Newtonian
Fluids
When the Reynolds number (see, for instance, [5]) goes beyond a threshold, ﬂows
become turbulent. Turbulent ﬂows are characterized by exhibiting ﬂuctuating ve-locity ﬁelds which also produce ﬂuctuations in the other ﬁelds: density, pressure,
temperature and composition. Moreover, the scale of these ﬂuctuations is so small
that it makes impossible its numerical simulation, due to the big size of the re-
quired computational mesh. Thus we are led to use turbulence models which consist
of equations (algebraic or partial diﬀerential equations) allowing us to determinethe so-called turbulent viscosity . In this way we are able to simulate the macro-
scopic ﬂow. One of the most used turbulence models is the k−/epsilon1model which will
be introduced below (see, for instance, [7]).
14.1 Turbulence Models
We suppose there are two scales: the large one corresponds to the macroscopic ﬂowwe are interested in simulating and the small one corresponds to the ﬂuctuations.In practice, we are not interested in knowing these ﬂuctuations in detail except
for the fact that they strongly inﬂuence the large scale. It is this inﬂuence that
should be determined by using turbulence models .
Let us decompose the velocity ﬁeld as
v=/angbracketleftv/angbracketright+v
/prime=V+v/prime, (14.1)
where /angbracketleft/angbracketrightis aﬁlter(i.e., some particular average) giving the main part of the
velocity ﬁeld and v/primeis the ﬂuctuation with respect to it.

106 Chapter 14. Turbulent Flow of Incompressible Newtonian Fluids
For the sake of simplicity, let us suppose ρ0andηare constant. Then the
incompressible Navier-Stokes equations (13.18) can be written in the form
ρ0∂v
∂t+ρ0div(v⊗v) + grad π−η∆v=0. (14.2)
By replacing (14.1) in (14.2) and then taking the average of the obtained
equation it is not diﬃcult to get
ρ0∂V
∂t+ρ0div(V⊗V) + gradΠ −η∆V+ρ0div/angbracketleftv/prime⊗v/prime/angbracketright=0, (14.3)
where π=Π+ π/prime, Π being the average of π. The tensor
τ=−ρ0/angbracketleftv/prime⊗v/prime/angbracketright (14.4)
is called the Reynolds stress tensor.
On the other hand, the incompressibility condition (13.2) yields
divV=0. (14.5)
In order to have a closed problem it is necessary to be able to express tensor
τas a function of the averaged ﬁelds Vand/or Π. The Boussinesq hypothesis
consists in assuming that this tensor has the form
τ=1
3tr(τ)I+2ηTD(V), (14.6)
where
D(V)=gradV+g r a d Vt
2(14.7)
andηTis called eddy dynamic viscosity ; it has to be determined by so-called
closure models .
Since we are dealing with incompressible ﬂows, the isotropic term tr( τ)I
does not need to be computed because it can simply be absorbed into the mean
pressure Π which is then replaced by Π∗=π−1
3tr(τ).
Thus we are interested in determining ηT. For this purpose several models
exist. Let us mention, among others, algebraic models like the Smagorinsky model :
ηT=ρ0l2
0√
2|D(V)|, (14.8)
where l0is the mixing length for sub-grid scales, or two equation models like the
celebrated k−/epsilon1model.

14.2. The k−/epsilon1Model 107
14.2 The k−/epsilon1Model
In this model the eddy dynamic viscosity is assumed to be of the form
ηT=ρ0cµk2
/epsilon1, (14.9)
where k:=−1
2ρ0tr(τ)=1
2/angbracketleft|v/prime|2/angbracketrightis the turbulent kinetic energy ,/epsilon1is the turbulent
dissipation rate
/epsilon1:=η
ρ0/angbracketleft|gradv/prime|2/angbracketright, (14.10)
andcµis a constant parameter to be chosen below.
Scalar ﬁelds kand/epsilon1are obtained from the following system of partial diﬀer-
ential equations:
ρ0∂k
∂t+ρ0V·gradk−ηT
2|gradV+ (grad V)t|2
−div/parenleftbigg
(η+ηT
δk)grad k/parenrightbigg
+ρ0/epsilon1=0, (14.11)
ρ0∂/epsilon1
∂t+ρ0V·grad/epsilon1−c1
2ρ0k|gradV+ (grad V)t|2
−div/parenleftbigg
(η+ηT
δ/epsilon1)grad /epsilon1/parenrightbigg
+c2ρ0/epsilon12
k=0.(14.12)
The constants in this system are obtained from experiments. In particular they
must be consistent with the time decay of turbulence and the logarithmic wall law
at boundary layer . The following values are often proposed: cµ=0.09,c1=0.126,
c2=1.92,δk=1a n d δ/epsilon1=1.3.
Finally we summarize the whole system of equations modelling turbulent
ﬂows of Newtonian incompressible ﬂuids and write them in a more compact way:
ρ0˙V+∇Π∗−2d i v( [ η+ηT]D(V)) = 0 , (14.13)
divV=0, (14.14)
ρ0˙k−div/parenleftbigg
[η+ηT
δk]gradk/parenrightbigg
−2ηT|D(V)|2+ρ0/epsilon1=0, (14.15)
ρ0˙/epsilon1−div/parenleftbigg
[η+ηT
δ/epsilon1]grad/epsilon1/parenrightbigg
−2c1ρ0k|D(v)|2+c2ρ0/epsilon12
k=0, (14.16)
where the dot stands for the material time derivative with respect to the mean
velocity ﬁeld V.

Chapter 15
Mixtures of Coleman-Noll
Fluids
In this chapter we consider the ﬂow of a mixture of (possibly chemically reacting)
species. More precisely, we suppose that the mixture consists of Coleman-Noll
ﬂuids as those studied in Chapter 10. As we will see, a mixture of Coleman-Noll ﬂuids is not a Coleman-Noll ﬂuid so we must adapt to mixtures the theory
developed in the previous chapters.
15.1 General Deﬁnitions
Let us consider the ﬂow of a mixture of Nreacting species Ei,1≤i≤N. We assume
that each species is a Coleman-Noll ﬂuid and use the following notations for species
Ei:
1.ρi:d e n s i t y ,
2.Mi: molecular mass,
3.vi:v e l o c i t y ,
4.πi: (partial) pressure,
5.ei: speciﬁc internal energy,
6.si: speciﬁc entropy,
7.hi: speciﬁc enthalpy,
8.ψi: speciﬁc Helmholtz free energy,
9.Gi: speciﬁc Gibbs free energy,
10.cv,i: speciﬁc heat at constant volume,

110 Chapter 15. Mixtures of Coleman-Noll Fluids
11.cπ,i: speciﬁc heat at constant pressure.
We also assume local thermodynamic equilibrium which implies that, at each ( x, t),
the temperature of all species is the same.
Deﬁnition 15.1.1. Thedensity of the mixture is deﬁned by
ρ=N/summationdisplay
i=1ρi. (15.1)
Deﬁnition 15.1.2. Themass fraction of species Eiis deﬁned by
Yi=ρi
ρ. (15.2)
Lemma 15.1.3. The following equality holds
N/summationdisplay
i=1Yi=1. (15.3)
Proof.N/summationdisplay
i=1Yi=N/summationdisplay
i=1ρi
ρ=/summationtextN
i=1ρi
ρ=ρ
ρ=1. /square
Deﬁnition 15.1.4. Thevelocity of the mixture is deﬁned by
v=N/summationdisplay
i=1Yivi, (15.4)
while the diﬀusion velocity of species Eiis deﬁned by
Vi=vi−v. (15.5)
Deﬁnition 15.1.5. We introduce the following ﬁelds for the mixture.
•speciﬁc internal energy:
e=N/summationdisplay
i=1Yiei, (15.6)
•speciﬁc entropy:
s=N/summationdisplay
i=1Yisi, (15.7)
•speciﬁc enthalpy:
h=N/summationdisplay
i=1Yihi, (15.8)

15.1. General Deﬁnitions 111
•speciﬁc Helmholtz free energy:
ψ=N/summationdisplay
i=1Yiψi, (15.9)
•speciﬁc Gibbs free energy:
G=N/summationdisplay
i=1YiGi, (15.10)
•speciﬁc heat at constant volume:
cv=N/summationdisplay
i=1Yicv,i, (15.11)
•pressure (Dalton’s law)
π=N/summationdisplay
i=1πi. (15.12)
From the deﬁnition above, we can see that the speciﬁc internal energy cannot
be written as a function of ν,sandponly. Thus the mixture itself is not a Coleman-
Nollﬂuid.
However, according to (15.6) we have
e=ˆe(ρ, θ, Y 1,…,Y N)=N/summationdisplay
i=1Yiˆei(ρYi,θ) (15.13)
and then
/parenleftbigg∂e
∂θ/parenrightbigg
ρ,Y1,…,Y N=N/summationdisplay
i=1Yi/parenleftbigg∂ei
∂θ/parenrightbigg
ρi=N/summationdisplay
i=1Yicv,i=cv. (15.14)
Thus, the deﬁned speciﬁc heat at constant volume for the mixture satisﬁes the same
relation as for one single Coleman-Noll ﬂuid, i.e., it is also the partial derivative
of the speciﬁc internal energy with respect to θ, at constant density and mass
fractions.
Deﬁnition 15.1.6. We deﬁne concentration of species Eito be the ﬁeld cideﬁned
by
ci=ρi
Mi. (15.15)
Remark 15.1.7.If the molecular mass Miis given in g/mol ,t h e n ciis given in
k−mol/m3.

112 Chapter 15. Mixtures of Coleman-Noll Fluids
Deﬁnition 15.1.8. We deﬁne the concentration of the mixture by
c=N/summationdisplay
i=1ci. (15.16)
Deﬁnition 15.1.9. We deﬁne molecular mass of the mixture to be the scalar ﬁeld
Msuch that
1
M=N/summationdisplay
i=1Yi
Mi. (15.17)
The following Lemma gives a justiﬁcation to this Deﬁnition.
Lemma 15.1.10. We have
c=ρ
M. (15.18)
Proof.
c=N/summationdisplay
i=1ρi
Mi=N/summationdisplay
i=1ρYi
Mi=ρN/summationdisplay
i=1Yi
Mi=ρ
M.
/square
Deﬁnition 15.1.11. We deﬁne molar fraction of species Eito be the non-dimension-
al scalar ﬁeld Xigiven by
Xi=ci
c. (15.19)
The next Lemma relates the mass fraction to the molar fraction .
Lemma 15.1.12. The following equality holds:
Yi
Mi=Xi
M. (15.20)
Proof.Yi
Mi=ρi/ρ
Mi=ci
ρ=ci
cM=Xi
M. /square
Deﬁnition 15.1.13. The scalar ﬁeld nigiven by
ni=Yi
Mi(15.21)
represents the k-moles of species Eiper kg of mixture .
We notice that niis related to cibyci=ρniand to XibyXi=Mni.
Lemma 15.1.14. Letn:=/summationtextN
i=1ni.T h e n
n=1
M. (15.22)
Proof.N/summationdisplay
i=1ni=N/summationdisplay
i=1Yi
Mi=1
M. /square

15.2. Mixture of Perfect Gases 113
15.2 Mixture of Perfect Gases
We consider the particular case where the mixture consists of perfect gases. We
recall that this means each species Eiobeys the state law
πi=ρiRiθ, (15.23)
where Ri=R
Mi.
Proposition 15.2.1. The state equation (15.23) can be written in the following
equivalent forms:
1.πi=ciRθ, (15.24)
2.πi=ρniRθ. (15.25)
Proof. πi=ρiRiθ=ρiR
Miθ=ciRθ=ρYi
MiRθ=ρniRθ. /square
Proposition 15.2.2. The following state law holds for the mixture:
π=cRθ. (15.26)
Proof. By using Dalton’s law and (15.24) we get
π=N/summationdisplay
i=1πi=N/summationdisplay
i=1ciRθ=cRθ. (15.27)
/square
Remark 15.2.3.We observe that the state equation for the mixture is similar to
the one for a perfect gas. However the mixture itself is not a perfect gas because
R=R
Mis not constant; actually, it depends on the composition of the mixture
through the mass fractions (see (15.17)).
Proposition 15.2.4. The following equalities hold:
1. πi
π=ci
c, (15.28)
2. πi
π=Xi, (15.29)
3. πi
π=ni
n. (15.30)

114 Chapter 15. Mixtures of Coleman-Noll Fluids
Proof. By using (15.24), (15.26) and (15.19) we obtain
π
πi=cRθ
ciRθ=c
ci=1
Xi.
Then, from (15.21) and (15.22), we get
π
πi=c
ci=ρ
M
ρYi
Mi=1
M
Yi
Mi=n
ni. /square
Now we write the speciﬁc Gibbs free energy as a function of θ,π,n1,…,n N
and compute its partial derivatives with respect to the two ﬁrst variables. For this
purpose, we notice that (15.30) yields
πi=πni
/summationtextN
j=1nj, (15.31)
which allows us to express partial pressures in terms of the above independent
variables. Thus the speciﬁc Gibbs free energy of the mixture can be written as a
function of θ, π, n 1,…,n N,n a m e l y ,
ˆG(θ,π,n 1,…,n N)=N/summationdisplay
i=1MiniˆGi(θ,πi)=N/summationdisplay
i=1MiniˆGi(θ,π n i/(N/summationdisplay
j=1nj)).
(15.32)
Proposition 15.2.5. The partial derivatives of ˆGwith respect to θandπare given
by
1./parenleftBigg
∂G
∂θ/parenrightBigg
π,nj=−s, (15.33)
2./parenleftBigg
∂G
∂π/parenrightBigg
θ,nj=ν. (15.34)
Proof. By using (10.129) and (10.130) we have
1./parenleftBigg
∂G
∂θ/parenrightBigg
π,nj=N/summationdisplay
i=1Mini/parenleftBigg
∂Gi
∂θ/parenrightBigg
πi=N/summationdisplay
i=1Mini(−si)=−s, (15.35)
2./parenleftBigg
∂G
∂π/parenrightBigg
θ,nj=N/summationdisplay
i=1Mini/parenleftBigg
∂Gi
∂π/parenrightBigg
θ=N/summationdisplay
i=1Mini/parenleftBigg
∂Gi
∂πi/parenrightBigg
θ/parenleftbigg∂πi
∂π/parenrightbigg
θ,nj
=N/summationdisplay
i=1Miniνini
n=N/summationdisplay
i=1Yi1
ρYini
n=1
ρ=ν. (15.36)
/square

15.2. Mixture of Perfect Gases 115
We recall that, for perfect gases, the speciﬁc Gibbs free energy is given by
(see (12.26))
ˆGi(θ,πi)=ˆG0
i(θ)+Riθln (πi
π0), (15.37)
ˆG0
i(θ)b e i n gt h e speciﬁc Gibbs free energy of formation at temperature θand
pressure π0.
Then the speciﬁc Gibbs free energy for the mixture is
ˆG(θ,π,n 1,…,n N)=N/summationdisplay
i=1niMiˆGi(θ,πi)=N/summationdisplay
i=1niMi/parenleftBigg
ˆG0
i(θ)+Riθln (πi
π0)/parenrightBigg
=N/summationdisplay
i=1ni/parenleftBigg
MiˆG0
i(θ)+Rθln (π
π0)/parenrightBigg
+RθN/summationdisplay
i=1niln (ni
n),(15.38)
where we have used (15.30).
Remark 15.2.6.In most books the standard speciﬁc Gibbs free energy of formation
is given in J/k-mol, that is, they include values of
ˆµ0
i(θ)=MiˆG0
i(θ), (15.39)
rather than ˆG0
i(θ), for π0=1a t m .
Finally we have the following expression for the Gibbs free energy of a mixture
of perfect gases:
ˆG(θ,π,n 1,…,n N)=ˆG0(θ,n1,…,n N)+Rθln (π
π0)+RθN/summationdisplay
i=1niln (ni
n),(15.40)
w h e r ew eh a v eu s e dt h ef a c tt h a t
N/summationdisplay
i=1niR=R
M=R, (15.41)
and introduced the speciﬁc Gibbs free energy of formation of the mixture at tem-
perature θand pressure π0by
ˆG0(θ,n1,…,n N): =N/summationdisplay
i=1niˆµ0
i(θ). (15.42)
Deﬁnition 15.2.7. We denote by chemical potential of species Eithe scalar ﬁeld
µideﬁned by µi(x, t)=ˆµi(θ(x, t),π(x, t),n1(x, t),…,n N(x, t)), where
ˆµi(θ,π,n 1,…,n N)=∂ˆG
∂ni(θ,π,n 1,…,n N). (15.43)

116 Chapter 15. Mixtures of Coleman-Noll Fluids
Proposition 15.2.8. We have the following expressions for the chemical potential
of species Ei:
1. µi=µ0
i+Rθln (π
π0)+Rθln (ni
n), (15.44)
2. µi=µ0
i+Rθln (πi
π0). (15.45)
Proof.
∂ˆG
∂ni(θ,π,n 1,…,n N)=ˆ µ0
i(θ)+Rθln (π
π0)+Rθln (ni
n)
+Rθ/parenleftBiggN/summationdisplay
j=1njδijn−nj
n2n
nj/parenrightBigg
=ˆµ0
i(θ)+Rθln (π
π0)+Rθln (ni
n), (15.46)
because
N/summationdisplay
j=1δijn−nj
n=n−/summationtextN
j=1nj
n=0, (15.47)
from which (15.44) follows.
To get (15.45) it is enough to replace in (15.44) ni/nbyπi/π(see (15.30)). /square
Corollary 15.2.9. The following expression for the speciﬁc Gibbs free energy of the
mixture holds:
ˆG(θ,π,n 1,…,n N)=N/summationdisplay
i=1niˆµi(θ,π,n 1,…,n N). (15.48)
Proof. It follows from the fact that
ˆGi(θ,πi)=M−1
i/parenleftbigg
ˆµ0
i(θ)+Rθln(πi
π0)/parenrightbigg
=M−1
iˆµi(θ,π,n 1,…,n N),(15.49)
which is an immediate consequence of (15.37) and (15.45). /square
Now we consider other thermodynamic variables and compute their partial
derivatives. Let us start with the speciﬁc internal energy. We have the following
Proposition 15.2.10./parenleftBigg
∂e
∂ni/parenrightBigg
ν,s,n j=µi. (15.50)

15.2. Mixture of Perfect Gases 117
Proof. From the deﬁnition of Gwe have
e=G+θs−π/ρ (15.51)
and then
ˆe(ν,s,n 1,…,n N)=ˆG(ˆθ(ν,s,n 1,…,n N),ˆπ(ν,s,n 1,…,n N),n1,…,n N)
+sˆθ(ν,s,n 1,…,n N)−νˆπ(ν,s,n 1,…,n N).
(15.52)
By taking the partial derivative with respect to niwe get
/parenleftBigg
∂e
∂ni/parenrightBigg
ν,s=/parenleftBigg
∂G
∂θ/parenrightBigg
π,nj/parenleftBigg
∂θ
∂ni/parenrightBigg
ν,s,n j+/parenleftBigg
∂G
∂π/parenrightBigg
θ,nj/parenleftBigg
∂π
∂ni/parenrightBigg
ν,s,n j
+/parenleftBigg
∂G
∂ni/parenrightBigg
θ,ν,n j+s/parenleftBigg
∂θ
∂ni/parenrightBigg
ν,s,n j−ν/parenleftBigg
∂π
∂ni/parenrightBigg
ν,s,n j. (15.53)
By using (15.33) and (15.34) in the above equality, the result is easily obtained. /square
In a similar way, we can compute the partial derivatives below.
Proposition 15.2.11. We have
1./parenleftBigg
∂e
∂s/parenrightBigg
ν,nj=θ,/parenleftBigg
∂e
∂ν/parenrightBigg
s,nj=−π,/parenleftBigg
∂e
∂ni/parenrightBigg
ν,s,n j=µi, (15.54)
2./parenleftBigg
∂h
∂s/parenrightBigg
π,nj=θ,/parenleftBigg
∂h
∂π/parenrightBigg
s,nj=ν,/parenleftBigg
∂h
∂ni/parenrightBigg
π,s,n j=µi, (15.55)
3./parenleftBigg
∂ψ
∂θ/parenrightBigg
ν,nj=−s,/parenleftBigg
∂ψ
∂ν/parenrightBigg
θ,nj=−π,/parenleftBigg
∂ψ
∂ni/parenrightBigg
ν,θ,n j=µi,(15.56)
4./parenleftBigg
∂s
∂e/parenrightBigg
ν,nj=1
θ,/parenleftBigg
∂s
∂ν/parenrightBigg
e,nj=−π
θ,/parenleftBigg
∂s
∂ni/parenrightBigg
ν,e,n j=−µi
θ.(15.57)

Chapter 16
Chemical Reactions in a Stirred
Tank
In this chapter we consider a mixture of reacting Coleman-Noll ﬂuids in a stirred
tankB0. The latter means that ﬁelds do not depend on the space variable but only
on time. We also assume that the tank is closed so that its total mass is conserved
and hence density is constant.
16.1 Chemical Kinetics. The Mass Action Law
Let us assume the mixture undergoes a chemical process involving Lreactions:
νl
1E1+···+νl
NEN−→λl
1E1+···+λl
NEN,1 ≤l≤L.
Coeﬃcients νl
iandλl
iare called stoichiometric coeﬃcients .
The evolution of concentrations of diﬀerent species is governed by a system
of ordinary diﬀerential equations, namely,
dci
dt=L/summationdisplay
l=1(λl
i−νl
i)δl, (16.1)
where δlis the velocity of the l-th chemical reaction. According to the mass action
law,i ti sg i v e nb y
δl=klN/productdisplay
j=1cνl
j
j, (16.2)
klbeing a function of temperature through the Arrhenius law :
kl(θ)=Blθαle−El
Rθ. (16.3)

120 Chapter 16. Chemical Reactions in a Stirred Tank
In the previous expression, Blis the frequency (or pre-exponential) factor ,αlthe
pre-exponential exponent andEltheactivation energy . In many cases constant αl
is null.
Remark 16.1.1.Since the exponentEl
Rθis non-dimensional, θl=El
Rhas dimension
of temperature. Actually, it is called activation temperature .
In the particular case of combustion reactions, Elas well as Blare very large.
Then, for values of temperature below θl,kland, correspondingly, the velocity of
reaction, δl, are very small. On the contrary, for temperatures of the order of θlor
higher, this parameter becomes very large which means that reaction progresses
very rapidly.
By replacing expression (16.2) into (16.1) we get the nonlinear system of
ordinary diﬀerential equations
dci
dt=L/summationdisplay
l=1(λl
i−νl
i)Blθαle−El
RθN/productdisplay
j=1cνl
j
j. (16.4)
In order to solve it we need to specify the initial concentrations of species,
ci(0) = ci,init, (16.5)
as well as another equation allowing us to determine the evolution of the mixture
temperature. For this purpose several diﬀerent assumptions can be made concern-
ing the heat exchange between the tank and its environment. Here we suppose
there is a system of heat supplied to the tank (see Deﬁnition 1.4.1). This leads to
the following form of the energy conservation principle:
d
dt/integraldisplay
B0ρe dV =Q, (16.6)
where the function of time Q(t)i st h e heat rate supplied into the tank B0at time
t(see Deﬁnition 1.4.2).
Since ρis constant and edoes not depend on the space variable x, we deduce
from (16.6)
de
dt=1
mQ, (16.7)
mbeing the (constant) mass of the mixture. We notice that m=ρvol(B0).
Moreover, from (15.6) we deduce the equation:
e=N/summationdisplay
i=1Yiˆei(ρi,θ), (16.8)
where
e(t)=einit+1
m/integraldisplayt
0Q(s)ds, (16.9)

16.1. Chemical Kinetics. The Mass Action Law 121
einitbeing the given initial speciﬁc internal energy of the mixture. The “algebraic”
equation (16.8) allows us to compute temperature along the time.
Alternatively, one can obtain an ordinary diﬀerential equation for θas follows.
By replacing (16.8) in (16.7) we get
N/summationdisplay
i=1dYi
dtˆei(ρi,θ)+ρN/summationdisplay
i=1Yi∂ˆei
∂ρi(ρi,θ)dYi
dt+N/summationdisplay
i=1Yiˆcv,i(ρi,θ)dθ
dt=1
mQ,
(16.10)
because ρis constant. We recall that ˆ cv,i(ρi,θ)=∂ˆei
∂θ(ρi,θ)i st h e speciﬁc heat at
constant volume of species Ei. According to (15.11), the speciﬁc heat at constant
volume of the mixture is
cv=ˆcv(ρ, θ, Y 1,…,Y N)=N/summationdisplay
i=1Yiˆcv,i(ρYi,θ). (16.11)
Then equation (16.10) can be written as
N/summationdisplay
i=1dYi
dt/bracketleftbigg
ρYi∂ˆei
∂ρi(ρi,θ)+ˆei(ρYi,θ)/bracketrightbigg
+ˆcv(ρ, θ, Y 1,…,Y N)dθ
dt=1
mQ.
(16.12)
Now we rewrite (16.4) in terms of the mass fractions rather than the concentra-
tions. We have
dYi
dt=Mi
ρL/summationdisplay
l=1(λl
i−νl
i)Blθαle−El
RθN/productdisplay
j=1/parenleftbiggρYj
Mj/parenrightbiggνl
j
. (16.13)
In order to obtain a system of ordinary diﬀerential equations in normal form we
replace (16.13) in (16.12). It yields
dθ
dt=−1
ρˆcv(ρ, θ, Y 1,…,Y N)N/summationdisplay
i=1Mi/braceleftbigg/bracketleftbigg
ρYi∂ˆei
∂ρi(ρi,θ)+ˆei(ρYi,θ)/bracketrightbigg
×⎡
⎣L/summationdisplay
l=1(λl
i−νl
i)Blθαle−El
RθN/productdisplay
j=1/parenleftbiggρYj
Mj/parenrightbiggνl
j⎤
⎦⎫
⎬
⎭+1
mˆcv(ρ, θ, Y 1,…,Y N)Q.
(16.14)
The system (16.13), (16.14) can now be integrated for the initial conditions (16.5)
and
θ(0) = θinit, (16.15)
θinitbeing the initial temperature of the mixture.

122 Chapter 16. Chemical Reactions in a Stirred Tank
16.2 Conservation of Chemical Elements
Since atoms are conserved in chemical reactions, we can write linear algebraic
equations establishing this fact. In this way, we get a reduction in the number of
equations of the ordinary diﬀerential system (16.13).
Let us suppose there are Kdiﬀerent elements involved in species, named Ak,
1≤k≤K. Let the formula of species Eibe
Ei=(A1)a1i···(AK)aKi,1≤i≤N. (16.16)
Thus akidenotes the number of atoms of element Akin a molecule of species Ei.
In terms of mass, there are akik-atoms of element Akper k-mol of species Ei.
At time twe have a composition of the mixture deﬁned, for instance, in terms
ofn1,…,n N.T h e n akinigives the k-atoms of element Akin species Eiper kg of
mixture.
LetZk,init be the number of k-atoms of element Akper kg of mixture at
initial time. Since this quantity is conserved, we must have that
N/summationdisplay
i=1akini=N/summationdisplay
i=1akiYi
Mi=Zk,init,1≤k≤K, (16.17)
at any time.
In fact we can prove that any solution of the ordinary diﬀerential system
(16.1) satisﬁes (16.17) at any time. Indeed, ﬁrstly equations (16.17) can be written
in terms of species concentrations, in the form
N/summationdisplay
i=1akici=ρZk,init,k=1,…,K. (16.18)
Let us multiply (16.1) by ak,iand then add all equations for i=1,…,N .W e
have
N/summationdisplay
i=1d
dt(akici)=N/summationdisplay
i=1aki/parenleftBiggL/summationdisplay
l=1(λl
i−νl
i)δl/parenrightBigg
=L/summationdisplay
l=1δl/parenleftBiggN/summationdisplay
i=1aki(λl
i−νl
i)/parenrightBigg
,
k=1,…,K. (16.19)
Moreover, since atoms of each element are conserved in the l-th reaction, we must
have
N/summationdisplay
i=1akiλl
i=N/summationdisplay
i=1akiνl
i,k=1,…,K (16.20)
and then (16.19) yields
d
dt/parenleftBiggN/summationdisplay
i=1akici/parenrightBigg
=0,k=1,…,K. (16.21)

16.3. Reacting Mixture of Perfect Gases 123
Hence/summationtextN
i=1akici,k=1,…,K are constant in time so equal to their respective
initial values.
Constraints (16.18) can be considered as a linear system of equations for
ci,i=1,…,N, which allows us to write Kof them in terms of the remaining
N−K. Thus we can eliminate the Kcorresponding equations in (16.1).
16.3 Reacting Mixture of Perfect Gases
Let us suppose all components of the mixture are perfect gases. In this case the
speciﬁc internal energy of each component is a function of temperature only (see
Proposition 12.1.2) and then (16.8) becomes
N/summationdisplay
i=1Yiˆei(θ)=e. (16.22)
The speciﬁc internal energy of species Eiis now given by
ˆei(θ)=e0
i+/integraldisplayθ
θ0ˆcv,i(s)ds, (16.23)
where e0
i=ˆei(θ0) is the speciﬁc internal energy of formation of species Eiat
temperature θ0. If this temperature is 25oC,t h e n e0
iis called the standard speciﬁc
internal energy of formation of species Ei.
By replacing (16.23) in (16.22) we obtain
N/summationdisplay
i=1Yi[e0
i+/integraldisplayθ
θ0ˆcv,i(s)ds]=e, (16.24)
withestill given by (16.9).
We can include this equation into the ordinary diﬀerential system (16.13)
to get a closed problem. It allows us to compute the temperature and the mass
fractions of species along time.
Instead, we can also use the ordinary diﬀerential equation (16.14) which, for
a mixture of perfect gases, becomes
dθ
dt=−1
ρˆcv(θ,Y1,…,Y N)N/summationdisplay
i=1Miˆei(θ)⎡
⎣L/summationdisplay
l=1(λl
i−νl
i)Blθαle−El
RθN/productdisplay
j=1/parenleftbiggρYj
Mj/parenrightbiggνl
j⎤
⎦
+1
mˆcv(θ,Y1,…,Y N)Q.
(16.25)
Usually, equation (16.22) is written in terms of enthalpy rather than internal
energy. We recall that
hi=ei+πi
ρi=ei+Riθ, (16.26)

124 Chapter 16. Chemical Reactions in a Stirred Tank
withhigiven by
hi=ˆhi(θ)=h0
i+/integraldisplayθ
θ0ˆcπ,i(s)ds, (16.27)
andh0
i=ˆhi(θ0) being the speciﬁc enthalpy of formation of species Eiat tem-
perature θ0. If this temperature is 25oC,t h e n h0
iis called the standard speciﬁc
enthalpy of formation .
Thestandard speciﬁc enthalpy of formation of chemical elements is taken to
be zero. Thus the standard speciﬁc enthalpy of formation represents the change in
enthalpy that accompanies the formation of one kg of the species from its elements
at the standard speciﬁed temperature, when each reactant and each product is in
its standard state.
By using (16.26) and (16.27), equation (16.22) can be rewritten in the form
N/summationdisplay
i=1Yi[h0
i+/integraldisplayθ
θ0ˆcπ,i(s)ds−Riθ]=e. (16.28)
Alternatively, by taking the derivative of this equation with respect to time, we
can obtain an ordinary diﬀerential equation for θ. Indeed, ﬁrstly we have
N/summationdisplay
i=1dYi
dt(h0
i+/integraldisplayθ
θ0ˆcπ,i(s)ds−Riθ)+N/summationdisplay
i=1Yi[ˆcπ,i(θ)−Ri]dθ
dt=1
mQ,
(16.29)
from which it follows that
N/summationdisplay
i=1dYi
dt(h0
i+/integraldisplayθ
θ0ˆcπ,i(s)ds−Riθ)+[ˆcπ(θ,Y1,…,Y N)−ˆR(Y1,…,Y N)]dθ
dt=1
mQ,
(16.30)
where
ˆcπ(θ,Y1,…,Y N): =N/summationdisplay
i=1Yiˆcπ,i(θ) (16.31)
is thespeciﬁc heat at constant pressure of the mixture andˆR(Y1,…,Y N)i si t s gas
constant given by
ˆR(Y1,…,Y N)=R
M(Y1,…,Y N), (16.32)
M(Y1,…,Y N) being the molecular mass of the mixture (see Deﬁnition 15.1.9).
Finally we replace (16.13) in (16.30) to get
dθ
dt=−1
ρ[ˆcπ(θ,Y 1,…,Y N)−ˆR(Y1,…,Y N)]NX
i=1Mij»
h0
i+Zθ
θ0ˆcπ,i(s)ds−Riθ–
×"LX
l=1(λl
i−νl
i)Blθαle−El
RθNY
j=1„ρYj
Mj«νl
j#)
+Q
m[ˆcπ(θ,Y 1,…,Y N)−ˆR(Y1,…,Y N)].

Chapter 17
Chemical Equilibrium of a
Reacting Mixture of PerfectGases in a Stirred Tank
In this Chapter we study the chemical equilibrium of a reacting mixture of perfect
gases in a stirred tank .
17.1 The Least Action Principle for the Gibbs
Free Energy
In this Section we assume the tank is closed both to mass and energy transfer. At
initial time we introduce a certain amount of each species Ei. Then they react and
attain chemical equilibrium (see, for instance, [9]). The problem consists in deter-
mining the composition of the mixture as well as its pressure and temperature.
The data are the initial composition deﬁned by ni,init,i=1,…,N ,( f r o mw h i c h
we can easily compute the corresponding numbers Zk,init,k=1,…,K, giving
the k-atoms of element Akper kg of mixture), and the initial speciﬁc internal
energy of the mixture.
We notice that, since density does not change with time, it is also known at
equilibrium and hence we could write the pressure as a function of temperature
and composition through the state law .
The main result in chemical equilibrium theory is the least action principle
for the free energy:
The system is in equilibrium if and only if the speciﬁc Gibbs free energy,
ˆG(θ,π,n 1,…,n N), attains a minimum subject to the following constraints:

126 Chapter 17. Chemical Equilibrium of a Reacting Mixture
•mass
N/summationdisplay
i=1akini=Zk,init:=N/summationdisplay
i=1akini,init,1≤k≤K, (17.1)
•energy
N/summationdisplay
i=1Mini[h0
i+/integraldisplayθ
θ0ˆcπ,i(s)ds−Riθ]=einit, (17.2)
•state law
π=ρRN/summationdisplay
i=1niθ, (17.3)
•positivity
ni≥0,i=1,…,N. (17.4)
Remark 17.1.1.We recall that, in this constrained optimization problem, ρ,einit
andZk,init,k=1,…,K are known.
Alternatively, if θis given instead of einit, then we remove constraint (17.2).
Remark 17.1.2.In order to compute the equilibrium state, the above constrained
optimization problem must be solved by mathematical programming techniques.
It is important to notice that we do not need to know the chemical reactions
involved, but only the species that are present in the mixture at equilibrium. Inwhat follows, we suppose that these chemical reactions are also given and show
that, in this case, the equilibrium problem can be more easily solved.
17.2 Equilibrium for a Set of Reversible Reactions,
Equilibrium Constants
For the sake of clarity we ﬁrst consider a particular case consisting of only one
reversible reaction. Moreover, we assume that θandπare given so we can remove
constraints (17.2) and (17.3) from the above optimization problem characterizingthe equilibrium state. Accordingly, the density is not imposed which means that
the volume of the tank may change. In fact, it can be obtained from the state law
(17.3) after the composition of the mixture has been determined by solving the
minimization problem.
In the framework of Section 16, one reversible reaction corresponds to two
reactions ( L= 2). Let us introduce
λ
i:=ν2
i=λ1
i, (17.5)
νi:=λ2
i=ν1
i. (17.6)

17.2. Equilibrium for a Set of Reversible Reactions, Equilibrium Constants 127
By using this notation, the reversible reaction can be written as
ν1E1+···+νNEN−→←λ1E1+···+λNEN, (17.7)
so that we have
δ1=k1N/productdisplay
j=1cνj
jand δ2=k2N/productdisplay
j=1cλj
j (17.8)
for the velocities of the forward and backward reactions, respectively. The ordinary
diﬀerential system (16.1) has to be replaced by
ρdni
dt=(λi−νi)(k1N/productdisplay
j=1cνj
j−k2N/productdisplay
j=1cλj
j). (17.9)
By introducing
δ=(k1N/productdisplay
j=1cνj
j−k2N/productdisplay
j=1cλj
j), (17.10)
equation (17.9) yields
1
(λi−νi)dni
dt=δ
ρ. (17.11)
By integrating this system we deduce
n1(t)−n1,init
(λ1−ν1)=···=nN(t)−nN,init
(λN−νN)=ξ(t), (17.12)
where ni,init,i=1,…,N correspond to the initial composition of the mixture
andξ, the reaction extent ,i sg i v e nb y
ξ(t)=/integraldisplayt
0δ(s)
ρ(s)ds. (17.13)
The interest of (17.12) is that we can write the ni,i=1,…,N, in terms of one
single function ξ. Thus the speciﬁc Gibbs free energy is given by
G=ˆg(ξ): =ˆG(θ,π,n 1,init+(λ1−ν1)ξ,…,n N,init +(λN−νN)ξ).(17.14)
In this way, the mass conservation equations (17.1) are automatically satisﬁed.
Then the only constraint remaining is (17.4) and so the equilibrium composition
can be obtained by solving the following one-dimensional constrained optimization
problem:
min
(λi−νi)ξ+ni,int≥0ˆg(ξ). (17.15)

128 Chapter 17. Chemical Equilibrium of a Reacting Mixture
If ˆgattains a minimum at ξ, two possibilities exist: either ( λi−νi)ξ+ni,int=0
for some ior ˆg/prime(ξ) = 0. The latter can be written as follows:
ˆg/prime(ξ)=N/summationdisplay
i=1∂ˆG
∂ni(θ,π,n 1,…,n N)dni
dξ=N/summationdisplay
i=1(λi−νi)ˆµi(θ,π,n 1,…,n N)=0,
(17.16)
withni=ni,init+(λi−νi)ξ, i=1,…,N.
Deﬁnition 17.2.1. The function
ˆa(θ,π,n 1,…,n N)=−N/summationdisplay
i=1(λi−νi)ˆµi(θ,π,n 1,…,n N) (17.17)
is called chemical aﬃnity of reaction (17.7).
Since species Eiare perfect gases we can replace the expression for ˆ µigiven
by (15.45) in (17.16). It becomes
N/summationdisplay
i=1(λi−νi)/parenleftbigg
ˆµ0
i(θ)+Rθln (πi
π0)/parenrightbigg
=0. (17.18)
Let us introduce K0
π,t h e equilibrium constant based on partial pressures for
chemical reaction (17.7). It is given as a function of temperature by
K0
π(θ)=e x p/parenleftBigg
−1
RθN/summationdisplay
i=1(λi−νi)ˆµ0
i(θ)/parenrightBigg
. (17.19)
Then, from (17.18),
K0
π(θ)=N/productdisplay
i=1/parenleftbiggπi
π0/parenrightbiggλi−νi
. (17.20)
The quantity
∆G0(θ)=N/summationdisplay
i=1(λi−νi)ˆµ0
i(θ) (17.21)
is called change of speciﬁc free energy in the reaction at temperature θand pres-
sureπ0. Then (17.19) yields
K0
π(θ)=e x p/parenleftbigg
−1
Rθ∆G0(θ)/parenrightbigg
. (17.22)

17.2. Equilibrium for a Set of Reversible Reactions, Equilibrium Constants 129
We can deﬁne another equilibrium constant based on concentrations. For this
purpose, we rewrite (17.20) making use of (15.24). We obtain
K0
π(θ)=N/productdisplay
i=1/parenleftbiggciRθ
π0/parenrightbiggλi−νi
=/parenleftbiggRθ
π0/parenrightbiggPN
i=1(λi−νi)N/productdisplay
i=1cλi−νi
i. (17.23)
By introducing the equilibrium constant based on concentrations ,
K0
c(θ)=K0
π(θ)/parenleftbiggRθ
π0/parenrightbiggPN
i=1(νi−λi)
, (17.24)
we deduce from (17.23)
K0
c(θ)=N/productdisplay
i=1cλi−νi
i. (17.25)
Finally we recall that we can choose π0to be the standard pressure , i.e., 1 atm. In
this case it can be removed from the computations above as far as partial pressures
are also given in atm.
We come back to the solution of the equilibrium problem. Since the change of
s p e c i ﬁ cf r e ee n e r g y in the reaction can be computed from the speciﬁc free energy of
species by (17.21), we can obtain the equilibrium constant K0
πfrom (17.22). Then,
using (15.30) and (17.12) in (17.20) we get
K0
π(θ)=N/productdisplay
i=1/parenleftbiggπni
π0n/parenrightbiggλi−νi
=N/productdisplay
i=1/parenleftBigg
π[ni,init+(λi−νi)ξ]
π0/summationtextN
i=1[ni,init+(λi−νi)ξ]/parenrightBiggλi−νi
=/parenleftbiggπ
π0/parenrightbiggPN
i=1(λi−νi)N/productdisplay
i=1/parenleftBigg
ni,init+(λi−νi)ξ
/summationtextN
i=1[ni,init+(λi−νi)ξ]/parenrightBiggλi−νi
.(17.26)
This equation can be solved for ξ.L e t ξ∗be a solution. If ξ∗satisﬁes all of the
constraints then it allows us to compute the composition at equilibrium by means
of (17.12).
Now we can consider the more general case of Jreversible reactions between
species E1,…,E N:
νj
1E1+···+νj
NEN−→←λj
1E1+···+λj
NEN,j=1,…,J . (17.27)
We recall that each of them leads to two reactions in the framework introduced
in Section 16. We denote by ξjtheextent of the j-th reversible reaction given by
(17.27). Then, in a similar way to (17.12), we can obtain
ni=ni,init+J/summationdisplay
j=1(λj
i−νj
i)ξj,i=1,…,N. (17.28)

130 Chapter 17. Chemical Equilibrium of a Reacting Mixture
Hence the speciﬁc Gibbs free energy can be written in terms of the extents ξj,j=
1,…,J instead of ni,i=1,…,N .L e tˆgbe deﬁned by
ˆg(ξ1,…,ξ J):=ˆG⎛
⎝θ,π,n 1,init+J/summationdisplay
j=1(λj
1−νj
1)ξj,…,n N,init+J/summationdisplay
j=1(λj
N−νj
N)ξj⎞
⎠.
(17.29)
Since now the ni,i=1,…,N are restricted to those of the form (17.28), the
mass conservation constraints (17.1) automatically hold. Indeed,
N/summationdisplay
i=1akini=N/summationdisplay
i=1aki[ni,init+J/summationdisplay
j=1(λj
i−νj
i)ξj]
=Zk,init +J/summationdisplay
j=1N/summationdisplay
i=1aki(λj
i−νj
i)ξj=Zk,init,1≤k≤K, (17.30)
because of (16.20).
Then the equilibrium problem is equivalent to the following constrained op-
timization problem:
min
ni,init+PJ
j=1(λj
i−νj
i)ξj≥0ˆg(ξ1,…,ξ J). (17.31)
Let us suppose that the solution to this problem does not saturate any positivity
constraint, i.e., the corresponding ni,i=1,…,N are all strictly positive. Then
the partial derivatives of ˆ gmust vanish, that is,
∂ˆg
∂ξj(ξ1,…,ξ J)=N/summationdisplay
i=1∂ˆG
∂ni(θ,π,n 1,…,n N)dni
dξj
=N/summationdisplay
i=1(λj
i−νj
i)ˆµi(θ,π,n 1,…,n N)=0,j=1,…,J , (17.32)
withni=ni,init+/summationtextJ
j=1(λj
i−νj
i)ξj,i=1,…,N . This is a system of Jequations
for the Junknowns ξj,j=1,…,J .
Since species Eiare perfect gases, we can replace in equations (17.32) the
expression for ˆ µigiven by (15.45). They become
N/summationdisplay
i=1(λj
i−νj
i)/parenleftbigg
ˆµ0
i(θ)+Rθln (πi
π0)/parenrightbigg
=0,j=1,…,J . (17.33)
Let us introduce K0
π,j,t h eequilibrium constant based on partial pressures for the
j-th chemical reaction in (17.27). It is given as a function of temperature by
K0
π,j(θ)=e x p/parenleftBigg
−1
RθN/summationdisplay
i=1(λj
i−νj
i)ˆµ0
i(θ)/parenrightBigg
. (17.34)

17.2. Equilibrium for a Set of Reversible Reactions, Equilibrium Constants 131
Then, from (17.33),
K0
π,j(θ)=N/productdisplay
i=1/parenleftbiggπi
π0/parenrightbiggλj
i−νj
i
,j=1,…,J . (17.35)
The quantity
∆G0,j(θ)=N/summationdisplay
i=1(λj
i−νj
i)ˆµ0
i(θ) (17.36)
is called change of speciﬁc free energy in the j-th reaction at temperature θand
pressure π0. Using (17.21) in (17.34) we get
K0
π,j(θ)=e x p/parenleftbigg
−1
Rθ∆G0,j(θ)/parenrightbigg
. (17.37)
As in the case of one reversible reaction, we can deﬁne other equilibrium
constants based on concentrations. For this purpose, we rewrite (17.35) making
use of (15.24). We obtain
K0
π,j(θ)=N/productdisplay
i=1/parenleftbiggciRθ
π0/parenrightbiggλj
i−νj
i
=/parenleftbiggRθ
π0/parenrightbiggPN
i=1(λj
i−νj
i)N/productdisplay
i=1cλj
i−νj
i
i. (17.38)
By introducing the equilibrium constant based on concentrations for the j-th reac-
tion,
K0
c,j(θ)=K0
π,j(θ)/parenleftbiggRθ
π0/parenrightbiggPN
i=1(νj
i−λj
i)
, (17.39)
we deduce from (17.38)
K0
c,j(θ)=N/productdisplay
i=1cλj
i−νj
i
i. (17.40)
We come back to the solution of the equilibrium problem. Since the change of
s p e c i ﬁ cf r e ee n e r g y of the reactions can be computed from the speciﬁc free energy of
species by (17.36), we can obtain the equilibrium constants K0
π,jby using (17.37).
Then we use (15.30) and (17.28) in (17.35) to write
K0
π,j(θ)=N/productdisplay
i=1/parenleftbiggπni
π0n/parenrightbiggλj
i−νj
i
=N/productdisplay
i=1/parenleftBigg
π[ni,init+/summationtextJ
j=1(λj
i−νj
i)ξj]
π0/summationtextN
i=1[ni,init+/summationtextJ
j=1(λj
i−νj
i)ξj]/parenrightBiggλj
i−νj
i
=/parenleftbiggπ
π0/parenrightbiggPN
i=1(λj
i−νj
i)N/productdisplay
i=1/parenleftBigg
ni,init+/summationtextJ
j=1(λj
i−νj
i)ξj
/summationtextN
i=1[ni,init+/summationtextJ
j=1(λj
i−νj
i)ξj]/parenrightBiggλj
i−νj
i
.
(17.41)
This system of equations can be solved for ξ1,…,ξ Jand then the composition at
equilibrium can be computed by means of (17.28).

132 Chapter 17. Chemical Equilibrium of a Reacting Mixture
17.3 The Stoichiometric Method
An important conclusion of the previous section is that, when the chemical reac-
tions are known, we can eliminate the mass conservation constraints and reduce
the number of unknowns in the equilibrium problem. Our present goal is to show
that, even if we do not know these chemical reactions, it is possible to introduce
some virtual ones which also allow us to reduce the number of unknowns and toeliminate the mass constraints in the equilibrium minimization problem.
LetAbe the matrix A=(a
ki)k=1,…,K, i =1,…,N. Then (16.17) can be written
as
An=Zinit. (17.42)
wheren=(n1,…,n N)tandZinit=(Z1,init,…,Z K,init)t. Let us denote by R the
rank of matrix A, that is, the maximum number of independent rows or columns.
Then the dimension of the kernel ofAis given by
J=d i mK e r ( A)=N−R. (17.43)
Since
Zinit=Aninit, (17.44)
then the linear system (17.42) is compatible which implies that
rank(A|Zinit)=R. (17.45)
Furthermore its general solution is
n=ninit+R/summationdisplay
j=1ξjγj, (17.46)
where {ξj,j=1,…,J }are arbitrary real numbers and the set of vectors in RN,
{γj,j=1,…,J }, is a basis of the kernel of Awhich is nothing but the subspace
ofRNof the solutions of the homogeneous linear system An=0 .
In order to obtain a basis for Ker( A) we can use the next algorithm. Since
rank(A)=Rthere must exist an invertible square matrix Pof order K,a n ds u c h
thatB=PAhas the block form
B=/parenleftbigg
IRC
00/parenrightbigg
, (17.47)
IRbeing the identity matrix of order RandCa non-null rectangular matrix of
order R×J.M a t r i x Bcan be obtained by Gaussian elimination.
Then an admissible choice for {γj,j=1,…,J }are the columns of the
N×Jmatrix Gdeﬁned by
G=/parenleftbigg
−C
IJ/parenrightbigg
. (17.48)

17.3. The Stoichiometric Method 133
Indeed, we have
AG=P−1B/parenleftbigg
−C
IJ/parenrightbigg
=P−1/parenleftbigg
IRC
00/parenrightbigg/parenleftbigg
−C
IJ/parenrightbigg
=P−1/parenleftbigg
0
0/parenrightbigg
=( 0 ).(17.49)
Now we come back to the general situation where {γj,j=1,…,J }is any
basis of Ker( A). Let us consider the matrix or rank Jobtaining by putting these
vectors as columns, i.e.,
Γ=/parenleftbig
γ1|…|γJ/parenrightbig
. (17.50)
We notice that Γ = Gin the previous discussion. This matrix is called full sto-
ichiometric matrix and deﬁnes a so-called full set of chemical reactions .T h e s e
reactions are
0=γj
1E1+···+γj
NEN,j=1,…,J . (17.51)
One can consider as reactants those species Eifor which γj
iare negative and as
products those for which these coeﬃcients are positive. Comparing with (17.7) we
have
γj
i=λj
i−νj
i,i=1,…,N, j =1,…,J . (17.52)
Thus, in order to solve the equilibrium problem, ﬁrstly we determine vectors
γj,j=1,…,J . Secondly, we replace the expressions for ni,i=1,…,N given
by (17.46) in the Gibbs free energy, ˆG(θ,π,n 1,…,n N) and in the constraints
(17.2)–(17.4). Thirdly, we minimize the obtained function
ˆg(θ,π,ξ 1,…,ξ J)= :ˆG(θ,π,n 1,init+J/summationdisplay
j=1γj
1ξj,…,n N,init +J/summationdisplay
j=1γj
Nξj) (17.53)
under the constraints (17.2)–(17.4), with nireplaced by
ni,init+R/summationdisplay
j=1ξjγj
i,i=1,…,N. (17.54)
For this purpose it is convenient to compute the partial derivatives of ˆ g.As t r a i g h t –
forward application of the chain rule leads to the following results:
Proposition 17.3.1. We have
1./parenleftBigg
∂g
∂θ/parenrightBigg
π,ξj=/parenleftBigg
∂G
∂θ/parenrightBigg
π,ni. (17.55)
2./parenleftBigg
∂g
∂π/parenrightBigg
θ,ξj=/parenleftBigg
∂G
∂π/parenrightBigg
θ,ni. (17.56)

134 Chapter 17. Chemical Equilibrium of a Reacting Mixture
3./parenleftBigg
∂g
∂ξl/parenrightBigg
θ,π,ξ j=N/summationdisplay
i=1γl
iˆµi(θ,π,n 1,init+J/summationdisplay
j=1γj
1ξj,…,n N,init +J/summationdisplay
j=1γj
Nξj).
(17.57)
The quantity
ˆal(θ,π,ξ 1,…,ξ J)=−N/summationdisplay
i=1γl
iˆµi(θ,π,ninit+J/summationdisplay
j=1ξjγj) (17.58)
is the chemical aﬃnity of the l-th virtual reaction (see (17.17)).

Chapter 18
Flow of a Mixture of Reacting
Perfect Gases
In this chapter we establish the equations for general ﬂows of a mixture of reacting
perfect gases. Thus we remove the assumption of stirred tank and from now on
all functions characterizing the system depend on position and time, i.e., they are
spatial ﬁelds . This is the case, in particular, for those deﬁning the composition of
the mixture.
We consider Nsuperposed motions Xi,i=1,…,N, ofNbodies Bi,i=
1,…,N (the species Ei) such that all of them occupy the same position at each
timet, i.e.,
Bi
t=Bt∀i=1,…,N. (18.1)
Hence, each species has its own spatial velocity viand we can deﬁne the velocity
of the mixture ,v, by (15.4) and the diﬀusion velocity of species Ei,Vi, by (15.5).
18.1 Mass Conservation Equations
Themass conservation equation of species Eiis
∂ρi
∂t+d i v ( ρivi)=ωi+mi, (18.2)
where ωiandmiare the source of mass for species Eidue to chemical reactions
and to an evaporating or combusting condensed phase, respectively. An example
of the latter is a liquid or solid fuel in a combustion chamber. According to (16.1),
ωiis given by
ωi=MiL/summationdisplay
l=1(λl
i−νl
i)δl, (18.3)

136 Chapter 18. Flow of a Mixture of Reacting Perfect Gases
where δlis the velocity of the l-th reaction (see (16.2)). In its turn, term miwould
be determined by a model for the condensed phase.
In order to get a conservation equation for the total mass of the mixture we
add (18.2) from i=1t o N:
N/summationdisplay
i=1[∂ρi
∂t+d i v ( ρivi)] =∂
∂t(N/summationdisplay
i=1ρi)+d i v (N/summationdisplay
i=1ρivi)=N/summationdisplay
i=1ωi+N/summationdisplay
i=1mi.(18.4)
We prove that the ﬁrst term on the right-hand side is null. Indeed,
N/summationdisplay
i=1ωi=N/summationdisplay
i=1MiL/summationdisplay
l=1(λl
i−νl
i)δl=L/summationdisplay
l=1δlN/summationdisplay
i=1Mi(λl
i−νl
i)=0, (18.5)
because the total mass is conserved in each chemical reaction, i.e.,
N/summationdisplay
i=1Miλl
i=N/summationdisplay
i=1Miνl
i,l=1,…,L. (18.6)
By using this, and deﬁnitions (15.1) and (15.4), (18.4) yields
∂ρ
∂t+d i v ( ρv)=m:=N/summationdisplay
i=1mi. (18.7)
Now we come back to the mass conservation equation for species (18.2).
We do not intend to compute the velocity of each species so, in order to get a
closed system, we have to write the diﬀusion velocity in terms of any other ﬁelds
remaining in the model. This can be done by using Fick’s law :
Vi=−Digrad(ln Yi)=−Di
YigradYi, (18.8)
where Diis the mass diﬀusion coeﬃcient of species Eiinto the mixture.
By replacing (18.8) in (18.2) after using (15.5) we get
∂
∂t(ρYi)+d i v ( ρYiv)−div(ρDigradYi)=ωi+mi, (18.9)
which is the mass conservation equation for species Eiinconservative form. By
multiplying (18.7) by Yiand then subtracting from (18.9) we can also obtain the
non-conservative form:
ρ˙Yi−div(ρDigradYi)=ωi+mi−mYi. (18.10)

18.2. Motion Equation 137
18.2 Motion Equation
Thebalance of linear and angular momentum for the mixture can be written as
(1.27) and (1.28) where now the body force bincludes all external body forces as,
for instance, gravity forces or the rate of linear momentum supplied to the mixture
by a condensed phase.
Moreover, we assume a constitutive law for Tsimilar to that for a Coleman-
Noll ﬂuid, namely
T=−πI+l(D), (18.11)
where πis related to the partial pressures of species by Dalton’s law (see (15.12))
andl(D), the viscous part of the Cauchy stress tensor, is given by (10.29).
Then we deduce the following motion equation inconservative form:
∂(ρv)
∂t+d i v ( ρv⊗v) + grad π=d i v ( l(D)) +b. (18.12)
By multiplying (18.7) by vand then subtracting from (18.12), we get the
non-conservative form:
ρ˙v+g r a d π=d i v ( l(D)) +b−mv. (18.13)
18.3 Energy Conservation Equation
We assume the energy conservation law (1.46) where we recall that faccounts for
the external contribution of heat, in particular from the condensed phase. Then, in
terms of the speciﬁc internal energy , the conservative form of the energy equation
is
∂(ρe)
∂t+d i v ( ρev)=T·D−divq+f. (18.14)
From the deﬁnitions of internal energy andenthalpy of the mixture and the Dalton
lawit is easy to deduce an equation for the mixture analogous to (10.35). It allows
writing the energy equation in terms of enthalpy, namely
∂(ρh)
∂t+d i v ( ρhv)−˙π=l(D)·D−divq+f. (18.15)
Moreover, the constitutive law for the conductive heat ﬂux qis taken to be
q=−kgradθ+N/summationdisplay
i=1ρiVihi, (18.16)
where the second term on the right-hand side represents the enthalpy ﬂux due to
the diﬀusion velocity of species. We make use of Fick’s law to replace the diﬀusion

138 Chapter 18. Flow of a Mixture of Reacting Perfect Gases
velocities in the above expression. This leads to
∂(ρh)
∂t+div( ρhv)−∂π
∂t−div(πv)=l(D)·D−div(−kgradθ−ρN/summationdisplay
i=1DihigradYi)+f.
(18.17)
Equation (18.17) can be written in a more compact form in which tempera-
ture does not appear in an explicit way. For this, we have to make the following
assumptions:
1.The mass diﬀusion coeﬃcient is the same for all species, i.e., Di=D,i=
1,…,N.
2.The Lewis number of the mixture is equal to 1. More precisely, we can replace
the thermal conductivity, k,b yρDcπin (18.16).
Then we have
ρDgradh=ρDgrad/parenleftBiggN/summationdisplay
i=1Yihi/parenrightBigg
=ρDN/summationdisplay
i=1gradYihi+ρDN/summationdisplay
i=1Yigradhi
=ρDN/summationdisplay
i=1gradYihi+ρDN/summationdisplay
i=1Yicπ,igradθ=ρDN/summationdisplay
i=1gradYihi+ρDcπgradθ,
(18.18)
and hence
q=−ρDgradh. (18.19)
Now we replace this expression for qin equation (18.15). It yields
∂(ρh)
∂t+d i v ( ρhv)−˙π=l(D)·D+d i v ( ρDgradh)+f. (18.20)
Sometimes it is more convenient to handle this energy conservation equation
innon-conservative form. For this we multiply (18.7) by hand subtract from
(18.20). Since we have
ρ˙h+h∂ρ
∂t+hdiv(ρv)=ρ∂h
∂t+ρv·gradh+h∂ρ
∂t+hdiv(ρv)=∂(ρh)
∂t+div( ρvh),
(18.21)
it yields
ρ˙h−˙π=l(L)·D+d i v ( ρDgradh)+f−mh. (18.22)
Sometimes equation (18.20) is written in terms of the speciﬁc thermal en-
thalpy rather than the speciﬁc (total) enthalpy . For this we recall that, because of
(15.8) and (16.27), the speciﬁc enthalpy of the mixture is given by
h=N/summationdisplay
i=1Yih0
i+hT, (18.23)

18.3. Energy Conservation Equation 139
where we recall that h0
i=ˆhi(θ0)i st h e enthalpy of formation of species Eiat
temperature θ0andhTis the thermal enthalpy of the mixture with respect to
thisreference temperature . The latter is given, in terms of temperature and mass
fractions, by
hT=ˆhT(θ,Y1,…,Y n)=N/summationdisplay
i=1Yi/integraldisplayθ
θ0cπ,i(s)ds. (18.24)
By using (18.23) we obtain
∂
∂t(ρh)=∂
∂t(ρhT)+N/summationdisplay
i=1/parenleftBig
h0
i∂
∂t(ρYi)/parenrightBig
, (18.25)
div(ρvh)=d i v ( ρvN/summationdisplay
i=1Yih0
i)+d i v ( ρvhT), (18.26)
−div(ρDgradh)= −div/parenleftBigg
ρDgradhT/parenrightBigg
−N/summationdisplay
i=1h0
idiv/parenleftBig
ρDgradYi/parenrightBig
.(18.27)
Let us multiply the mass conservation equation for species Ei, (18.9), by h0
i
and then add for i=1,…,N .W eg e t
N/summationdisplay
i=1h0
i∂
∂t(ρYi)+N/summationdisplay
i=1h0
idiv(ρvYi)−N/summationdisplay
i=1h0
idiv(ρDgradYi)=N/summationdisplay
i=1h0
i(ωi+mi).
(18.28)
Then we add equations (18.25), (18.26) and (18.27), and use (18.28). We get
∂
∂t(ρh)+d i v ( ρvh)−div(ρDgradh)=∂
∂t(ρhT)+d i v ( ρvhT)
−div(ρDgradhT)+N/summationdisplay
i=1h0
i(ωi+mi). (18.29)
Now, we use this equality in (18.20) to obtain a partial diﬀerential equation for
thermal enthalpy inconservative form:
∂
∂t(ρhT)+div( ρvhT)−˙π−div(ρDgradhT)=l(D)·D+f−N/summationdisplay
i=1h0
i(ωi+mi).(18.30)
In order to obtain the corresponding non-conservative equation it is enough
to subtract the mass conservation equation (18.7) multiplied by hTfrom (18.30).
It yields
ρ˙hT−˙π−div(ρDgradhT)=l(D)·D+f−N/summationdisplay
i=1h0
i(ωi+mi)−hTm.(18.31)

140 Chapter 18. Flow of a Mixture of Reacting Perfect Gases
Finally, we summarize the equations to be satisﬁed by any thermodynamic
process of areacting mixture of perfect gases . Under the assumptions above con-
cerning the mass diﬀusion coeﬃcients and Lewis number, they are, in conservative
form,
∂ρ
∂t+d i v ( ρv)=m, (18.32)
∂(ρv)
∂t+d i v ( ρv⊗v) + grad π=d i v ( l(D)) +b,(18.33)
∂(ρh)
∂t+d i v ( ρhv)−˙π−div(ρDgradh)=l(D)·D+f, (18.34)
∂(ρYi)
∂t+d i v ( ρYiv)−div(ρDgradYi)=ωi+mi,i=1,…,N, (18.35)
together with
π=ρRθ, (18.36)
R=R
M=RN/summationdisplay
i=1Yi
Mi, (18.37)
l(D)=2 ηD+ξdivvI, (18.38)
h=N/summationdisplay
i=1Yi[ˆhi(θ0)+/integraldisplayθ
θ0ˆcπ,i(s)ds], (18.39)
ωi=MiL/summationdisplay
l=1(λl
i−νl
i)δl,i=1,…,N, (18.40)
δl=klN/productdisplay
j=1/parenleftbiggρYi
Mi/parenrightbiggνl
j
,l=1,…,L. (18.41)
18.4 Conservation of Elements
In Section 16.2 we have deﬁned some linear combinations of the mass fractions of
species which are conserved in a stirred tank ,n a m e l y
Zk=N/summationdisplay
i=1akini=N/summationdisplay
i=1akiYi
Mi,1≤k≤K. (18.42)
They represent the k-atoms of element Akper kg of mixture.
The situation is not the same in the case of a general ﬂow: these ﬁelds are not
conserved anymore at each particular position xbecause the mixture is moving

18.5. Equilibrium Chemistry 141
and element mass fractions may be diﬀerent from one point to another. Neverthe-
less, they satisfy partial diﬀerential equations similar to (18.35) but without the
reaction terms ωion the right-hand side. In other words, they are transported like
passive scalars .
In order to obtain these equations, let us multiply (18.35) by aki/Miand
then add from i=1,…,N .W eg e t
∂
∂t(ρZk)+d i v ( ρZkv)−div(ρDgradZk)=N/summationdisplay
i=1aki
Mimi,k=1,…,K, (18.43)
where we have used the fact that (see (16.19))
N/summationdisplay
i=1aki
Miωi=0,k=1,…,K. (18.44)
Thus, in a ﬁrst step, we can solve these partial diﬀerential equations for Zk,k=
1,…,K . Next we can construct, for each ( x, t), the linear system (cf. (18.42))
N/summationdisplay
i=1akiYi(x, t)
Mi=Zk(x, t), 1≤k≤K. (18.45)
In general, this system is under-determined because K<N so it cannot be solved
to obtain the mass fractions of species. Nevertheless, it allows us to replace K
nonlinear partial diﬀerential equations in the system (18.35) by Klinear algebraic
equations, which are much easier to solve.
18.5 Equilibrium Chemistry
Sometimes chemical reactions are very fast compared to the residence time so we
can assume local equilibrium chemistry at each point xand time t. As we will
show below, in this case numerical computations are highly reduced because none
of equations (18.35) needs to be solved. Instead, it is enough to solve the (linear)
uncoupled equations (18.43) to determine ﬁelds Zk.
Indeed, according to the previous analysis in Section 17, the local equilibrium
problem to be solved at each ( x, t) is the following:
Findθ(x, t),π(x, t),n1(x, t), …,n N(x, t)minimizing the speciﬁc Gibbs free
energy
ˆG(θ,π,n 1,…,n N) (18.46)
under the following constraints:
•mass
N/summationdisplay
i=1akini=Zk(x, t),1≤k≤K, (18.47)

142 Chapter 18. Flow of a Mixture of Reacting Perfect Gases
•enthalpy
N/summationdisplay
i=1Mini[h0
i+/integraldisplayθ
θ0ˆcπ,i(s)ds]=h(x, t), (18.48)
•state law
π=ρ(x, t)RN/summationdisplay
i=1niθ, (18.49)
•positivity
ni≥0,i=1,…,N, (18.50)
where ﬁelds ρ(x, t),h(x, t)a n dZk(x, t),k=1,…,K are the solution of the partial
diﬀerential equations system (18.32), (18.33), (18.34) and (18.43).
18.6 The Case of Low Mach Number
For low Mach number, one can show that the spatial ﬂuctuations of pressure are
small compared to the pressure itself. In this case pressure may be considered as
spatially constant in the state equation while, in the motion equation, the term
gradπhas to be retained. Moreover, sometimes the mean pressure is given as a
function of time and its value, to be denoted by ¯ π=¯π(t) ,c a nb eu s e di nt h es t a t e
equation as well as in the speciﬁc Gibbs free energy . Then the problem to be solved
is the following:
For each (x, t), to ﬁnd θ(x, t),n1(x, t),···,nN(x, t)minimizing the speciﬁc
Gibbs free energy
ˆG(θ,¯π(t),n1,…,n N) (18.51)
under the constraints:
•mass
N/summationdisplay
i=1akini=Zk(x, t),1≤k≤K, (18.52)
•enthalpy
N/summationdisplay
i=1Mini[h0
i+/integraldisplayθ
θ0ˆcπ,i(s)ds]=h(x, t), (18.53)

18.6. The Case of Low Mach Number 143
•positivity
ni≥0,i=1,…,N, (18.54)
where ﬁelds h(x, t)a n d Zk(x, t),k=1,…,K , are solution of partial diﬀerential
equations (18.34) and (18.43). In the former, ˙ πmust be replaced by ˙¯π,w h i c hi s
given.
Remark 18.6.1.The rest of the equations, namely, (18.32), (18.33) and the state
equation
¯π=ρRθ, (18.55)
are solved independently in order to determine density, velocity and pressure.
More precisely, after computation of θandYi,i=1,…,N ,f r o mt h ea b o v e
equilibrium problem , we can use (18.55) to obtain density . Then we solve (18.32),
(18.33) to get velocity andpressure . Actually, equation (18.32) can be viewed as
a constraint on the velocity allowing us to determine pressure in equation (18.33)
as aLagrange multiplier . In this sense, problem (18.32), (18.33) is quite similar to
theincompressible Navier-Stokes equations (13.18) and (13.2).

Chapter 19
The Method of Mixture
Fractions
We introduce a method based on the conservation of chemical elements which is
very useful to reduce the number of equations to be solved.
19.1 General Equations
We restrict ourselves to the steady state c ase. This means that ﬁelds do not depend
on time and therefore equations (18.32)–(18.35) have to be replaced by
div(ρv)=m, (19.1)
div(ρv⊗v) + grad π=d i v ( l(D)) +b, (19.2)
div(ρhv)−v·gradπ−div(ρDgradh)=l(D)·D+f, (19.3)
div(ρYiv)−div(ρDgradYi)=ωi+mi,i=1,…,N. (19.4)
Similarly, instead of equations (18.43) for Zk,w eh a v e
div(ρZkv)−div(ρDgradZk)=N/summationdisplay
i=1akimi
Mi,1 ≤k≤K. (19.5)
We are going to see that, in some particular cases, namely, if boundary con-
ditions have the particular form described below, then the solution of partial dif-
ferential equations (19.5) can be obtained in an advantageous way.

146 Chapter 19. The Method of Mixture Fractions
Let us suppose the motion takes place in a ﬁxed domain B0, the boundary
of which, Γ, consists of several parts: the inlets Γ1,…,ΓP,t h eoutlet Γoand the
rest
Γw=Γ\(∪P
p=1Γp∪Γo), (19.6)
which will be called the wall.
We make the following assumptions:
1.There are Ssources deﬁned by scalar ﬁelds ϕs,s=1,…,S , such that
mi(x)=S/summationdisplay
s=1bisϕs(x),i=1,…,N. (19.7)
Function ϕsrepresents the rate of total mass per unit volume provided by
thes-th source to the gas phase. Thus we assume that
N/summationdisplay
i=1bis=1,s=1,…,S , (19.8)
and hence
S/summationdisplay
s=1ϕs=S/summationdisplay
s=1N/summationdisplay
i=1bisϕs=N/summationdisplay
i=1S/summationdisplay
s=1bisϕs=N/summationdisplay
i=1mi=m. (19.9)
2.Boundary conditions for equations (19.4) are of the following form:
Yi=Yi,ponΓp,p=1,…,P, i =1,…,N, (19.10)
∂Yi
∂n=0onΓo∪Γw,i=1,…,N, (19.11)
for some given constants Yi,p.
As a consequence of this, boundary conditions for equations (19.5) are
Zk=Zk,pon Γ p,p=1,…,P, k =1,…,K, (19.12)
∂Zk
∂n=0 o n Γ o∪Γw,k=1,…,K, (19.13)
where constants Zk,pdenote the k-atoms of element Akper kg of mixture entering
the domain through inletΓp. They can be computed from the mass fractions of
species at Γ p,Yi,p,b y
Zk,p=N/summationdisplay
i=1akiYi,p
Mi,p=1,…,P, k =1,…,K. (19.14)

19.1. General Equations 147
In order to solve (19.5), let us deﬁne the scalar steady ﬁelds fp(x),p=1,…,P ,
to be the solution of the partial diﬀerential equations
div(ρfpv)−div(ρDgradfp)=0 i n B0, (19.15)
fp(x)= δnpon Γ n,n=1,…,P, (19.16)
∂fp
∂n=0 o n Γ o∪Γw, (19.17)
and the scalar ﬁelds gs,s=1,…,S , as the solution of the boundary value prob-
lems
div(ρgsv)−div(ρDgradgs)= ϕsinB0, (19.18)
gs(x)=0 o n Γ n,n=1,…,P, (19.19)
∂gs
∂n=0 o n Γ o∪Γw. (19.20)
Since
N/summationdisplay
i=1akimi(x)
Mi=S/summationdisplay
s=1(N/summationdisplay
i=1akibis
Mi)ϕs(x), (19.21)
it is straightforward to see that the solution of equation (19.5) is given by
Zk(x)=P/summationdisplay
p=1Zk,pfp(x)+S/summationdisplay
s=1αksgs(x)∀x∈B0,k=1,…,K, (19.22)
where αksis deﬁned by
αks:=N/summationdisplay
i=1akibis
Mi. (19.23)
Moreover, we have
P/summationdisplay
p=1fp(x)+S/summationdisplay
s=1gs(x)=1∀x∈B0. (19.24)
Indeed, by adding (19.15) from p=1,…,P , together with (19.18) for s=1,…,S ,
we obtain
div/parenleftBigg
ρv(P/summationdisplay
p=1fp+S/summationdisplay
s=1gs)/parenrightBigg
−div/parenleftBigg
ρDgrad(P/summationdisplay
p=1fp+S/summationdisplay
s=1gs)/parenrightBigg
=S/summationdisplay
s=1ϕs=m. (19.25)

148 Chapter 19. The Method of Mixture Fractions
Furthermore, by adding the boundary conditions, we get
P/summationdisplay
p=1fp+S/summationdisplay
s=1gs=1o nΓ p,p=1,…,P, (19.26)
∂(/summationtextP
p=1fp+/summationtextS
s=1gs)
∂n=0o nΓ o∪Γw. (19.27)
Finally, by using (19.1), it is straightforward to see that w(x) = 1 is the unique
solution of the boundary-value problem
div(ρwv)−div(ρDgradw)=m, (19.28)
w=1o nΓ p,p=1,…,P, (19.29)
∂w
∂n=0o nΓ o∪Γw, (19.30)
from which (19.24) follows.
Scalar ﬁelds fpandgsare called mixture fractions .Inlets andsources are
called streams .W en o t i c et h a t ,i ft h e number of streams ,P+S,i sl e s st h a n
thenumber of chemical elements ,K, then it is advantageous to solve the partial
diﬀerential equations for fpandgs, instead of those for Zk, and then to compute
the latter by (19.22). Moreover, it is enough to solve P+S−1 of them and then
to get the remaining stream by using (19.24).
Finally we emphasize the fact that we do not require that each part of the
boundary Γpbe connected. Actually, each Γ pmay consist of several diﬀerent phys-
icalinlets. We only need that input gases have the same uniform composition at
these inlets (cf. (19.10)).
19.2 Examples
In this section we consider some simple particular cases:
1. P=1, S=0 (one stream).
f1(x)≡1,Z k(x)=Zk,1. (19.31)
2. P=2, S=0 (two streams).
f2(x)=1−f1(x),Z k(x)=Zk,1f1(x)+Zk,2(1−f1(x)).(19.32)
3. P=1, S=1 (two streams).
g1(x)=1−f1(x),Z k(x)=Zk,1f1(x)+αk1(1−f1(x)).(19.33)

19.3. The Adiabatic Case 149
4. P=2, S=1 (three streams).
g1(x)=1−f1(x)−f2(x), (19.34)
Zk(x)=Zk,1f1(x)+Zk,2f2(x)+αk1(1−f1(x)−f2(x)). (19.35)
5. P=1, S=2 (three streams).
g2(x)=1−f1(x)−g1(x), (19.36)
Zk(x)=Zk,1f1(x)+αk1g1(x)+αk2(1−f1(x)−g1(x)). (19.37)
19.3 The Adiabatic Case
Moreover, under the assumptions below, the speciﬁc enthalpy can also be obtained
from the mixture fractions instead of solving (19.3) directly.
Let us suppose that
H1.pressure can be considered as spatially constant (isobaric process),
H2.viscous dissipation l(D)·Dmay be neglected,
H3.body heating, f,i sn u l l ,
H4.there is no heat ﬂux by conduction through Γ0∪Γw.
In this case we say that the ﬂow is adiabatic . We recall that the ﬁrst assumption
holds for low Mach number ﬂows (see (18.6)). Moreover, we notice that assumption
H3is not compatible with the existence of a condensed phase.
Under H1toH4the energy equation (19.3) becomes
div(ρhv)−div(ρDgradh)=0. (19.38)
Furthermore, if boundary conditions on inlets are
h=hpon Γ p,p=1,…,P, (19.39)
for some constants hp,p=1,…,P , then the speciﬁc enthalpy of the mixture is
given by
h(x)=P/summationdisplay
p=1hpfp(x)∀x∈B0. (19.40)
19.4 The Case of Equilibrium Chemistry
In this section we still suppose steady state. Furthermore, we assume local equi-
librium chemistry andisobaric process in the sense that, except in the motion
equation, pressure can be taken equal to a given constant ¯ π.

150 Chapter 19. The Method of Mixture Fractions
We will see that we can make some previous computations allowing us to
save computational eﬀort in solving the whole model for the reacting ﬂow of a
mixture.
Indeed, we notice that giving the mixture fractions f1(x),…,f P(x),g1(x),
…,g S(x)a n dt h e speciﬁc enthalpy h(x), at any point x, we can determine tem-
perature, density and composition of the mixture at equilibrium, at this point.
For this purpose, we ﬁrst determine the element mass fractions Zk,k=1,…,K ,
by (19.22) and then temperature and composition are obtained by solving the
equilibrium chemistry. The latter is done by minimizing
ˆG(θ,¯π,n1,…,n N), (19.41)
under the constraints
•mass
N/summationdisplay
i=1akini=Zk(x)=P/summationdisplay
p=1Zk,pfp(x)+S/summationdisplay
s=1αksgs(x), 1≤k≤K, (19.42)
•enthalpy
N/summationdisplay
i=1Mini[h0
i+/integraldisplayθ
θ0ˆcπ,i(s)ds]=h(x), (19.43)
•positivity
ni≥0,i=1,…,N. (19.44)
Next the density of the mixture at point xcan be obtained from the state
lawby
ρ(x)=¯π
R(x)θ(x)(19.45)
with
R(x)=RN/summationdisplay
i=1Yi(x)
Mi(19.46)
andYi(x)=Mini(x).
Thus, for given f1,…,f P,g1,…,g Sandhwe can compute θ,ρandYi,i=
1,…,N without solving any partial diﬀerential equation, just by solving the above
constrained optimization problem .

19.4. The Case of Equilibrium Chemistry 151
Let us suppose that speciﬁc enthalpy is between two given values hminand
hmax. Then, in a ﬁrst step, we may tabulate the “response” functions
ˆθ:A−→R (19.47)
(f1,…,f P−1,g1,…,g S,h)−→ θ=ˆθ(f1,…,f P,g1,…,g S,h),
ˆρ:A−→R (19.48)
(f1,…,f P−1,g1,…,g S,h)−→ ρ=ˆρ(f1,…,f P,g1,…,g S,h),
ˆYi:A−→R (19.49)
(f1,…,f P−1,g1,…,g S,h)−→ Yi=ˆYi(f1,…,f P,g1,…,g S,h),
where
A=[ 0,1]P+S−1×[hmin,hmax]. (19.50)
Then these tables are used as follows: we compute f1(x),…,f P−1(x),g1(x),
…,g S(x)( o r f1(x),…,f P(x),g1(x),…,g S−1(x)) and h(x) by solving partial
diﬀerential equations (19.15)–(19.20) and (19.3), and, after that, we determine
θ(x),Yi(x),i=1,…,N andρ(x) from the tables. Then this density is replaced
in (19.1) and (19.2) which are solved to get velocity and pressure.
Since ρandvappear in these partial diﬀerential equations, some iterative
procedure needs to be introduced. In practice (19.15), (19.18) and (19.3), withtheir respective boundary conditions , are solved by using numerical methods and,
usually, the steady state is obtained as the limit of an evolutionary process as time
goes to inﬁnity, which is solved step by step by discretization schemes.
We notice that the number of partial diﬀerential equations to be solved is
P+Swhich is usually much smaller than N+1a n de v e nt h a n K+1 .
Finally, we notice that, under the additional assumptions H1toH4,h(x)c a n
be obtained by using (19.40) instead of solving the partial diﬀerential equation
(19.3). Thus, h(x) is also given by means of a response function of the mixture
fractions, namely
h(x)=ˆh(f
1,…,f P−1): =P/summationdisplay
p=1hpfp(x). (19.51)
This also implies that response functions ˆθ,ˆρandˆYic a nb ed e ﬁ n e di nt e r m so f
f1,…,f P−1andg1,…,g Sonly.

Chapter 20
Turbulent Flow of Reacting
Mixtures of Perfect Gases,
The PDF Method
Throughout this chapter we make the assumptions introduced at the beginning
of Chapter 19, namely, (19.7) and the particular boundary conditions (19.10) and
(19.11).
We recall that, when the ﬂow is turbulent, the velocity induces ﬂuctuations in
the remaining physical variables: density, pressure, temperature and composition.
However, as in Chapter 14, we are interested in determining the mean values
of these ﬁelds. An approach to compute them is the following: instead of writingpartial diﬀerential equations to directly determine these mean values, as we did for
velocity and pressure in Chapter 14, we consider the mixture fractions as random
variables with some prescribed probability density functions (PDF). Then, since
density, temperature and mass fractions are functions of these mixture fractions
through the response functions (19.47)–(19.49), we can get their mean values at
each particular point x∈B
0by using classical results in probability theory .
20.1 Elements of Probability
Let us ﬁrst recall some basic notions on random variables and vectors. We consider
aprobability space , that is, a triple (Ω ,F,P) consisting of a set Ω, a σ-ﬁeldMof
subsets of Ω and a probability measure Pon (Ω,F).
Deﬁnition 20.1.1. Arandom variable Yis a function Y:Ω−→Rwith the
property that the set [ Y≤y]: ={ω∈Ω:Y(ω)≤y}belongs to F∀y∈R.

154 Chapter 20. Turbulent Flow of Reacting Mixtures of Perfect Gases
Deﬁnition 20.1.2. Thedistribution function of a random variable Yis the function
Φ:R−→[0,1] given by
Φ(y)=P([Y≤y]). (20.1)
Deﬁnition 20.1.3. The random variable Yis said to be continuous if its distribution
function can be expressed as
Φ(y)=/integraldisplayy
−∞φ(s)ds,∀y∈R (20.2)
for some integrable function φ:R−→[0,∞) called the probability density function
ofY.
Deﬁnition 20.1.4. Themean value orexpectation of a continuous random variable
Ywith density function φis the number
¯Y=E[Y]: =/integraldisplay∞
−∞yφ(y)dy, (20.3)
whenever this integral exists.
Deﬁnition 20.1.5. Thevariance of a continuous random variable Ywith probability
density function φis the number
σ=E[(Y−E[Y])2] (20.4)
whenever it exists.
Deﬁnition 20.1.6. Arandom vector Y=(Y1,…,Y N) is a mapping Y:Ω−→
RPsuch that [ Y≤y]: ={ω∈Ω:Yi(ω)≤yi,i=1,…,P }belongs to F
∀y=(y1,…,y P)∈RP.
Deﬁnition 20.1.7. The(joint) distribution function of a random vector Yon the
probability space (Ω ,F,P) is the function Φ : RP−→[0,1],
Φ(y)=P([Y≤y]). (20.5)
Deﬁnition 20.1.8. A random vector is (jointly) continuous if their joint distribution
function can be expressed as
Φ(y1,…,y P)=/integraldisplayy1
−∞···/integraldisplayyP
−∞φ(s1,···,sP)ds1…d s P∀(y1,…,y P)∈RP.
(20.6)
Deﬁnition 20.1.9. The random variables Y1,…,Y Pare called independent if the
events [ Y1≤y1],…,[YP≤yP] are independent ∀(y1,…,y P)∈R, i.e.,
P(P/intersectiondisplay
i=1[Yi≤yi]) =P/productdisplay
i=1P([Yi≤yi]). (20.7)

20.2. The Mixture Fraction/PDF Method 155
Proposition 20.1.10. Let us assume that Y1,…,Y Pare independent continuous
random variables with density functions φ1,…,φ P, respectively. Then the random
vector Y=(Y1,…,Y P)is continuous and its joint density function, φis given by
φ(y1,…,y P)=φ1(y1)···φP(yP).
Proposition 20.1.11. Let(Y1,…,Y P)be a random vector and F:RP−→Ra
continuous function. Then the mapping
ω∈Ω−→F(Y1(ω),…,Y P(ω))∈R (20.8)
is a random variable. Furthermore its expectation is given by
E[F(Y1,…,Y P)] =/integraldisplay∞
−∞···/integraldisplay∞
−∞F(y1,…,y P)φ(y1,···,yP)dy1…d y P.
(20.9)
Corollary 20.1.12. Under the assumptions of the previous Proposition if, further-
more, Y1,…,Y Pare independent continuous random variables with density func-
tionsφ1,…,φ N, respectively, then the expectation of F(Y1,…,Y P)is given by
E[F(Y1,…,Y P)] =/integraldisplay∞
−∞···/integraldisplay∞
−∞F(y1,…,y P)φ1(y1)···φP(yP)dy1…d y P.
(20.10)
20.2 The Mixture Fraction/PDF Method
Let us come back to the turbulent reacting ﬂow. We postulate that, at each
point x∈B0,t h emixture fractions f1(x),…,f P−1(x)a n d g1(x),…,g S(x)( o r
f1(x),…,f P(x),g1(x),…,g S−1(x)) are independent continuous random variables
with density functions φ1(x,·),…,φ P−1(x,·)a n d ψ1(x,·),…,ψ S(x,·), respec-
tively, which are completely determined by their respective expectations ¯f1(x),…,
¯fP−1(x)a n d¯ g1(x),…,¯gS(x), and variances σ1(x),…,σ P−1(x)a n d ε1(x),…,
εS(x). Moreover, we assume that these expectations and variances are the solu-
tions of the following so-called Favre averaged partial diﬀerential equations (see
also Chapter 14):
div(ρ¯fpV)−div([ρD+ηT
δf]grad ¯fp)=0, (20.11)
¯fp=δpnon Γ n,n=1,…,P, (20.12)
∂¯fp
∂n=0o nΓ o∪Γw, (20.13)
div(ρσpV)−div([ρD+ηT
δf]gradσp)−2ηT
δf|grad¯fp|2+Cdρ/epsilon1
kσp=0, (20.14)
σp=0o nΓ n,n=1,…,P, (20.15)
∂σp
∂n=0o nΓ o∪Γw, (20.16)

156 Chapter 20. Turbulent Flow of Reacting Mixtures of Perfect Gases
div(ρ¯gsV)−div([ρD+ηT
δg]grad¯gs)=ϕs, (20.17)
¯gs=0o nΓ n,n=1,…,P, (20.18)
∂¯gs
∂n=0o nΓ o∪Γw, (20.19)
div(ρεsV)−div([ρD+ηT
δg]gradεs)−2ηT
δg|grad¯gs|2+Cdρ/epsilon1
kεs=0, (20.20)
εs=0o nΓ n,n=1,…,P, (20.21)
∂εs
∂n=0o nΓ o∪Γw, (20.22)
where σfandσgare the turbulent Prandtl/Schmidt numbers and the turbulence
constant Cdis taken to be 2.
Concerning density functions φp(x,·)a n d ψs(x,·), they are usually chosen
of the same form. Several possibilities are considered in the bibliography, two of
which are given below (see [6]):
1.The double delta function
φi(x, y)=⎧
⎨
⎩0.5i f y=¯fi(x)−/radicalbig
σi(x),
0.5i f y=¯fi(x)+/radicalbig
σi(x),
0o t h e r w i s e .(20.23)
2.Theβ-function
φi(x, y)=yαi(x)−1(1−y)βi(x)−1
/integraltext1
0sαi(x)−1(1−s)βi(x)−1ds, (20.24)
where
αi(x)=¯fi(x)/bracketleftbigg¯fi(x)(1−¯fi(x))
σi(x)−1/bracketrightbigg
(20.25)
and
βi(x)=( 1 −¯fi(x))/bracketleftbigg¯fi(x)(1−¯fi(x))
σi(x)−1/bracketrightbigg
. (20.26)
Similar expressions can be taken for ψs(x,·),s=1,…,S .
Now, in order to compute the local mean values of density, temperature and
mass fractions of species, we use (20.10) with Freplaced by the response functions

20.2. The Mixture Fraction/PDF Method 157
ˆρ,ˆθandˆYi,i=1,…,P, as deﬁned in (19.47)–(19.49). We have
¯ρ(x)=/integraldisplay1
0···/integraldisplay1
0/integraldisplay1
0···/integraldisplay1
0ˆρ(y1,…,y P,z1,…,z S,h(x))
φ1(x, y1)···φP(x, yP)ψ1(x, z1)···ψS(x, zS)dy1···dyPdz1···dzS,
(20.27)
¯θ(x)=/integraldisplay1
0···/integraldisplay1
0/integraldisplay1
0···/integraldisplay1
0ˆθ(y1,…,y P,z1,…,z S,h(x))
φ1(x, y1)···φP(x, yP)ψ1(x, z1)···ψS(x, zS)dy1···dyPdz1···dzS,
(20.28)
¯Yi(x)=/integraldisplay1
0···/integraldisplay1
0/integraldisplay1
0···/integraldisplay1
0ˆYi(y1,…,y P,z1,…,z S,h(x))
φ1(x, y1)···φP(x, yP)ψ1(x, z1)···ψS(x, zS)dy1···dyPdz1…d z S,
(20.29)
∀i=1,…,N, where his the solution of the turbulent version of the energy
equation (19.3), namely
div(ρhV)−div([ρD+ηT
δh]gradh)=l(D)·D+f. (20.30)
We recall that the response functions can be previously tabulated .
N o ww eg i v eas c h e m a t i ci d e ao nh o wt os o l v et h ew h o l ep r o b l e m .W ed i s t i n –
guish several steps:
1.Tabulate the mean values of density, temperature and mass fractions as func-
tions of the mean values and variances of the mixture fractions and the spe-
ciﬁc total enthalpy (the latter only in the case where (19.40) cannot be used) .
The data for this step are the following:
•The mass fractions of species at each inlet Γ p:
Yi,p,i=1,…,N, p =1,…,P. (20.31)
•Either the values of hpor the values of θpat inlets, if assumptions H1
toH4hold, so that (19.40) can be applied to determine the speciﬁc
enthalpy .
•The mean pressure ¯ π.
This task consists of the following ones:
(a) Tabulate the response functions (19.47)–(19.49):

158 Chapter 20. Turbulent Flow of Reacting Mixtures of Perfect Gases
i. Compute the mass fractions of elements at inlets by
Zk,p=N/summationdisplay
i=1akiYi,p
Mi,k=1,…,K, p =1,…,P, (20.32)
and then the right-hand side of (19.42), namely,
P/summationdisplay
p=1Zk,pfp+S/summationdisplay
s=1αksgs,k=1,…,K, (20.33)
for the chosen values of f1,…,f P−1andg1,…,g S(orf1,…,f P,
g1,…,g S−1).
ii. If (19.40) can be used and the values of temperature at inlets are
given, then compute hpby
hp=P/summationdisplay
i=1Yi,p/parenleftBigg
hi(θ0)+/integraldisplayθp
θ0ˆcπ,i(s)ds/parenrightBigg
,p=1,…,P. (20.34)
iii. If (19.40) can be used, compute the right-hand side of (19.43),
namely,
h=P/summationdisplay
p=1hpfp. (20.35)
iv. Compute θandn1,…,n Nas the solution of the equilibrium prob-
lem(19.41)–(19.44).
v. Compute the molecular mass of the mixture by
M=1
/summationtextN
i=1ni(20.36)
and then ρby
ρ=¯πM
Rθ. (20.37)
(b) Use the tables for the response functions calculated above to obtain
tables of mean values of density, temperature and mass fractions, against
the mean values and variances of mixture fractions, and the speciﬁc
enthalpy if (19.40) cannot be used. More precisely, for chosen values
of numbers ¯f1,…,¯fP−1,σ1,…,σ P−1and ¯g1,…,¯gS,ε1,…,ε S, build
the PDF functions φ1,…,φ P,ψ1,…,ψ Sand then use (20.27), (20.28),
(20.29), with (19.47), (19.48), (19.49).
2.Solve the turbulent ﬂow equations (which are compressible versions of
(14.13) –(14.16) ), to determine the mean velocity Vand pressure Π.

20.2. The Mixture Fraction/PDF Method 159
3.Compute the mean values of mixture fractions ¯f1,…,¯fP−1and¯g1,…,¯gS,
and their variances σ1,…,σ P−1andε1,…,ε Sby solving equations (20.11) –
(20.16) and(20.17) –(20.22) ,a n d hby(20.30) , for the case where we cannot
use(19.40) .
4.For each point x∈B0,determine the mean values of ρ,θandYi,i=1,…,N ,
at point xby using (20.27), (20.28) and (20.29), and the tables obtained in
step 1.
Remark 20.2.1.We notice that all of the equations to be solved in steps 2 and 3
need the density ﬁeld. However, this ﬁeld is calculated in step 4 making use of the
mean values and variances of the mixture fractions. This means that some iterativeprocess should be introduced in order to solve the whole coupled problem.

Appendix A
Vector and Tensor Algebra
A.1 Vector Space. Basis
Areal vector space is an algebraic structure consisting of a set Vendowed with
two operations. One of them is an internal operation denoted by + and satisfying
the following properties:
i)Associativity .
a+(b+c)=(a+b)+c∀a,b,c.
ii)Existence of a neutral element 0such that
a+0=0+a=a∀a.
iii)Existence of a symmetric element :f o re a c h athere exists bsuch that
a+b=b+a=0.
iv)Commutativity .
a+b=b+a∀a,b.
Thus, Vwith the + operation is a commutative group.
The second operation is an external one, the product by real numbers, satis-
fying the following properties:
v)λ(a+b)=λa+λb.
vi)λ(µa)=(λµ)a.
vii) (λ+µ)a=λa+µa.

162 Appendix A. Vector and Tensor Algebra
viii) 1a=a.
Elements of Vare called vectors .
A set of vectors {ei}N
i=1is said to be
•linearly independent if
N/summationdisplay
i=1λiei=0⇒λi=0∀i∈{1,…,N }, (A.1)
•asystem of generators if
∀v∈V∃λi,i=1,…,N, such that v=N/summationdisplay
i=1λiei, (A.2)
•abasisif it is both linearly independent and a system of generators.
IfB={e1, …, eN}is a basis of V,t h e ne a c h v∈Vcan be written as a
linear combination,
v=N/summationdisplay
i=1viei,
in a unique way. The vi∈R,i=1,…,N, are called coordinates ofvwith respect
to basis B.
Let us consider two bases in V,{ei}N
i=1and{Ei}N
i=1. In particular, vectors
Eican be written as linear combinations of vectors {ei}N
i=1,n a m e l y ,
El=N/summationdisplay
i=1Cliei. (A.3)
Now, let vbe any vector in V. Let us denote by ve
iandvE
i,i=1,…,N ,t h e
coordinates of vwith respect to basis {ei}N
i=1and{Ei}N
i=1, respectively. If we
denote by vethe column vector of the former and by vEthe column vector of the
latter, it is easy to see that
vE=[ C ]ve, (A.4)
where [C] is the matrix [C] ij=C ijwith C ijgiven in (A.3).

A.2. Inner Product 163
A.2 Inner Product
A mapping ϕ:V×V− → Ris called an inner product if the following properties
hold:
1)ϕis bilinear :
1.1)ϕ(λ1v1+λ2v2,w)=λ1ϕ(v1,w)+λ2ϕ(v2,w),
1.2)ϕ(v,λ1w1+λ2w2,w)=λ1ϕ(v,w1)+λ2ϕ(v,w2).
2)ϕis symmetric :
ϕ(v,w)=ϕ(w,v).
3)ϕis positive deﬁnite :
ϕ(v,v)>0∀v/negationslash=0.
Finite dimensional vector spaces with an inner product are called Euclidean
vector spaces . For them we can deﬁne a norm by
|v|=ϕ(v,v)1
2.
In what follows we write v·winstead of ϕ(v,w).
Two vectors vandware said to be orthogonal ifv·w=0 .
Ab a s i s B={e1, …,eN}is said to be orthogonal ifei·ej=0∀i/negationslash=jand
is said to be orthonormal if
ei·ej=δij=/braceleftbigg1i f i=j,
0i f i/negationslash=j.
IfBis orthonormal, then the i-th coordinate of a vector vis given by
vi=v·ei.
Moreover, the matrix to change coordinates with respect to two orthonormal basis
is orthogonal, namely,
[C][C]t=[ I ], (A.5)
where [I] is the identity matrix.

164 Appendix A. Vector and Tensor Algebra
A.3 Tensors
Atensor of order pis any p-linear mapping
T:←p→
V×…×V/mapsto→R.
The set of tensors of order pis denoted by Lp(V). When endowed with the two
algebraic operations:
•Sum
(f+g)(v)=f(v)+g(v)∀v∈V,
•Product by scalars
(λf)(v)=λf(v),∀λ∈R∀v∈V,
it is a vector space.
Given a basis B={e1, …,eN}ofV, we can associate to any f∈Lp(V)t h e
p-dimensional array
fi1,…,i p:=f(ei1, …,eip),1≤i1, …, i p≤N. (A.6)
These numbers are called coordinates of fwith respect to basis B.At e n s o ri s
called isotropic if its coordinates do not depend on the basis.
Forp=1 ,t h es p a c e L1(V) of ﬁrst order tensors is nothing but the dual space
ofVwhich is isomorphic to V.E a c h f∈L1(V) is identiﬁed to the unique v∈V
such that
f(w)=v·w∀w∈V.
For any given orthonormal basis, B, one can show that the coordinates of tensor
f, deﬁned by A.6 with p= 1, coincides with the coordinates of vector v.M o r e o v e r ,
the only isotropic ﬁrst order tensor is f≡0.
Forp=2 ,t h es p a c e L2(V) is isomorphic to Lin, the space of endomorphisms
ofV.At e n s o r f∈L2(V) is identiﬁed to the unique endomorphism S∈Linsuch
that
f(v,w)=Sw·v∀v,w∈V.
It is easy to see that, given a basis, the entries of the matrix associated with the
endomorphism Sare the coordinates of fgiven by (A.6) with p=2 .M o r e o v e r ,
one can show that fis isotropic if and only if S=QSQt∀Q∈Orth+.
Forp=4 ,t h es p a c e L4(V) is isomorphic to L(Lin,Lin ), the space of en-
domorphisms of Lin.I n d e e d ,e a c h f∈L4(V) can be identiﬁed to the unique
endomorphism of Lin,l, such that
f(v1,v2,v3,v4)=l(v3⊗v4)·(v1⊗v2)∀vi∈V,i=1,…,4.
In what follows, due to the above identiﬁcations, the elements of Linwill be
called (second order) tensors and the endomorphisms of Linwill be called fourth
order tensors.

A.3. Tensors 165
Second Order Tensors
Letaandbbe two vectors in V. The tensor
(a⊗b)v:=v·ba
is called the tensor product ofaandb.
Ifeis a unit vector, then
(e⊗e)v=(v·e)e
is the orthogonal projection of von the straight line generated by e.M o r e o v e r ,
(I−e⊗e)v=v−(v·e)e, (A.7)
is the projection of von the orthogonal plane to e.
IfB={e1, …, eN}, the set of tensors {ei⊗ej:1≤i, j≤N}is a basis
ofLin.M o r ep r e c i s e l y ,w eh a v e
S=N/summationdisplay
i,j=1Sijei⊗ej,
where
Sij=Sej·ei
are the coordinates ofSwith respect to basis B.
We notice that the coordinates of a tensor Scan be arranged as a matrix,
[S]=⎛
⎜⎝S11… S 1N
………
SN1… S NN⎞
⎟⎠.
For example, the matrix of coordinates of tensor a⊗bis the matrix product
⎛
⎜⎜⎜⎝a1
a2
…
aN⎞
⎟⎟⎟⎠(b1b2···bN), (A.8)
where a=N/summationdisplay
i=1aieiandb=N/summationdisplay
i=1biei.
Let us see how tensor coordinates change when we change the basis in vector
space V.L e tS∈Linbe a tensor. Let us denote by Se
ijandSE
ij,i,j=1,…,N ,

166 Appendix A. Vector and Tensor Algebra
its coordinates with respect to basis {ei}N
i=1and{Ei}N
i=1, respectively. Then it is
not diﬃcult to see that the following relation holds:
[SE] = [C][ Se][C]t. (A.9)
In vector space Linwe can introduce another internal operation which is the
mapping composition: given two tensors SandT,STis the tensor deﬁned by
(ST)v=S(Tv).
It is easy to see that the matrix of coordinates of STis the product of those
corresponding to SandT, i.e.,
[ST]=[S][T]. (A.10)
Thetranspose tensor of Sis the unique tensor Stsatisfying
Sa·b=a·Stb∀a,b∈V.
A tensor is called symmetric ifS=Standskew ifS=−St. The subspace of
symmetric tensors will be denoted by Symand that of skew tensors by Skw.
We can also deﬁne the inner product of tensors by
S·T=t r (StT),
where tr denotes the trace operator which is the unique linear operator from Lin
intoRsatisfying
tr(a⊗b)=a·b.
We have
tr(S)= t r⎛
⎝N/summationdisplay
i,j=1Sijei⊗ej⎞
⎠=N/summationdisplay
i,j=1Sijei·ej=N/summationdisplay
i,j=1Sijδij=N/summationdisplay
i=1Sii,
and hence
S·T=N/summationdisplay
i,j=1SijTij.
IfT∈SymandW∈Skwthen it easy to show that T·W=0 .

A.3. Tensors 167
Any tensor Sis the sum of one symmetric tensor Eand one skew tensor W.
More precisely, EandWare deﬁned by
E=1
2/parenleftbig
S+St/parenrightbig
, (A.11)
W=1
2/parenleftbig
S−St/parenrightbig
. (A.12)
Tensor Eis called the symmetric part ofSwhile Wis called the skew part ofS.
Furthermore, this decomposition is unique. More precisely,
Lin=Sym⊥/circleplusdisplay
Skw.
The following equalities can be easily proved:
(a⊗b)(c⊗d)=(b·c)a⊗d. (A.13)
R·(ST)=(StR)·T=(RTt)·S∀R,S,T ∈Lin. (A.14)
1·S=t r (S). (A.15)
S·(a⊗b)=a·Sb. (A.16)
(a⊗b)(c⊗d)=(a·c)(b·d). (A.17)
LetN=3a n d {e1,e2,e3}be a positively oriented orthonormal basis in V.
We deﬁne the cross product of vectors aandbby
a×b=3/summationdisplay
i,j,k=1Eijkajbkei.
where
Eijk=⎧
⎨
⎩1i f ijkis an even permutation of (1 2 3) ,
−1i f ijkis an odd permutation of (1 2 3) ,
0o t h e r w i s e .
One can prove the following equality:
a×(b×c)=(a·c)b−(a·b)c (A.18)
showing that the cross product is not associative .
There is an isomorphism between the vector space Vand the vector space of
skew tensors. It is deﬁned by
V− → Skw
w−→ W

168 Appendix A. Vector and Tensor Algebra
withWv:=w×v∀v∈V. Vector wis called the axial vector associated to W.
At e n s o r Sisinvertible if there exists another tensor S−1, called the inverse
ofS, such that
SS−1=S−1S=I. (A.19)
At e n s o r Qisorthogonal if it preserves the inner product, that is,
Qu·Qv=u·v∀u,v∈V. (A.20)
A necessary and suﬃcient condition for Qto be orthogonal is that
QQt=QtQ=I, (A.21)
which is equivalent to
Qt=Q−1. (A.22)
In particular, det Qmust be 1 or −1.
Arotation is an orthogonal tensor with positive determinant. The set of
rotations is a subspace of Linwhich will be denoted by Orth+.
One can show that a tensor Sis isotropic if and only if it commutes with any
rotation, that is,
S=QSQt∀Q∈Orth+.
Furthermore (see, for instance, [5]), this condition is satisﬁed if and only if Sis
spherical , i.e.,S=ωIfor some scalar ω.
At e n s o r Sis called deviatoric if it is symmetric and traceless. We denote
byDevthe subspace of deviatoric tensors. Any symmetric tensor S∈Symcan
be uniquely decomposed as the sum of a spherical tensor and a deviatoric tensor.
Indeed, the tensor
SD=S−1
Ntr(S)I
is a deviatoric tensor, and we can rewrite this equality in the form
S=1
Ntr(S)I+SD.
Moreover the subspace of spherical tensors is orthogonal to the subspace of devi-
atoric tensors because I·S=t r ( S) = 0 for any S∈Dev. Hence, we have the
orthogonal direct sum decomposition
Sym=Sph⊥/circleplusdisplay
Dev,
where Sphdenotes the subspace of spherical tensors.
At e n s o r Sispositive deﬁnite if
v·Sv>0∀v/negationslash=0. (A.23)

A.3. Tensors 169
Theprincipal invariants of a tensor Sa r et h et h r e en u m b e r s
ı1(S)=t r ( S), (A.24)
ı2(S)=1
2[( tr(S))2−tr(S2)], (A.25)
ı3(S)=d e t ( S)=1
6[( tr(S))3+2 (t r ( S))2−3t r (S2)t r (S)]. (A.26)
The set of principal invariants of Swill be denoted by IS.
Fourth Order Tensors
LetT, S∈Lin. The fourth order tensor T⊗S∈L(Lin,Lin ) deﬁned by
(T⊗S)R=S·RT∀R∈Lin,
is called the tensor product ofTandS.
For the basis BofV,t h es e t
{(ei⊗ej)⊗(ek⊗el),1≤i,j,k,l ≤N}
is a basis of the space of fourth order tensors. In fact, for any fourth order tensor
lwe have
l=N/summationdisplay
i,j,k,l=1lijkl(ei⊗ej)⊗(ek⊗el),
where
lijkl=l(ek⊗el)·(ei⊗ej) (A.27)
are the coordinates oflwith respect to basis B.
One can show that lis isotropic (i.e., its coordinates are independent of the
basis) if and only if
l(QSQt)=Ql(S)Qt∀Q∈Orth+∀S∈Lin. (A.28)
The following result has important applications in continuum mechanics:
Proposition 1.3.1. Letl∈L(Lin,Lin )be a fourth order tensor. Then lis isotropic
if and only if there exist three scalars α, βandγsuch that
l(S)=αtr(S)I+βS+γSt. (A.29)
Proof. Iflis of the form (A.29) then it is isotropic because (A.28) holds. Indeed,
letQ∈Orth+.W eh a v e
l(QSQt)=αtr(QSQt)I+βQSQt+γ(QSQt)t(A.30)
=Q/parenleftbig
αtr(QSQt)I+βS+γS/parenrightbig
Qt=Ql(S)Qt, (A.31)

170 Appendix A. Vector and Tensor Algebra
because tr( QSQt)= t r ( S)∀Q∈Orth. The proof of this follows immediately by
using (A.14):
tr(QSQt)=I·QSQt=Qt·SQt=QQt·S=I·S=t r (S).
The proof of the converse result is much more diﬃcult and will be omitted
here (see [4]). /square
Corollary 1.3.2. Letl∈L(Lin,Lin )be a fourth order tensor. Then, if lis isotropic,
there exist two scalars λandµsuch that
l(E)=λtr(E)I+2µE∀E∈Sym.
Proof. It is enough to take λ=αandµ=β+γ
2in (A.29). /square
Corollary 1.3.3. The isotropic fourth order tensor lgiven by (A.29) satisﬁes l(S)∈
Sym∀S∈Linif and only if β=γin which case
l(S)=l/parenleftbiggS+St
2/parenrightbigg
∀S∈Lin.
Proof. It follows from the fact that tr( S)=t r ( St)∀S∈Lin. /square
It is immediate to see that the coordinates of lgiven by (A.29) are, in any
orthonormal basis,
lijkm=αδijδkm+βδikδmj+γδimδjk.
Indeed, according to (A.27),
lijkl=l(ek⊗el)·(ei⊗ej)=(αtr(ek⊗em)I+βek⊗em+γem⊗ek)·ei⊗ej
=αek·emei·ej+βek·eiem·ej+γem·eiek·ej, (A.32)
and the result follows.
A.4 The Aﬃne Space
Anaﬃne space is a triple ( E,V,+) consisting of a set E, a vector space Vand a
mapping
E×V+−→ E
(p,v)−→ p+v
satisfying the following property:
For any p, q∈Ethere exists a unique v∈Vsuch that p+v=q.
This vector vis usually denoted by−→pqorq−p.

A . 4 . T h eA ﬃ n eS p a c e 171
IfVis a Euclidean vector space, then the corresponding aﬃne space is called
Euclidean aﬃne space .
In a Euclidean aﬃne space we can deﬁne a distance by
E×Ed−→R
(p,q)−→ d(p,q)=|−→pq|.
Thus ( E,d) is a metric space and hence a topological space.
In a Euclidean aﬃne space, a cartesian coordinate frame is a pair
(o,{ei,i=1,…,N })w h e r e ois any ﬁxed point in Ecalled the origin and{ei,i=
1,…,N }is an orthonormal basis of the Euclidean vector space V.
For any point p∈Eits coordinates with respect to the above cartesian frame
are the numbers pi,i=1,…,N, such that
−→op=N/summationdisplay
i=1piei.

Appendix B
Vector and Tensor Analysis
Aﬁeldis any mapping deﬁned in a subset of Eand valued in R,VorLin,i nw h i c h
case it is called a scalar, vector or tensor ﬁeld, respectively.
B.1 Diﬀerential Operators
LetWbe a normed vector space (in practice W=R,V,L i n ). A mapping
f:R⊂E →W d e ﬁ n e di na no p e ns e t RofEis said to be diﬀerentiable at point
x∈Rif there exists a linear mapping X:V− →W such that
f(x+h)−f(x)−X(h)=o(h)a sh→0,
which means
lim
h→0/bardblf(x+h)−f(x)−X(h)/bardbl
/bardblh/bardbl=0.
If f is diﬀerentiable at x, then linear mapping Xis called the diﬀerential of
fa tx. It is denoted by D f(x).
Particular cases:
1) Let us assume W=R, i.e., f=ϕis a scalar ﬁeld diﬀerentiable at x.T h e n
Dϕ(x):V− → R
is a linear mapping and the unique vector ∇ϕ(x) such that
Dϕ(x)a=∇ϕ(x)·a∀a∈V
is called the gradient of ϕatx.
Ifϕis diﬀerentiable at all points in Rwe can deﬁne the vector ﬁeld

174 Appendix B. Vector and Tensor Analysis
R⊂E − → V
x −→ ∇ ϕ(x),
which is called the gradient ofϕ.
2) Let us consider the case where W=V, i.e., f = uis a vector ﬁeld. Then
Du(x):V− →V ,t h a ti s ,D u(x)∈Lin.
Ifuis diﬀerentiable at each point in R, we can deﬁne the tensor ﬁeld
R⊂E − → Lin
x −→ Du(x),
which is called the gradient ofu.
In order to denote this diﬀerential operator we will use ∇uor grad u,
instead of D u,
Now we introduce other diﬀerential operators. Given a diﬀerentiable vector
ﬁeldu, the scalar ﬁeld
divu:R⊂E − → R
x −→ divu(x)= :t r ( ∇u(x))
is called the divergence ofu.
LetS:R⊂E− → Linbe a smooth tensor ﬁeld. The divergence ofSat
point x∈Ris deﬁned as the unique vector, div S(x), such that
divS(x)·a=d i v ( Sta)(x)∀a∈V.
Letu:R⊂E− →V be a smooth vector ﬁeld. The curlofuatxis the
unique vector, curlu (x), such that,
curlu (x)×a=(∇u(x)−∇ut(x))a∀a∈V.
LetS:R⊂E− → Linbe a smooth tensor ﬁeld. We deﬁne the curlofSat
point xas the unique tensor curlS(x) such that
curlS(x)a=curl(Sta)(x)∀a∈V.
For a scalar ﬁeld ϕ:R⊂E− → R,i t sLaplacian is the scalar ﬁeld
∆ϕ(x)=d i v ( ∇ϕ)(x).

B.2. Curves and Curvilinear Integrals 175
For a vector ﬁeld u:R⊂E− →V ,i t sLaplacian is the vector ﬁeld
∆u(x)=d i v ( ∇u)(x).
The following equalities hold:
∇(ϕv)= ϕ∇v+v⊗∇ϕ,
div(ϕv)= ϕdivv+v·∇ϕ,
∇(v·w)=( ∇w)tv+(∇v)tw,
div(v⊗w)= vdivw+(∇v)w,
div(Stv)= S·∇v+v·divS,
div(ϕS)= ϕdivS+S∇ϕ,
div(v×w)= w·curlv −v·curlw ,
div∇vt=∇(divv),
curl(ϕv)= ∇ϕ×v+ϕcurlv ,
curl(v×w)= ∇vw−wdivv+vdivw−∇wv,
divcurlv =0,
curl∇ϕ=0,
∆v=∇(divv)−curl(curlv ),
∆(∇ϕ)= ∇(∆ϕ),
div(∆v)=∆ ( d i v v),
curl(∆v)=∆ ( curlv ),
∆(ϕψ)= ϕ∆ψ+ψ∆ϕ+2∇ϕ·∇ψ.
B.2 Curves and Curvilinear Integrals
A (regular) curve c in Ris a smooth map
c:[ 0,1]−→ R
such that ˙ c(σ)/negationslash=0∀σ∈[0,1].
The curve is closed if c(0) = c(1).
Thelength of c is the number

176 Appendix B. Vector and Tensor Analysis
length(c) =/integraldisplay1
0|˙c(σ)|dσ.
Letv:R⊂E− →V be a continuous vector ﬁeld. The curvilinear integral of
valong c is the scalar deﬁned by
/integraldisplay
cv(x)·dx=/integraldisplay1
0v(c(σ))·˙c(σ)dσ.
Similarly, let S:R⊂E− → Linbe a continuous tensor ﬁeld. The curvilinear
integral ofSalong c is the vector deﬁned by
/integraldisplay
cS(x)dx=/integraldisplay1
0S(c(σ))˙c(σ)dσ.
Ifvis the gradient of a scalar ﬁeld ϕ,t h e n
/integraldisplay
cv(x)·dx=/integraldisplay
c∇ϕ(x)·dx=/integraldisplay1
0∇ϕ(c(σ))·˙c(σ)dσ
=/integraldisplay1
0d
dσ(ϕ◦c)(σ)dσ=ϕ(c(1)) −ϕ(c(0)). (B.1)
In particular/integraltext
c∇ϕ(x)·dx= 0 whenever c is closed.
A subset R⊂E issimply connected if any closed curve in Rcan be contin-
uously deformed to a point without leaving R.
Anopen region is any connected subset of E. The closure of an open region
will be called a closed region .W ed e s i g n a t eb y regular region any closed region
with smooth boundary ∂R.
LetRbe a closed region and Φ a ﬁeld deﬁned in R.W es a yt h a tΦi sC1(R)i f
it is continuously diﬀerentiable in the interior of R, and Φ and ∇Φ have continuous
extensions to all of R.
We end this section on vector calculus by recalling some fundamental results.
Theorem 2.2.1. Potential Theorem. Letvbe a smooth vector ﬁeld on an open or
closed simply connected region Rand assume that,
curlv =0.
Then there is a C2scalar ﬁeld ϕ:R− → Rsuch that,
v=∇φ.

B.3. Gauss’ and Green’s Formulas. Stokes’ Theorem 177
B.3 Gauss’ and Green’s Formulas. Stokes’ Theorem
Theorem 2.3.1. Divergence Theorem. LetRbe a bounded regular region and let
ϕ:R− → R,v:R− →V andS:R− → Linbe smooth ﬁelds. Then,
1./integraldisplay
∂RϕndAx=/integraldisplay
R∇ϕdVx,
2./integraldisplay
∂Rv⊗ndAx=/integraldisplay
R∇vdVx,
3./integraldisplay
∂Rv·ndAx=/integraldisplay
RdivvdVx,
4./integraldisplay
∂RSndAx=/integraldisplay
RdivSdVx.
Theorem 2.3.2. Green’s formulas. LetRbe a bounded regular region and let ϕ, ψ:
R− → R,v,w:R− →V andS:R− → Linbe smooth ﬁelds. Then,
1./integraldisplay
∂R∂ϕ
∂nψdAx=/integraldisplay
R∇ϕ·∇ψdVx+/integraldisplay
R∆ϕψdVx,
2./integraldisplay
∂RSn·vdAx=/integraldisplay
RS·∇vdVx+/integraldisplay
Rv·divSdVx,
3./integraldisplay
∂R(Sn)⊗vdAx=/integraldisplay
RS∇vtdVx+/integraldisplay
RdivS⊗vdVx,
4./integraldisplay
∂Rv·nψdAx=/integraldisplay
RdivvψdVx+/integraldisplay
Rv·∇ψdVx,
5./integraldisplay
∂R(w·n)vdAx=/integraldisplay
R(divw)vdVx+/integraldisplay
R(∇v)wdVx,
6./integraldisplay
∂R(v×w)·ndAx=/integraldisplay
Rw·curlv dVx−/integraldisplay
Rv·curlw dVx.
Theorem 2.3.3. Stokes’ Theorem. Letv(respectively, S) be a smooth vector (re-
spectively, tensor) ﬁeld on an open set R⊂E .L e tSb eas m o o t hs u r f a c ei n R
bounded by c, supposed to be a closed curve. At each point of Swe choose a unit
normal vector nconsistent with the direction of traversing cin that it causes a
right-handed screw to advance along n.T h e n ,
/integraldisplay
Scurlv ·ndAx=/integraldisplay
cv·dlx,
/integraldisplay
S(curlS)tndAx=/integraldisplay
cSdlx.

178 Appendix B. Vector and Tensor Analysis
B.4 Change of Variable in Integrals
Theorem 2.4.1. Letfbe a deformation of a body Bandϕ:f(B)→Ras m o o t h
scalar ﬁeld. Let c(resp. S)b eas m o o t hc u r v e( r e s p .s u r f a c e )i n BandPap a r to f
B. Then we have
1./integraldisplay
f(c)ϕtdlx=/integraldisplay
cϕ◦fFkdlp,
2./integraldisplay
f(S)ϕndAx=/integraldisplay
Sϕ◦fdetFF−tmdAp,
3./integraldisplay
f(c)ϕdlx=/integraldisplay
cϕ◦f/bardblFk/bardbldlp,
4./integraldisplay
f(S)ϕdAx=/integraldisplay
Sϕ◦fdetF/bardblF−tm/bardbldAp,
5./integraldisplay
f(P)ϕdVx=/integraldisplay
Pϕ◦fdetFdVp,
wherek(resp.t) denotes a unit tangent vector to the curve c(resp.f(c))a n dm
(resp.n) denotes a unit normal vector to surface S(resp.f(S)).
B.5 Transport Theorems
Theorem 2.5.1. LetXbe a motion of a body B.L e t c,SandPbe a curve, a
surface and a part of B.L e tϕ:T→Randw:T→ V be a scalar ﬁeld and a
vector ﬁeld, respectively. Then we have
1.d
dt/integraldisplay
ctϕtdlx=/integraldisplay
ct(˙ϕt+ϕLt)d lx,
2.d
dt/integraldisplay
StϕndAx=/integraldisplay
St/parenleftbig
˙ϕn+ϕdivvn−ϕLtn/parenrightbig
dAx,
3.d
dt/integraldisplay
PtϕdVx=/integraldisplay
Pt(˙ϕ+ϕdivv)d V x=/integraldisplay
Pt(ϕ/prime+ div( ϕv)) dV x
=/integraldisplay
Ptϕ/primedVx+/integraldisplay
∂Ptϕv·ndAx,
4.d
dt/integraldisplay
ctw⊗tdlx=/integraldisplay
ct(˙w⊗t+w⊗Lt)d lx,
5.d
dt/integraldisplay
ctw·tdlx=/integraldisplay
ct/parenleftbig˙w·t+Ltw·t/parenrightbig
dlx,
6.d
dt/integraldisplay
Stw⊗ndAx=/integraldisplay
St/parenleftbig˙w⊗n+d i vvw⊗n−w⊗Ltn/parenrightbig
dAx,

B.6. Localization Theorem 179
7.d
dt/integraldisplay
Stw·ndAx=/integraldisplay
St(˙w·n+d i vvw·n−∇vw·n)d A x
=/integraldisplay
St(w/prime·n+d i vwv·n+c u r l ( w×v)·n)d A x,
8.d
dt/integraldisplay
PtwdVx=/integraldisplay
Pt(˙w+d i vvw)d V x=/integraldisplay
Pt(w/prime+d i v (w⊗v)) dV x
=/integraldisplay
Ptw/primedVx+/integraldisplay
∂Ptwv·ndAx,
wheretdenotes a unit tangent vector to ctandnis a unit normal vector to St.
B.6 Localization Theorem
Theorem 2.6.1. LetΦbe a continuous scalar or vector valued mapping deﬁned in
an open set R⊂E .I f /integraldisplay
BΦd V x≥0
for every closed ball B⊂R,t h e n Φ≥0inR.H e n c e ,i f
/integraldisplay
BΦd V x=0
for every closed ball B⊂R,t h e n Φ≡0inR.
B.7 Diﬀerential Operators in Coordinates
B.7.1 Cartesian Coordinates
Deﬁnition
f:Ω=R3→R3(B.2)
f(u1,u2,u3)=(u1,u2,u3). (B.3)
Basis
•Contravariant
g1=e1,g2=e2,g3=e3. (B.4)
•Covariant
g1=e1,g2=e2,g3=e3. (B.5)
•Physical
e1,e2,e3. (B.6)

180 Appendix B. Vector and Tensor Analysis
Christoﬀel symbols of the second kind
Γ1
jk=⎛
⎝000
000
000⎞
⎠, (B.7)
Γ2
jk=⎛
⎝000
000
000⎞
⎠, (B.8)
Γ3
jk=⎛
⎝000
000
000⎞
⎠. (B.9)
Diﬀerential Operators in the Physical Basis
•Gradient of scalar ﬁeld
∇φ=∂φ
∂x1e1+∂φ
∂x2e2+∂φ
∂x3e3. (B.10)
•Gradient of a vector ﬁeld
∇w=∂w1
∂x1e1⊗e1+∂w1
∂x2e1⊗e2+∂w1
∂x3e1⊗e3
+∂w2
∂x1e2⊗e1+∂w2
∂x2e2⊗e2+∂w2
∂x3e2⊗e3
+∂w3
∂x1e3⊗e1+∂w3
∂x2e3⊗e2+∂w3
∂x3e3⊗e3. (B.11)
•Convective term
(∇w)w=/braceleftbigg
w1∂w1
∂x1+w2∂w1
∂x2+w3∂w1
∂x3/bracerightbigg
e1
+/braceleftbigg
w1∂w2
∂x1+w2∂w2
∂x2+w3∂w2
∂x3/bracerightbigg
e2
+/braceleftbigg
w1∂w3
∂x1+w2∂w3
∂x2+w3∂w3
∂x3/bracerightbigg
e3. (B.12)
•Curl of a vector ﬁeld
curlw=/parenleftbigg∂w3
∂x2−∂w2
∂x3/parenrightbigg
e1
+/parenleftbigg∂w1
∂x3−∂w3
∂x1/parenrightbigg
e2
+/parenleftbigg∂w2
∂x1−∂w1
∂x2/parenrightbigg
e3. (B.13)

B.7. Diﬀerential Operators in Coordinates 181
•Divergence of a vector ﬁeld
divw=∂w1
∂x1+∂w2
∂x2+∂w3
∂x3. (B.14)
•Divergence of a tensor ﬁeld
divS=/bracketleftbigg∂S11
∂x1+∂S12
∂x2+∂S13
∂x3/bracketrightbigg
e1
+/bracketleftbigg∂S21
∂x1+∂S22
∂x2+∂S23
∂x3/bracketrightbigg
e2
+/bracketleftbigg∂S31
∂x1+∂S32
∂x2+∂S33
∂x3/bracketrightbigg
e3. (B.15)
•Laplacian of a scalar ﬁeld
∆φ=∂2φ
∂x2
1+∂2φ
∂x2
2+∂2φ
∂x2
3. (B.16)
•Laplacian of a vector ﬁeld
∆w=∆w1e1+∆w2e2+∆w3e3. (B.17)
Measure elements for integration
•Line
dx1,dx2,dx3. (B.18)
•Surface
dx2δx3,dx1dx3,dx1dx2. (B.19)
•Volume
dx1dx2dx3. (B.20)

182 Appendix B. Vector and Tensor Analysis
B.7.2 Cylindrical Coordinates
Deﬁnition
f:Ω=( 0 ,∞)×(0,2π)×(−∞,∞)→R3(B.21)
f(r, θ, z)=(rcosθ,rsinθ,z). (B.22)
Basis
•Contravariant ⎧
⎨
⎩g1(r, θ, z)=c o s θe1+s i nθe2,
g2(r, θ, z)=−rsinθe1+rcosθe2,
g3(r, θ, z)=e3.(B.23)
•Covariant ⎧
⎨
⎩g1(r, θ, z)=c o s θe1+s i nθe2,
g2(r, θ, z)=−1
rsinθe1+1
rcosθe2,
g3(r, θ, z)=e3.(B.24)
•Physical⎧
⎨
⎩er(r, θ, z)=c o s θe1+s i nθe2,
eθ(r, θ, z)=−sinθe1+c o s θe2,
ez(r, θ, z)=e3.(B.25)
Christoﬀel symbols of the second kind
Γ1
jk=⎛
⎝000
0−r0
000⎞
⎠, (B.26)
Γ2
jk=⎛
⎝01
r0
1
r00
000⎞
⎠, (B.27)
Γ3
jk=⎛
⎝000
000
000⎞
⎠. (B.28)
Diﬀerential operators in the physical basis
•Gradient of scalar ﬁeld
∇φ=∂φ
∂rer+1
r∂φ
∂θeθ+∂φ
∂zez. (B.29)

B.7. Diﬀerential Operators in Coordinates 183
•Gradient of a vector ﬁeld
∇w=∂wr
∂rer⊗er+/bracketleftbigg1
r∂wr
∂θ−1
rwθ/bracketrightbigg
er⊗eθ+∂wr
∂zer⊗ez
+∂wθ
∂reθ⊗er+/bracketleftbigg1
r∂wθ
∂θ+1
rwr/bracketrightbigg
eθ⊗eθ+∂wθ
∂zeθ⊗ez
+∂wz
∂rez⊗er+1
r∂wz
∂θez⊗eθ+∂wz
∂zez⊗ez.(B.30)
•Convective term
∇ww=/braceleftbigg
wr∂wr
∂r+1
rwθ∂wr
∂θ+wz∂wr
∂z−(wθ)2
r/bracerightbigg
er
+/braceleftbigg
wr∂wθ
∂r+1
rwθ∂wθ
∂θ+wz∂wθ
∂z+wrwθ
r/bracerightbigg
eθ
+/braceleftbigg
wr∂wz
∂r+1
rwθ∂wz
∂θ+wz∂wz
∂z/bracerightbigg
ez. (B.31)
•Curl of a vector ﬁeld
curlw=/parenleftbigg1
r∂wz
∂θ−∂wθ
∂z/parenrightbigg
er
+/parenleftbigg
−∂wz
∂r+∂wr
∂z/parenrightbigg
eθ
+/parenleftbigg1
r∂
∂r(rwθ)−1
r∂wr
∂θ/parenrightbigg
ez. (B.32)
•Divergence of a vector ﬁeld
divw=1
r∂
∂r(rwr)+1
r∂wθ
∂θ+∂wz
∂z. (B.33)
•Divergence of a tensor ﬁeld
divS=1
r/bracketleftbigg∂
∂r(rSrr)+∂Srθ
∂θ+r∂Srz
∂z−1
rSθθ/bracketrightbigg
er
+/bracketleftbigg∂Sθr
∂r+1
r∂Sθθ
∂θ+∂Sθz
∂z+1
rSrθ+1
rSθr/bracketrightbigg
eθ
+1
r/bracketleftbigg∂
∂r(rSzr)+∂Szθ
∂θ+r∂Szz
∂z/bracketrightbigg
ez. (B.34)

184 Appendix B. Vector and Tensor Analysis
•Laplacian of a scalar ﬁeld
∆φ=1
r/bracketleftbigg∂
∂r/parenleftbigg
r∂φ
∂r/parenrightbigg
+∂
∂θ/parenleftbigg1
r∂φ
∂θ/parenrightbigg
+∂
∂z/parenleftbigg
r∂φ
∂z/parenrightbigg/bracketrightbigg
=1
r∂
∂r/parenleftbigg
r∂φ
∂r/parenrightbigg
+1
r2∂2φ
∂θ2+∂2φ
∂z2. (B.35)
•Laplacian of a vector ﬁeld
∆w=/bracketleftbigg
∆wr−1
r2wr−2
r2∂wθ
∂θ/bracketrightbigg
er+/bracketleftbigg
∆wθ−1
r2wθ+2
r2∂wr
∂θ/bracketrightbigg
eθ+∆wzez.
(B.36)
Measure elements for integration
•Line
dr, rdθ,dz. (B.37)
•Surface
rdθdz,drdz, rdrdθ. (B.38)
•Volume
rdrdθdz. (B.39)
B.7.3 Spherical Coordinates
Deﬁnition
f:Ω=( 0 ,∞)×(0,π)×(0,2π)→R3,
f(r, θ, ϕ)=(rsinθcosϕ, rsinθsinϕ, rcosθ).(B.40)
Basis
•Contravariant
⎧
⎨
⎩g1(r, ϕ, θ)=s i n θcosϕe1+s i nθsinϕe2+c o s θe3,
g2(r, ϕ, θ)=rcosθcosϕe1+rcosθsinϕe2−rsinθe3,
g3(r, ϕ, θ)=−rsinθsinϕe1+rsinθcosϕe2.(B.41)
•Covariant
⎧
⎨
⎩g1(r, ϕ, θ)=s i n θcosϕe1+s i nθsinϕe2+c o s θe3,
g2(r, ϕ, θ)=1
rcosθcosϕe1+1
rcosθsinϕe2−1
rsinθe3,
g3(r, ϕ, θ)=−1
rsinθsinϕe1+1
rsinθcosϕe2.(B.42)

B.7. Diﬀerential Operators in Coordinates 185
•Physical
⎧
⎨
⎩er(r, ϕ, θ)=s i n θcosϕe1+s i nθsinϕe2+c o s θe3,
eθ(r, ϕ, θ)=c o s θcosϕe1+c o s θsinϕe2−sinθe3,
eϕ(r, ϕ, θ)=−sinϕe1+c o s ϕe2.(B.43)
Christoﬀel symbols of the second kind
Γ1
jk=⎛
⎝00 0
0−r 0
00 −rsin2θ⎞
⎠, (B.44)
Γ2
jk=⎛
⎝01
r0
1
r00
00 −1
2sin 2θ⎞
⎠, (B.45)
Γ3
jk=⎛
⎝001
r
00c o t θ
1
rcotθ0⎞
⎠. (B.46)
Diﬀerential operators
•Gradient of a scalar ﬁeld
∇φ=∂φ
∂rg1+∂φ
∂θg2+∂φ
∂ϕg3=∂φ
∂rer+1
r∂φ
∂θeθ+1
rsinθ∂φ
∂ϕeϕ.(B.47)
•Gradient of a vector ﬁeld
∇w=∂wr
∂rer⊗er+/bracketleftbigg1
r∂wr
∂θ−1
rwθ/bracketrightbigg
er⊗eθ+/bracketleftbigg1
rsinθ∂wr
∂ϕ−1
rwϕ/bracketrightbigg
er⊗eϕ
+∂wθ
∂reθ⊗er+/bracketleftbigg1
r∂wθ
∂θ+1
rwr/bracketrightbigg
eθ⊗eθ+/bracketleftbigg1
rsinθ∂wθ
∂ϕ−1
rcotθwϕ/bracketrightbigg
eθ⊗eϕ
+∂wϕ
∂reϕ⊗er+1
r∂wϕ
∂θeϕ⊗eθ+/bracketleftbigg1
rsinθ∂wϕ
∂ϕ+1
rwr+1
rcotθwθ/bracketrightbigg
eϕ⊗eϕ.
(B.48)
•Convective term
(∇w)w=/braceleftBigg
wr∂wr
∂r+1
rwθ∂wr
∂θ+wϕ
rsinθ∂wr
∂ϕ−w2
θ+w2
ϕ
r/bracerightBigg
er
+/braceleftBigg
wr∂wθ
∂r+1
rwθ∂wθ
∂θ+wϕ
rsinθ∂wθ
∂ϕ+wrwθ
r−w2
ϕcotθ
r/bracerightBigg
eθ
+/braceleftbigg
wr∂wϕ
∂r+1
rwθ∂wϕ
∂θ+wϕ
rsinθ∂wϕ
∂ϕ+wrwϕ
r+wθwϕcotθ
r/bracerightbigg
eϕ.
(B.49)

186 Appendix B. Vector and Tensor Analysis
•Curl of a vector ﬁeld
curlw=1
rsinθ/parenleftbigg∂
∂θ(wϕsinθ)−∂wθ
∂ϕ/parenrightbigg
er
+/parenleftbigg1
rsinθ∂wr
∂ϕ−1
r∂
∂r(rwϕ)/parenrightbigg
eθ
+1
r/parenleftbigg∂
∂r(rwθ)−∂wr
∂θ/parenrightbigg
eϕ. (B.50)
•Divergence of a vector ﬁeld
divw=1
r2∂
∂r(r2wr)+1
rsinθ∂
∂θ(sinθwθ)+1
rsinθ∂wϕ
∂ϕ. (B.51)
•Divergence of a tensor ﬁeld
divS
=/bracketleftbigg∂Srr
∂r+1
r∂Srθ
∂θ+1
rsinθ∂Sϕr
∂ϕ+1
r(2Srr−Sθθ−Sϕϕ+Srθcotθ)/bracketrightbigg
er
+/bracketleftbigg∂Srθ
∂r+1
r∂Sθθ
∂θ+1
rsinθ∂Sθϕ
∂ϕ+1
r((Sθθ−Sϕϕ)cotθ+3Srθ)/bracketrightbigg
eθ
+/bracketleftbigg∂Sϕr
∂r+1
r∂Sθϕ
∂θ+1
rsinθ∂Sϕϕ
∂ϕ/bracketrightbigg
eϕ. (B.52)
•Laplacian of a scalar ﬁeld
∆ϕ=1
r2sinθ/bracketleftbigg∂
∂r/parenleftbigg
r2sinθ∂φ
∂r/parenrightbigg
+∂
∂θ/parenleftbigg
sinθ∂φ
∂θ/parenrightbigg
+∂
∂ϕ/parenleftbigg1
sinθ∂φ
∂ϕ/parenrightbigg/bracketrightbigg
=1
r2∂
∂r(r2∂φ
∂r)+1
r2sinθ∂
∂θ(sinθ∂φ
∂θ)+1
r2sin2θ∂2φ
∂ϕ2. (B.53)
•Laplacian of a vector ﬁeld
∆w=/bracketleftbigg
∆wr−2
r2wr−2
r2∂wθ
∂θ−2
r2wθcotθ−2
r2sinθ∂wϕ
∂ϕ/bracketrightbigg
er
+/bracketleftbigg
∆wθ+2
r2∂wr
∂θ−1
r2sin2θwθ−2
r2cosθ
sin2θ∂wϕ
∂ϕ/bracketrightbigg
eθ
+/bracketleftbigg
∆wϕ−1
r2sin2θwϕ+2
r2sinθ∂wr
∂ϕ+2
r2cosθ
sin2θ∂wθ
∂ϕ/bracketrightbigg
eϕ(B.54)
where the Laplacian operators in the right-hand side are given by (B.53).

B.7. Diﬀerential Operators in Coordinates 187
Measure elements for integration
•Line
dr, rdθ, rsinθdϕ. (B.55)
•Surface
r2sinθdθdϕ, rsinθdrdϕ, rdrdθ. (B.56)
•Volume
r2sinθdrdθdϕ. (B.57)

Appendix C
Some Equations of Continuum
Mechanics in Curvilinear
Coordinates
In this section we write some partial diﬀerential equations of continuum mechanics
in cylindrical and spherical coordinates.
C.1 Mass Conservation Equation
It is given by
∂ρ
∂t+d i v ( ρv)=0, (C.1)
where ρisdensity andvisvelocity .
•Cartesian coordinates
∂ρ
∂t+∂(ρv1)
∂x1+∂(ρv2)
∂x2+∂(ρv3)
∂x3=0. (C.2)
•Cylindrical coordinates
∂ρ
∂t+1
r∂
∂r(rρvr)+1
r∂(ρvθ)
∂θ+∂(ρvz)
∂z=0. (C.3)
•Spherical coordinates
∂ρ
∂t+1
r2∂
∂r(r2ρvr)+1
rsinθ∂
∂θ(sinθρvθ)+1
rsinθ∂(ρvϕ)
∂ϕ=0.(C.4)

190 Appendix C. Some Equations in Curvilinear Coordinates
C.2 Motion Equation
The general motion equation has the form
ρ(∂v
∂t+ (grad v)v)=d i v T+b, (C.5)
where Tis the Cauchy stress tensor andbis the body force .
•Cartesian coordinates
ρ/braceleftbigg∂v1
∂t+v1∂v1
∂x1+v2∂v1
∂x2+v3∂v1
∂x3/bracerightbigg
=/bracketleftbigg∂T11
∂x1+∂T12
∂x2+∂T13
∂x3/bracketrightbigg
+b1, (C.6)
ρ/braceleftbigg∂v2
∂t+v1∂v2
∂x1+v2∂v2
∂x2+v3∂v2
∂x3/bracerightbigg
=/bracketleftbigg∂T21
∂x1+∂T22
∂x2+∂T23
∂x3/bracketrightbigg
+b2, (C.7)
ρ/braceleftbigg∂v3
∂t+v1∂v3
∂x1+v2∂v3
∂x2+v3∂v3
∂x3/bracerightbigg
=/bracketleftbigg∂T31
∂x1+∂T32
∂x2+∂T33
∂x3/bracketrightbigg
+b3. (C.8)
•Cylindrical coordinates
ρ/braceleftbigg∂vr
∂t+vr∂vr
∂r+vθ
r∂vr
∂θ+vz∂vr
∂z−v2
θ
r/bracerightbigg
=1
r/bracketleftbigg∂
∂r(rTrr)+∂Trθ
∂θ+r∂Trz
∂z−1
rTθθ/bracketrightbigg
+br, (C.9)
ρ/braceleftbigg∂vθ
∂t+vr∂vθ
∂r+vθ
r∂vθ
∂θ+vz∂vθ
∂z+vrvθ
r/bracerightbigg
=/bracketleftbigg∂Tθr
∂r+1
r∂Tθθ
∂θ+∂Tθz
∂z+1
rTrθ+1
rTθr/bracketrightbigg
+bθ,(C.10)
ρ/braceleftbigg∂vz
∂t+vr∂vz
∂r+vθ
r∂vz
∂vθ+vz∂vz
∂z/bracerightbigg
=1
r/bracketleftbigg∂
∂r(rTzr)+∂Tzθ
∂θ+r∂Tzz
∂z/bracketrightbigg
+bz. (C.11)

C.3. Constitutive Law for Newtonian Viscous Fluids in Cooordinates 191
•Spherical coordinates
ρ/braceleftbigg∂vr
∂t+vr∂vr
∂r+1
rvθ∂vr
∂θ+vϕ
rsinθ∂vr
∂ϕ−v2
θ+v2
ϕ
r/bracerightBigg
=/bracketleftbigg∂Trr
∂r+1
r∂Trθ
∂θ+1
rsinθ∂Tϕr
∂ϕ+1
r(2Trr−Tθθ−Tϕϕ+Trθcotθ)/bracketrightbigg
+br,
(C.12)
ρ/braceleftbigg∂vθ
∂t+vr∂vθ
∂r+1
rvθ∂vθ
∂θ+vϕ
rsinθ∂vθ
∂ϕ+vrvθ
r−v2
ϕcotθ
r/bracerightBigg
=/bracketleftbigg∂Trθ
∂r+1
r∂Tθθ
∂θ+1
rsinθ∂Tθϕ
∂ϕ+1
r((Tθθ−Tϕϕ)cotθ+3Trθ)/bracketrightbigg
+bθ,
(C.13)
ρ/braceleftbigg∂vϕ
∂t+vr∂vϕ
∂r+1
rvθ∂vϕ
∂θ+vϕ
rsinθ∂vϕ
∂ϕ+vrvϕ
r+vθvϕcotθ
r/bracerightbigg
=/bracketleftbigg∂Tϕr
∂r+1
r∂Tθϕ
∂θ+1
rsinθ∂Tϕϕ
∂ϕ/bracketrightbigg
+bϕ.
(C.14)
C.3 Constitutive Law for Newtonian Viscous Fluids in
Cooordinates
In terms of the stretching D,i ti sg i v e nb y
T=2ηD+ξdivvI, (C.15)
with
D=1
2(gradv+g r a d vt). (C.16)

192 Appendix C. Some Equations in Curvilinear Coordinates
•Cartesian coordinates
T11=ξdivv+2η∂v1
∂x1, (C.17)
T22=ξdivv+2η∂v2
∂x2, (C.18)
T33=ξdivv+2η∂v3
∂x3, (C.19)
T12=T21=η/parenleftbigg∂v1
∂x2+∂v2
∂x1/parenrightbigg
, (C.20)
T13=T31=η/parenleftbigg∂v1
∂x3+∂v3
∂x1/parenrightbigg
, (C.21)
T23=T32=η/parenleftbigg∂v2
∂x3+∂v3
∂x2/parenrightbigg
, (C.22)
with
divv=∂v1
∂x1+∂v2
∂x2+∂v3
∂x3. (C.23)
•Cylindrical coordinates
Trr=ξdivv+2η∂vr
∂r, (C.24)
Tθθ=ξdivv+2η/parenleftbigg1
r∂vθ
∂θ+vr
r/parenrightbigg
, (C.25)
Tzz=ξdivv+2η∂vz
∂z, (C.26)
Trθ=Tθr=η/bracketleftbigg1
r∂vr
∂θ+r∂
∂r/parenleftBigvθ
r/parenrightBig/bracketrightbigg
, (C.27)
Trz=Tzr=η/parenleftbigg∂vr
∂z+∂vz
∂r/parenrightbigg
, (C.28)
Tθz=Tzθ=η/parenleftbigg∂vθ
∂z+1
r∂vz
∂θ/parenrightbigg
, (C.29)
with
divv=1
r∂
∂r(rvr)+1
r∂vθ
∂θ+∂vz
∂z. (C.30)

C.3. Constitutive Law for Newtonian Viscous Fluids in Cooordinates 193
•Spherical coordinates
Trr=ξdivv+2η∂vr
∂r, (C.31)
Tθθ=ξdivv+2η/parenleftbigg1
r∂vθ
∂θ+vr
r/parenrightbigg
, (C.32)
Tϕϕ=ξdivv+2η/parenleftbigg1
rsinθ∂vϕ
∂ϕ+vr
r+vθcotθ
r/parenrightbigg
, (C.33)
Trθ=Tθr=η/bracketleftbigg1
r∂vr
∂θ+r∂
∂r/parenleftBigvθ
r/parenrightBig/bracketrightbigg
, (C.34)
Trϕ=Tϕr=η/bracketleftbigg
r∂
∂r/parenleftBigvϕ
r/parenrightBig
+1
rsinθ∂vr
∂ϕ/bracketrightbigg
, (C.35)
Tθϕ=Tϕθ=η/bracketleftbigg1
rsinθ∂vθ
∂ϕ+sinθ
r∂
∂θ/parenleftBigvϕ
sinθ/parenrightBig/bracketrightbigg
, (C.36)
with
divv=1
r2∂
∂r(r2vr)+1
rsinθ∂
∂θ(sinθvθ)+1
rsinθ∂vϕ
∂ϕ. (C.37)

Appendix D
Arbitrary Lagrangian-Eulerian
(ALE) Formulations of theConservation Equations
In some ﬂuid-structure interaction problems, the domain where the motion of the
ﬂuid is taking place changes with time because the structure interacting withthe ﬂuid also moves. This is the case, for instance, of aeroelasticity problems.
Moreover, there are free boundary ﬂows for which the sets B
t, i.e., the positions
occupied by the body, change with time and are a priori unknown. In these twosituations, numerical methods based on so-called Arbitrary Lagrangian-Eulerian
(ALE) formulations can be very useful.
The main goal of this chapter is to write several ALE formulations of the
conservation laws.
D.1 ALE Conﬁguration
The situation is summarized in Figure D.1. Recall that Bis the reference or La-
grangian conﬁguration, Btis the Eulerian conﬁguration and Ωdenotes the ALE
conﬁguration. Y(·,t) is a motion transforming ΩintoBtat each time tand
Z(·,t)=[Y(·,t)]−1◦X(·,t).
Hence,
Y(Z(p,t),t)=X(p,t). (D.1)
Fields can be written in terms of Eulerian (spatial), Lagrangian (material) or ALE
conﬁguration. More precisely, if Φ is an Eulerian ﬁeld, we can deﬁne its Lagrangian

196 Appendix D. ALE Formulations of the Conservation Equations
Z(, ) t Y(, ) tX(, ) t
BBt
/c87
Figure D.1: Lagrangian, Eulerian and ALE conﬁgurations
and ALE descriptions by
Φm(p,t)=Φ (X(p,t),t),
Φa(z,t)=Φ (Y(z,t),t),
respectively. We recall that, in particular, vm=˙X.
Similarly, for a Lagrangian ﬁeld Ψ we can deﬁne its Eulerian and ALE coun-
terparts by
Ψs(x, t)=Ψ ( P(x, t),t),
Ψa(z,t)=Ψ ( PZ(z,t),t),
wherePis the reference map for motion X, i.e.,P(·,t)=[X(·,t)]−1,a n dPZ(z,t)=
[Z(·,t)]−1. Hence,
PZ(z,t)=P(Y(z,t),t).
In particular v=˙Xs.
We will use the notations grada,divaand∂
∂t=/primefor the corresponding dif-
ferential operators applied to ﬁelds in ALE conﬁguration. Let us denote wthe
spatial ﬁeld deﬁned by wa=∂Y
∂t. Field wais called the ALE velocity . We also
introduce d:=/parenleftbig∂Z
∂t/parenrightbig
a.
By taking the time derivative of (D.1) we obtain
∂X
∂t(p,t)=∂Y
∂t(Z(p,t),t) + gradaY(Z(p,t),t)∂Z
∂t(p,t). (D.2)
Letp=PZ(z,t). Then z=Z(p,t) and (D.2) yields
va(z,t)=wa(z,t)+G(z,t)d(z,t), (D.3)

D.2. Conservative ALE Form of Conservation Equations 197
where Gdenotes the tensor ﬁeld gradaY.
For a part PofB, let us denote by Pa
tthe subset of Ωgiven by
Pa
t=Z(P,t).
Then, from the Reynolds Transport Theorem we get
Theorem 4.1.1. LetϕandΦbe an ALE scalar ﬁeld and an ALE vector ﬁeld,
respectively. Then
d
dt/integraldisplay
Pa
tϕdVz=/integraldisplay
Pa
tϕ/primedVz+/integraldisplay
Pa
tdiva(ϕd)d V z, (D.4)
d
dt/integraldisplay
Pa
tΦdVz=/integraldisplay
Pa
tΦ/primedVz+/integraldisplay
Pa
tdiva(Φ⊗d)d V z. (D.5)
Corollary 4.1.2. We have
1. d
dt/integraldisplay
Pa
tϕdVz=/integraldisplay
Pa
tϕ/primedVz+/integraldisplay
Pa
tdiva/bracketleftbig
ϕG−1(va−wa)/bracketrightbig
dVz,(D.6)
2.d
dt/integraldisplay
Pa
tΦdVz=/integraldisplay
Pa
tΦ/primedVz+/integraldisplay
Pa
tdiva/braceleftbig
[Φ⊗(va−wa)]G−t/bracerightbig
dVz.(D.7)
Proof. It is an immediate consequence of (D.3). Indeed, we get d=G−1(va−wa)
and then
diva(Φ⊗d)=d i v a/bracketleftbig
Φ⊗G−1(va−wa)/bracketrightbig
=d i v a/braceleftbig
[Φ⊗(va−wa)]G−t/bracerightbig
. (D.8)
/square
D.2 Conservative ALE Form of Conservation Equations
Now we are in a position to obtain the ALE formulations of the conservation equa-
tions in conservative form . For this purpose we recall the conservation principles
in integral form and Eulerian coordinates, namely,
•Massd
dt/integraldisplay
PtρdVx=0. (D.9)
•Linear Momentum
d
dt/integraldisplay
PtρvdVx=/integraldisplay
∂PtTndAx+/integraldisplay
PtbdVx. (D.10)

198 Appendix D. ALE Formulations of the Conservation Equations
•Energy
d
dt/integraldisplay
PtρEdVx=/integraldisplay
∂PtTn·vdAx+/integraldisplay
Ptb·vdVx−/integraldisplay
∂Ptq·ndAx+/integraldisplay
PtfdVx.
(D.11)
The corresponding ALE formulations are obtained by making, ﬁrst, the change of
variable x=Y(z,t) in the integrals. Let J(z,t)=d e t G(z,t)a n d Pt=Y(Pa
t,t).
We have
•Massd
dt/integraldisplay
Pa
tρaJdVz=0. (D.12)
By using (D.11) for ϕ=ρaJwe obtain
/integraldisplay
Pa
t/braceleftbig
(ρaJ)/prime+d i v a/bracketleftbig
ρaJG−1(va−wa)/bracketrightbig/bracerightbig
dVz=0, (D.13)
and hence, from the Localization Theorem,
(ρaJ)/prime+d i v a/bracketleftbig
ρaJG−1(va−wa)/bracketrightbig
=0. (D.14)
•Linear momentum
d
dt/integraldisplay
Pa
tρaJdVz=/integraldisplay
∂Pa
tJTaG−tmdAz+/integraldisplay
Pa
tJbadVz. (D.15)
We transform this equality by using (D.7) for Φ = ρaJvaandGauss’ Theo-
rem.W eo b t a i n
/integraldisplay
Pa
t/bracketleftbig
(ρavaJ)/prime+d i v a/braceleftbig
ρaJ[va⊗(va−wa)]G−t/bracketrightbig/bracerightbig
dVz (D.16)
=/integraldisplay
Pa
tdiva/bracketleftbig
JTaG−t/bracketrightbig
dVz+/integraldisplay
Pa
tJbadVz, (D.17)
and then, the Localization Theorem yields
(ρavaJ)/prime+d i v a/braceleftbig
J[ρava⊗(va−wa)−Ta]G−t/bracerightbig
=Jba. (D.18)
•Energy
d
dt/integraldisplay
Pa
tρaEaJdVz=/integraldisplay
∂Pa
tJTava·G−tmdAz (D.19)
+/integraldisplay
Pa
tJba·vadVz−/integraldisplay
∂Pa
tJqa·G−tmdAz+/integraldisplay
Pa
tJfadVz. (D.20)

D.2. Conservative ALE Form of Conservation Equations 199
By using (D.11) for ϕ=ρaEaJandGauss’ Theorem , this equality becomes
/integraldisplay
Pa
t/braceleftbig
(ρaEaJ)/prime+d i v a/bracketleftbig
ρaEaJG−1(va−wa)/bracketrightbig/bracerightbig
dVz
=/integraldisplay
Pa
tdiva/parenleftbig
JG−1Tava/parenrightbig
dVz+/integraldisplay
Pa
tJba·vadVz
−/integraldisplay
Pa
tdiva/parenleftbig
JG−1qa/parenrightbig
dVz+/integraldisplay
Pa
tJfadVz, (D.21)
and, from the Localization Theorem ,
(ρaEaJ)/prime+d i v a/braceleftbig
JG−1[ρaEa(va−wa)−Tava+qa]/bracerightbig
=Jba·va+Jfa. (D.22)
Similar computations allow us to deduce the ALE formulation of the energy equa-
tion in terms of the internal energy, e=E−1
2|v|2,n a m e l y ,
(ρaeaJ)/prime+d i v a/braceleftbig
JG−1[ρaea(va−wa)+qa]/bracerightbig
=JTaG−t·gradava+Jfa. (D.23)
We notice that ALE partial diﬀerential equations (D.14), (D.18), (D.22) and (D.23)
hold in the ﬁxed domain Ω.
D.2.1 Mixed Conservative ALE Form of the Conservation
Equations
Now, we are going to deduce another conservative form to be called mixed form
because it includes ﬁelds in both ALE and spatial conﬁguration while only partial
diﬀerential operators in the spatial conﬁguration are involved. For this purpose we
will make extensive use of the following results:
Lemma 4.2.1. Letgbe a vector spatial ﬁeld and Sa tensor spatial ﬁeld. We have
diva/bracketleftbig
JG−1ga/bracketrightbig
=G−t·gradaga=J(divg)a, (D.24)
diva/bracketleftbig
JSaG−t/bracketrightbig
=J(divS)a. (D.25)
Proof. We use the following equalities:
diva(Rty)=R·graday+y·divaR, (D.26)
which holds for any smooth ALE tensor ﬁeld Rand vector ﬁeld y,a n d
diva(JG−t)=0 ( Piola’s identity). (D.27)

200 Appendix D. ALE Formulations of the Conservation Equations
Firstly, from (D.26) and (D.27) we get
diva/bracketleftbig
JG−1ga/bracketrightbig
=d i v a/bracketleftbig
JG−t/bracketrightbig
·ga+JG−t·gradaga=JG−t·gradaga.(D.28)
Moreover, by the chain rule ,
gradaga= (grad g)aG (D.29)
and hence
(gradg)a=g r a dagaG−1. (D.30)
From this equality we deduce
(divg)a= tr( grad g)a=t r/parenleftbig
gradagaG−1/parenrightbig
=G−t·gradaga. (D.31)
Then, equality (D.24) follows by using this equality in (D.28).
In order to prove (D.25) we recall that, from the deﬁnition of the divergence
operator of a tensor ﬁeld, we have,
diva(JSaG−t)·e=d i v a/bracketleftbig
JG−1St
ae/bracketrightbig
=d i v a/bracketleftbig
JG−t/bracketrightbig
·St
ae
+JG−t·grada(St
ae)=J/parenleftbig
div(Ste)/parenrightbig
a=J(divS)a·e, (D.32)
for all vectors e, where we have used (D.26), (D.27), and (D.24) for y=Steand
R=JG−1. /square
From this Lemma, and (D.14), (D.18) and (D.22) we easily get
(ρaJ)/prime+J(div [ρ(v−w)])a=0, (D.33)
(ρavaJ)/prime+J(div [ρv⊗(v−w)−T])a=Jba, (D.34)
(ρaEaJ)/prime+J(div [ρE(v−w)−Tv+q])a=Jba·va+Jfa, (D.35)
respectively.
D.3 Mixed Nonconservative Form of ALE Conservation
Equations
Finally, we want to obtain mixed nonconservative ALE formulations of the con-
servation equations.
•Mass. Firstly we have,
(ρaJ)/prime=ρ/prime
aJ+ρaJ/prime=ρ/prime
aJ+ρaJ(divw)a (D.36)

D.3. Mixed Nonconservative Form of ALE Conservation Equations 201
and
J(div [ρ(v−w)])a=J[(grad ρ)a·(va−wa)+ρa(div(v−w))a]
=JGt(grad ρ)a·G−1(va−wa)+Jρa(div(v−w))a
=Jgradaρa·d+Jρa(div(v−w))a.(D.37)
Replacing these expressions in (D.33) we get
J(ρ/prime
a+g r a daρa·d)+Jρa(divv)a= 0 (D.38)
and ﬁnally
˙ρa+ρa(divv)a=0, (D.39)
where ˙ ρadenotes the material time derivative of ALE ﬁeld ρawith respect
to motion Z.
•Momentum. Firstly, we have
(ρaJva)/prime=ρaJv/prime
a+(ρaJ)/primeva (D.40)
and
J(div [ρv⊗(v−w)])a=Jva(div [ρ(v−w)])a
+Jρa(gradv)a(va−wa). (D.41)
We add these equalities and then subtract the mass conservation equation
(D.33) multiplied by va.W eg e t
(ρaJva)/prime+J(div [ρv⊗(v−w)])a=Jρav/prime
a+Jρa(gradv)a(va−wa)
=Jρav/prime
a+Jρa(g r a dv)aGG−1(va−wa)=Jρav/prime
a+Jρagradavad
=Jρa˙va.
(D.42)
By replacing this equality in (D.34) we ﬁnally obtain
ρa˙va−(divT)a=ba. (D.43)
•Energy. Firstly we have
(ρaJEa)/prime=ρaJE/prime
a+(ρaJ)/primeEa (D.44)
and
J(div [ρE(v−w)])a=JEa(div [ρ(v−w)])a
+Jρa(grad E)a·(va−wa). (D.45)

202 Appendix D. ALE Formulations of the Conservation Equations
We add these equalities and then subtract the mass conservation equation
(D.33) multiplied by Ea.W eg e t
(ρaJEa)/prime+J(div [ρE(v−w)])a=JρaE/prime
a+Jρa(gradE)a·(va−wa)
=JρaE/prime
a+Jρa(grad E)a·GG−1(va−wa)=JρaE/prime
a+JρagradaEa·d
=Jρa˙Ea.
(D.46)
By replacing this equality in (D.35) we ﬁnally obtain
ρa˙Ea−(div [Tv])a+( d i v q)a=ba·va+fa. (D.47)

Bibliography
[1] A. Berm´ udez,Obtaining the linear equations for the small perturbations of a
ﬂow.Mat. Notae XLI(2001/02), 123–138.
[2] P. G. Ciarlet, Mathematical Elasticity. North Holland, 1988.
[3] B. D. Coleman, W. Noll, Thermodynamics of viscosity, heat conduction and
elasticity. Arch. for Rational Mech. and Anal., 13(1963), 167–178. Also in The
Foundations of Mechanics and Thermodynamics . Springer. Berlin, 1974.
[4] Z. H. Guo, The representation theorem for isotropic, linear asymmetric stress-
strain relations. J. Elasticity 13(1983), 121–124.
[5] M. E. Gurtin, An Introduction to Continuum Mechanics. Academic Press, New
York, 1981.
[6] K. K. Kuo, Principles of Combustion. John Wiley and Sons, Hoboken, N.J.,
2005.
[7] B. Mohammadi, O. Pironneau, Analysis of the K-Epsilon Turbulence Model.
John Wiley and Sons, 1994.
[8] W. Noll, Representations of certain isotropic tensor functions, Arch. Math. 21
(1970), 87–90.
[ 9 ] W .R .S m i t h ,R .W .M i s s e n , Chemical Reaction Equilibrium Analysis: Theory
and Algorithms. John Wiley and Sons, 1982.

Index
β-function, 156
absolute temperature, 11
acceleration, 3, 65
accumulated thermal expansion, 59
acoustic intensity, 91activation energy, 120
activation temperature, 120
aﬃne space, 170
ALE conﬁguration
energy equation, 198linear momentum equation, 198
mass equation, 198
velocity, 196
Arbitrary Lagrangian-Eulerian(ALE)
conﬁguration, 195
Arrhenius law, 119
associativity, 161
atoms, 122axial vector, 168
balance of linear and angular momen-
tum, 5
basis, 162
bilinear, 163
body, 1body force, 5, 137
body heat, 8
Boussinesq hypothesis, 106
Boussinesq model, 79
bulk viscosity, 64
calorically perfect, 95
cartesian coordinate frame, 171cartesian coordinates, 179
constitutive law for Newtonian
viscous ﬂuids, 192
mass conservation equation, 189
motion equation, 190
Cauchy’s hypothesis, 7Cauchy’s theorem, 8Cauchy-Green strain tensors, 2change of speciﬁc free energy, 128,
129, 131
characteristic temperature of vibra-
tion, 95
chemical aﬃnity, 128, 134chemical equilibrium, 125
chemical potential, 115
Clausius-Duhem inequality, 11closure models, 106coeﬃcient
of isothermal compressibility, 68of linear thermal expansion at
constant stress, 55
of volumetric thermal expansion
at constant pressure, 67
Coleman-Noll material, 18
commutative group, 161
commutativity, 161compatible, 132compressible Euler equations, 98compressible Navier-Stokes
equations, 98
concentration
of species E
i, 111
of the mixture, 112
conductive heat ﬂux, 137coordinates, 162, 165, 169

206 Index
Crocco’s theorem, 66
cross product, 167curl, 174curve, 175
closed, 175length, 175
curvilinear integral, 176cylindrical coordinates, 182
constitutive law for Newtonian
viscous ﬂuids, 192
mass conservation equation, 189motion equation, 190
Dalton’s law, 111
deformation, 1density, 150
in the motion, 4of the mixture, 110reference, 4
deviatoric, 168diﬀerentiable, 173diﬀerential, 173diﬀusion velocity, 110, 135
displacement, 2
dissipation inequality, 19dissipation rate, 24dissipative acoustics, 81distance, 171distribution function, 154divergence, 174divergence theorem, 177double delta function, 156dynamic viscosity, 64, 103
eddy dynamic viscosity, 106, 107
elastic ﬂuid, 96elasticity tensor, 50element mass fractions, 150endomorphisms, 1energy, 126energy equation
conservative, 137
linearized for thermoelasticity, 54
non-conservative, 138enthalpy, 64, 138, 142, 150
of formation, 139of the mixture, 110standard, 124
entropy, 11
of the mixture, 110
equilibrium constant, 129–131
based on concentrations, 129based on partial pressures, 128
Euclidean aﬃne space, 171Euclidean vector space, 163
Eulerian ﬁeld, 3
Eulerian ﬂuctuation, 90Eulerian ﬂuctuation of pressure, 79expectation, 154, 155extended symmetry group, 37extent, 127, 130
Favre averaged, 155
Fick’s law, 136, 137ﬁeld, 173ﬁlter, 105ﬁrst principle of thermodynamics, 8ﬂow
adiabatic, 149
ﬂuid, 61
energy conservation law, 64energy equation, 70motion equation, 64
Fourier’s law, 42
free energy, 12, 129
frequency factor, 120full stoichiometric matrix, 133
generalized Hooke’s law, 52
Gibbs free energy, 77, 114, 127, 130,
141, 142
of formation, 97, 115of the mixture, 111, 114standard, 97
gradient, 173
of the motion, 2
gravity force, 87
Green’s formulas, 177

Index 207
Green-Lagrange strain tensor, 2
Green-Saint Venant strain tensor, 2,
41
heat ﬂux, 8
heat rate, 8Helmholtz free energy, 12
of the mixture, 111
hyperelastic material
with heat conduction and viscos-
ity, 18
ideal ﬂuid, 103
ideal gases, 96incompressible Euler equations, 104
incompressible Navier-Stokes
equations, 106
incompressible Newtonian ﬂuid, 101
incremental methods, 57
inﬁnitesimal strain tensor, 2, 48inlets, 146
inner product, 163, 166
internal energy, 9
of formation, 123of the mixture, 110, 121
standard, 123
invertible, 168isobaric process, 149isotropic, 164
k-atoms, 122, 140, 146
k-moles, 112
kernel, 132
Lagrangian coordinates, 13
Lagrangian ﬁelds, 3
Lam´e’s coeﬃcients, 55
Laplacian, 174least action principle, 125Lewis number, 138
linearly independent, 162
local equilibrium chemistry, 149local equilibrium problem, 141
localization theorem, 179Mach number, 142, 149
mass, 126, 141, 142, 150
of the mixture, 120
mass action law, 119mass conservation equation, 135
conservative, 136
mass diﬀusion, 136mass diﬀusion coeﬃcient, 138mass distribution, 4mass fraction, 110, 140, 146
material body, 18
isotropic, 39
material ﬁelds, 3material frame-indiﬀerence principle,
28
material points, 1material time derivative, 6Mayer equation, 94mean value, 154
mechanical energy, 91
mechanical equilibrium, 77mixture fractions, 148, 150, 155molar fraction, 112molecular mass, 93
of the mixture, 112
motion, 1motion equation
conservative, 5, 137linear approximation of the , 52non-conservative, 5, 137
natural convection, 78
neutral element, 161non-dissipative acoustics, 85
norm, 163
observer change, 27
origin, 171orthogonal, 163, 168
orthonormal, 163
outlet, 146
passive scalars, 141
Pekeris equation, 87

208 Index
perfect gas, 93
coeﬃcient of isothermal
compressibility, 94
coeﬃcient of thermal expansion,
94
enthalpy, 93entropy, 94internal energy, 93sound speed, 94
speciﬁc heat at constant
pressure, 93
positive deﬁnite, 56, 163positivity, 126, 142, 143, 150potential theorem, 176power stress, 6pre-exponential exponent, 120pressure, 102
Lagrangian ﬂuctuation, 86of the mixture, 111standard , 129
principal invariants, 169
probability density functions (PDF),
153, 154
probability space, 153
random variable, 153
continuous, 154independent, 154
random vector, 154
(jointly) continuous, 154distribution function, 154
rank, 132rate of total mass per unit volume,
146
ratio of speciﬁc heats, 94real vector space, 161reference conﬁguration, 1reference map, 3region
closed, 176open, 176regular, 176
response mappings, 18rotation, 168Saint Venant-Kirchhoﬀ material, 41
Schmidt numbers, 156
second principle of thermodynamics,
11
second viscosity coeﬃcient, 64
simply connected, 176
skew, 166
skew part, 167Smagorinsky’s model, 106
small deformations, 57
sound speed, 72source of mass, 135
sources, 146
spatial ﬁelds, 3, 135
species, 109
density, 109enthalpy, 109
entropy, 109
Gibbs free energy, 109Helmholtz free energy, 109
internal energy, 109
molecular mass, 109
pressure, 109
speciﬁc heat at constant
pressure, 110
speciﬁc heat at constant volume,
109
velocity, 109
speciﬁc heat
at constant deformation, 34
at constant pressure, 69
of the mixture, 124
at constant volume, 67, 121
of the mixture, 111, 121
spherical, 168spherical coordinates, 184
constitutive law for Newtonian
viscous ﬂuids, 193
mass conservation equation, 189
motion equation, 191
stagnation enthalpy, 65, 98
state law, 125, 126, 142, 150
stirred tank, 119, 125, 140stoichiometric coeﬃcients, 119

Index 209
stoichiometric method, 132
Stokes’ theorem, 177
streams, 148
stress tensor
Boussinesq, 13Cauchy, 5First Piola-Kirchhoﬀ, 13Piola-Lagrange, 13Reynolds, 106Second Piola-Kirchhoﬀ, 13
surface force, 5surface heat, 8symmetric, 163, 166symmetric element, 161symmetric part, 167symmetry transformation, 37
system of forces, 5
system of generators, 162system of heat, 8, 120
tensor, 164
positive deﬁnite, 168
tensor of thermal expansion at cons-
tant stress, 51
tensor product, 165, 169the response of a material body is in-
dependent of the observer,
27
theorem of power expended, 6thermal conductivity, 42, 138thermal enthalpy, 139
thermodynamic pressure, 63
thermodynamic process, 17
adiabatic, 25Eulerian, 25homentropic, 66isentropic, 24isochoric, 101steady, 66
thermomechanical equilibrium, 78total energy, 8, 98trace, 166trajectory, 2transpose, 166turbulence models, 105
turbulent dissipation rate, 107
turbulent kinetic energy, 107turbulent Prandtl, 156
V¨ais¨al¨a-Brunt
frequency, 89
tensor, 88
variance, 154, 155vectors, 162
velocity, 3, 119
of the mixture, 110, 135
wall, 146
wave equation, 87

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