MATERIALE PLASTICE 51 No. 3 2014 http:www.revmaterialeplastice.ro 225Stopper Effects in Network Type Polymers [619165]

MATERIALE PLASTICE ♦ 51♦ No. 3 ♦ 2014 http://www.revmaterialeplastice.ro 225Stopper Effects in Network Type Polymers
VIOREL-PUIU PAUN1*, VALENTIN NEDEFF2, DAN SCURTU3, GABRIEL LAZAR2, VLAD GHIZDOVAT4, MARICEL AGOP5,
LAURA-GHEUCA SOLOVASTRU6*, RADU FLORIN POPA7
1 Politehnica University of Bucharest, F aculty of Applied Sciences, Physics Department, Bucharest 060042, Romania
2”Vasile Alecsandri” University of Bacau, Faculty of Engineering, 157 Marasesti Str., 600115 Bacau, România
3 “Gheorghe Asachi” Technical University, Fluid Mechanics Department,, 59A D. Mangeron Rd., 700050, Iasi, Romania
“ Alexandru Ioan Cuza” University of Iași, Faculty of Physics,11 Carol I Blv, 700506, Iasi, Romania
5 “Gheorghe Asachi” Technical University, Physics Department, 59A D. Mangeron Rd., 700050, Iasi, Romania
6 University of Medicine and Pharmacy “Gr. T. Popa”, Department of Dermatology, 16 University Str., 700115,Romania
7 University of Medicine and Pharmacy “Gr. T. Popa”, Surgery Department, 16 University Str., 700115, Iași, Romania
The specific parameters describing the flows of Bingham type rheological fluids through a circular pipe,
under the action of a pressure gradient in the direction of the movement are established, using the Non-
Standard Scale Relativity Theory approach. In such context, an analytical solution and a numerical application(for KELTAN 4200 — an etilene-co-propilene copolymer) are obtained. The friction effort has two components,
one of specific fluid gliding, the other of shearing, depending on transverse speed gradient. In the central area
of the fluid movements a particles agglomeration (fluid stopper) occurs, defined by a constantly movingstructure.
Keywords: rheology, friction effort, polymers, fluid stopper, speed
A great variety of materials is categorized as complex
fluids: polymers (elastomers, thermoplastics, andcomposites) [1], colloidal fluids, biological fluids (DNA –
who creates cells by means of a simple but very elegant
language and it is responsible for the remarkable way inwhich individual cells organize into complex systems like
organs and these organs form even more complex systems
like organisms -, proteins, cells, dispersions of biopolymers
and cells, human blood), foams, suspensions, emulsions,
gels, micelar and liquid-crystal phases, molten materials,etc. Therefore, fluids with non-linear viscous behaviours,
as well as viscoelastic materials are complex [2-3].
Particle dynamics in complex fluids is highly nonlinear.
For example, the formation of amorphous solids (glasses,
granular or colloids) do not comply with the physical
mechanism explaining solids crystallization. So, inamorphous solids, either lowering the temperature or
increasing the density, the dynamic process achieves a
level where the system cannot totally relax and thereforebecomes rigid. This phenomenon is known as glass
transition (when the temperature lowers) or jamming
transition (when density increases) [4,5]. Also, the stress
of a viscoelastic fluid, unlike the Newtonian fluid, depends
not only on the actually stress applied, but on the one applied
during previous deformation of the fluid [6].
In order to develop new theoretical models we must
admit that the complex fluids systems that display chaotic
behaviour are recognized to acquire self-similarity (space-time structures seem to appear) in association with strong
fluctuations at all possible space-time scales [1-3]. Then,
for temporal scales that are large with respect to the inverseof the highest Lyapunov exponent, the deterministic
trajectories are replaced by a collection of potential
trajectories and the concept of definite positions by that ofprobability density [7,8].
Since the non-differentiability appears as a universal
property of the complex fluids systems, it is necessary toconstruct a non-differentiable physics. In such conjecture,
* email: paun@physics.pub.ro; laura_solovastru@gmail.roby considering that the complexity of the interactionsprocesses is replaced by non-differentiability, it is no longer
necessary to use the whole classical “arsenal” of
quantities from the standard physics (differentiablephysics).
This topic was developed in the Scale Relativity Theory
(SRT) [7,8] and in the non-standard Scale Relativity Theory
(NSRT) [9-23]. In the framework of SRT or NSRT we
assume that the movements of complex fluids entitiestake place on continuous but non-differentiable curves
(fractal curves) so that all physical phenomena involved in
the dynamics depend not only on the space-timecoordinates but also on the space-time scales resolution.
From such a perspective, the physical quantities that
describe the dynamics of complex fluids may beconsidered fractal functions [7,8,]. Unlike the classical case
previously studied [24], the entities of the complex fluids
may be reduced to and identified with their owntrajectories, so that the complex fluids will behave as a
special interaction-less “fluid” by means of its geodesics
in a non-differentiable (fractal) space (Schrödinger orhydrodynamic forms).
In the present paper, we propose the NSRT approach to
analyze the complex fluids dynamics. Particularly, wedetermine the parameters that characterize the Bingham
type fluid flows, through a horizontal pipe with circular
section. The study contains an analytical solution and anumerical application, using the Navier-Stokes type
equations, from the NSRT approach, in cylindrical
coordinates and the friction effort for a Bingham type fluid.Our numerical mathematical model differs than other
models used to describe the Bingham fluids [25,26].
Experimental part
The polidispersed heterogeneous mixtures that have
fluid continuous phase can also have a discontinuousphase given by solid or fluid particles with different
properties, such as density, granulometry and shape. The

MATERIALE PLASTICE ♦ 51♦ No. 3 ♦ 2014 http://www.revmaterialeplastice.ro 226discontinuous phase of these heterogeneous mixtures, in
different working conditions (depositing, transport, phase
separation), do not have unitary behaviour that could
exactly be characterized [2-3]. In the majority of cases,the discontinuous phases in a heterogeneous
mixture with laminar flow through a circular pipe or linear
flow with flow direction change (bends, speed limiters,reductions) concentrate towards the axis of the pipe with
a distribution that is proportional to the size
particle[21,22].Therefore, these fluids are complexmolecular structures that do not obey Newton’s law, which
are called rheological fluids (complex fluids)[2-3]. The
Bingham fluids are included in this category[25,26].
The non-Newtonian fluids are found in many technical
applications in petroleum engineering, civil engineering,
environmental engineering, food engineering as muds,paints, cement pastes, slurries, food substances, etc.[25-
28].
Behaviours of the Bingham type complex fluids
When Bingham type fluid moves, the energetic and
kinematic characteristics of the flow differ than those ofthe Newtonian fluid. Thus, due to viscosity tangential
unitary effort variation law [25,26],
(1)
the distribution of speeds in transverse section of a pipecovers two sub-domains (figs. 1a,b).
In the central area of radius r ∈ [0; r
o], the unitary effort
is lower in value than the flow limit τo . Therefore the
Bingham type fluid moves as an apparently undistorted rigid
system, having the shape of a stopper with quasi-parallelwalls to those of the pipe.The solid stopper flows with
constant speed in the central area of the pipe, without
changing its structure. The radius r
o the two sub-domains
border, depend on the rheological characteristic of the
Bingham type fluid.
In sub-domain r ∈ [ro; R], the effort τ exceeds the
value τo and the Bingham type fluid flows (a layer with
finer particles and lower concentration appears).
From the energetic point of view, the motion of Bingham
plastic type fluid implies additional external forces as the
Newtonian fluids. Thus, supplementary energy
consumption is achieved.
The rheological behaviours of suspensions entail the
following aspects [25-26]:
– highlighting the factors related with system’s structure
(composition, viscosity of the two phases, dimensional
distribution of the particles, the nature of the stabilizer, etc.);
– viscosity dependence on shear speed;- viscosity dependence on sample history or, more
specifically, on time effect as a consequence of structuring
and de-structuring processes, known as rheopexy andthixotropy.
At lower concentrations[1], the rheological behaviours
of a suspension are Newtonian if the dispersion medium isalso a Newtonian one. The Newtonian behaviours imply
the following characteristics:
-the only tension that is generated in the simple shear
flow is shear tension τ, the differences between two normal
tensions being null;
– shear viscosity is independent on shear speed;
– viscosity remains constant during the searing process,
while tension tends to zero, as soon as shear stops;
-according to the relation between viscosity tangential
effort τ and the speed gradient dv/dz , respectively to the
shear speed dν/dt, fluids can be classified in Newtonian
and non-Newtonian ones[27-29];
-Non-Newtonian fluids are those fluids that do not obey
Newton’s viscosity law, i.e. they do not any longer submit
to proportionality between the tangential effort τ and the
speed gradient dv/dz .
Similarly with the Newton law, the rheological
behaviours of some real fluids are described by the
relation[1-3, 30]:
(2)
where kα is the apparent viscosity coefficient and n an
integer number. For n = 1 and ka= η from (2) the Newton’s
law results.
Rheological fluids have certain characteristics such as:
– elasticity and delayed elasticity;- plasticity and dynamic plasticity;
– viscosity, relaxation, post effect;
– thixotropy, rheopexy.
Results and discussions
Mathematical model
It is well known that in the NSRT approach [10, 13, 14],
the dynamics of the complex fluids are described by the
fractal operator:
(3)
where:
(4)
is the complex velocity, VD is the differentiable and
independent scale resolution velocity, VF is the non-
differentiable and dependent scale resolution,
is the
convective term,
is the dissipative term, D is
the ”diffusion” coefficient associated to fractal-non-fractal
transition, dt is the time scale resolution and DF s the fractal
dimension. For DF any definition can be used (Kolmogorov
fractal dimension, Hausdorff-Besikovici fractal dimension
etc. [7,8]), but once accepted such a definition for DF, it has
to be constant over the entire analysis of the complex fluiddynamics. In this conditions the Navier-Stokes type
equation becomes:
(5)
with
Fig. 1a,b. Pressure gradient flow of a
rheological fluid through a circular
pipe(a). Speed and effort diagrams(b)

MATERIALE PLASTICE ♦ 51♦ No. 3 ♦ 2014 http://www.revmaterialeplastice.ro 227
or, even more, by separating the real part and the imaginary
one (i.e. separation on resolution scales):
(6 a,b)
In the relations (3)-(5 a,b) p is the pressure, ρ is the
density, ν is the kinematic viscosity, D is the generalized
kinematic viscosity and f is the specific force. If the motions
at fractal scale resolution are irrotational, i.e.
(7)
the equation (6b), by integration and choosing a null valuefor the integration constant, takes the form of a continuitytype equations
(8)
where η= ρv is the dynamic viscosity. Particularly, for ρ
= const. the equations (6a) and (8) become:

(9 a,b)
The momentum equations (6a), together with the
continuity type equation (8) allow the solving of movement
problems of real fluids if the limit conditions are known.
Let us consider the unidirectional flow, with speed VD ≡
v of a fluid through a cylindrical pipe, with radius R, under
the action of a pressure gradient (figs. 1a,b). Thus the
dynamics will be analyzed using equations (9 a,b).
Under these circumstances,
If we neglect the specific forces,
and using the continuity equation in the imposed
conditions, the dynamic equilibrium equation results:
(10)
Considering the expression of the friction effort for
Bingham type fluid(1), it results:

(11)
The flow domain can be separated into two distinct sub-
domains. In sub-domain (1) for r ∈ [ro; R], where the shear
speed is high so the fluid tends to have Newtonianbehaviours. In sub-domain (2) for r ∈ [0; r
o], the fluid
constantly moves as a solid stopper so,in this area, the
deformation tension τo was not exceeded and the fluid is
not sheared.Let us calculate the relations of flowing speed in the
two areas, under the action of the pressure gradient,
through the boundary conditions, both for the flowing speedand the shear speed dv
z/dr.
The solution of equation (10) is:
(12)
where c1 and c2 are integration constants. The values of
these constants are established by the following boundary
conditions:
i) for r = ro, i.e. on the stopper borderline, dvz/dr = 0, so
we will have:
ii) for r = R i.e. at the wall, vz(R) = 0.
The speed vz(r) in sub-domain (1) for the r ∈ [ro; R] has
the following expression:
(19)
with Δp < 0 (pressure drop along the direction of motion).
In order to determine the radius ro of the stopper, it is
taken into account a cylinder of radius ro placed inside the
pipe, which is in equilibrium under pressure and shearforces action (fig. 2).(13)
Fig. 2. Equilibrium under pressure and shear forces action
for a cylinder with radius r0 placed inside the pipe(16)(15)(14)
(18)(17)~
From the dynamic equilibrium equation of pressure and
friction forces on the stopper (of radius r0 and length l), i.e.:
(20)
for the radius ro of the fluid stopper,the following expression
results:
(21)

MATERIALE PLASTICE ♦ 51♦ No. 3 ♦ 2014 http://www.revmaterialeplastice.ro 228The movement speed of the fluid stopper is obtained
imposing in relation (19) the condition r = ro. We find the
relation:
(22)
Numerical application
Let us analyse the rheological fluid of the KELTAN 4200
(an etilene-co-propilene copolymer) [31,32]) that flows in
a circular pipe with radius R = 1.5×10-3 [m] and length l =
15 x 10-2[m]. At the temperature of T H ≈ 303 [K] the
dynamic coefficient of viscosity is η = 77.4 [Pas] and the
gliding effort is τ0 = 60 [N/m2].
The numerical simulation of the fluid flows takes into
account the radius variation of fluid stopper r0, of speed
variation of the fluid stopper vzo and the speed variation of
the fluid flows in relation to the current radius r. The average
flow speed as a function of pressure drop Δp will be
established. These numerical simulations were realizedfor the following pressure drops: Δp
1 = 5.103[N/m2]; Δp2=1.104[N/m2]; Δp3 = 3.104[N/m2]; Δp4 = 5.104[N/m2]; Δp5= 7.104[N/m2]; Δp6 = 9.104[N/m2].
The obtained results are presented in figures 3-5.Conclusions
Using the Non-Standard Scale Relativity Theory
approach, the specific parameters describing the flows ofBingham type rheological fluids through a circular pipe,
under the action of a pressure gradient in the direction of
the movement are established.
As we intended, an analytical solution is obtained.
Moreover, it shows explicitly that the friction effort has two
components, one of specific fluid gliding, and the other ofshearing, depending on transverse speed gradient. In the
central area of the fluid movements a particles
agglomeration (fluid stopper) occurs, defined by aconstantly moving structure.
For a polymer (rheological fluid of Keltan 4200 (an
etilene-co-propilene copolymer) 3%), a numericalapplication is presented. The most important observations
are gathered in the following sentences.
First, we find that an increase of the pressure drop
implies a decrease of the fluid stopper radius and the flow
speed of fluid stopper is proportional with the increase of
the pressure drop. Also, the average flow speed grows withthe increase of pressure drop.
Second, in the case of a constant pressure drop, the
local fluid speed between the fluid stopper and wallsdecreases with radius, in the interval [r
0; R]. In practice, for
a constant radius, in the stopper fluid – wall pipe border
area the local speed grows with the increase of pressuredrop.
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Manuscript received: 23.06.2014