Combine e ects of electro kinetic, variable viscosity and partial slip [600393]

Combine eects of electro kinetic, variable viscosity and partial slip
on peristaltic MHD
ow through a
exible channel
R. E. Abo-Elkhair1, Kh. S. Mekheimer2and A. M. A. Moawad3
1;2;3Mathematical Department, Faculty of Science (Men)
Al-Azhar University, Nasr City 11884, Cairo, Egypt
1E-mail:[anonimizat]
2E-mail: kh [anonimizat]
3E-mail:ali [anonimizat]
Abstract
This article addresses, eects of viscosity variation, wall slip conditions and electro kinetic
eects ( electro osmosis ) on electro-magneto hydrodynamic peristaltic
ow in a symmetric
exible
channel. Low Reynolds number and long wavelength assumptions have been used. The
ow is
measured in the wave frame of reference moving with a uniform velocity. The systematic results
are obtained for the velocity and pressure gradient distribution. We are solved Poisson Boltzmann
equation to obtain the (electrical double layer) EDL potential distribution. The pressure rise data is
extracted numerically. The diagrammatic sketch eect of various observing parameters on pressure
rise, frictionless force and axial velocity are drawing manually. The extension and contraction
phenomenon of the whole bolus is also displayed in the end.
Keywords: Peristaltic transport; Electromagnetohydrodynamic
ow ; electro osmosis; Vari-
able viscosity; Velocity slip.
1 Introduction
Electro-osmotic transport lies in the fact that when a surface is brought into contact with an
electrolyte solution, normally it takes a net charge. Due to the electro neutrality principle, the liquid
takes on an opposite charge in (EDL) the electric double layer near the supercies. The electric
double layer has an internal moving layer and the outer layer diuse. One of the rst attempts in this
context was carried out by Burgreen and Nakachee [1]. They have theoretically analyzed the electro-
kinetic
ow in ultrane capillary slits. Theoretical and numerical studies have been investigated in
electro-osmotic
ow [2]- [3]. Electro-osmotic
ow of bio-
uid is caused by Coulomb force under the
action of an external electric potential along the microchannel. Micro
uidic devices are also often
used to analyze bio-
uids for agent detection in the form of micro-biological sensor, DNA sample
analysis, separation of species and biochemical reactions in various physiological process, micro
electro-mechanical system (MEMS). The electro-osmotic
ow is one of the most important factors
in the optimization of electrophoretic separations [4]. Nevertheless, the study of micro
uidics is
very important in many technological processes and applications, involving detection, separation,
analysis of chemical and biological samples. Also, electro-kinetics based micro-
uidics have been
frequently utilized as one of the ecient mechanisms for manipulating and controlling liquid
ows in
micro devices [5]. Moreover, peristaltic mechanism can provide faster rate of micro-
uidic transport
and is happens to propel bio-
uids through the circulatory system of human beings. Chakraborty
[6] has rst obtained the employment of an axial electric eld and studied theoretically the rate of
micro
uidic transport in the peristaltic micro tubes.
In physiology, peristaltic mechanism plays an indispensable role in transporting physiological

uids inside living bodies. This mechanism has been developed for many industrial purposes such
as, blood pumps in heart lungs machine, etc. Several theoretical, numerical and experimental in-
vestigations [7]-[26] have been done after the seminal work of Latham[27]. Being motivated by the
above mentioned studies, the aim of the present work is to study the eects of viscosity variation,
wall slip conditions and electro kinetic eects (electro osmosis) on electro-magneto hydrodynamic

2
peristaltic
ow in a
exible channel. Mathematical model and potential prole are discussed in
section 2.1 and 2.2. Under the assumptions of long wave length and low Reynolds number approxi-
mation the solution of the proposed model is addressed in sections 2.3. In Section 2.4, perturbation
analysis has been used and the pressure rise and frictionless forms are obtained. Computational
and diagrammatic sketch results are discussed in section 3. It is believed that the results presented
here will nd prospective applications in the study of various
uid mechanical problems associated
with the gastrointestinal tract, Membranes inside the pleural cavity, and the walls of the capillaries
and small blood vessels.
2 Mathematical model and the governing equations
2.1 Problem formulation
Let us consider the peristaltic motion of bio-
uid (as a blood model) in a two-dimensional channel
by considering an incompressible viscous
uid in the presence of electric eld Eas well as magnetic
eldBof strength B0. Let us take ( X;Y ) as system coordinates of the position of a
uid particle, X
being measured in the direction of wave propagation and Yin the transverse direction. We assume
that the ow is symmetric about the central line of the channel i,e, Y= 0. LetY=H(X;T) and
Y=H(X;T) be the upper and lower boundaries of the channel respectively as shown in Fig. 1.
Due to the symmetry of the channel, it is sucient to examine the ow characteristics in the domain
0y. The geometry of the wall surface (gure 1) is dened as
H(X;T) =h+"cos[2
(XCT)]: (1)
Figure 1: Physical sketch of the problem.
Where,Tis time,"is the wave amplitude, his the mean distance of the wall from the central
axis,is the wave length and Cis the velocity with which the innite train of sinusoidal wave
progresses along the wall in the positive X-direction. The
ow which is unsteady in the xed
coordinates (laboratory frame) ( X,Y) appears steady in the moving coordinates (the wave frame)
(x0,y0) which travels in the positive X-direction with velocity C.
The relations between the two frames are:
x0=XCT; y0=Y; u0=UC; v0=V; (2)
where (U,V) and (u0,v0) are components of velocity in the xed and moving frames respectively.

3
2.2 Potential prole
The electric potential within the channel is described by the well-known Poisson-Boltzmann equa-
tion:
r2=e
; (3)
in which is the electro-osmotic potential function, eis the density of the total ionic charge and
is the dielectric constant. For any binary
uid consisting two kinds of ions of equal and opposite
charges, the density of the total ionic charge, e, is given by
=ze(n+n); (4)
in whichn+andnare the number densities of positive and negative ions, respectively, and are
given by the Boltzmann distribution (considering no EDL overlap):
n=n0exp(ze
kBTav); (5)
wherezis the valence of ions, eis the electronic charge, kBis the Boltzmann constant, Tavis the
local absolute temperature of the electrolytic solution and n0is the bulk volume concentration of
positive or negative ions which is independent of the surface electrochemistry. This distribution of
ionic concentration appears to be valid when there is no axial gradient of the ionic concentration
within the micro channel and the
ow Peclet number is suciently small. The physical extent of
the EDL is represented by a length scale known as the Debye length which is dened as
 = (ez)1(kBTav
2n0)1
2; (6)
Further we assume that the wall zeta potential ( 25mV) is small enough so that Debye-Huckel
linearization approximation can be applied.
Let us now introduce a normalized electro-osmotic potential function  with zeta potential of
the medium in the form
 =
; (7)
Also, the conguration of the peristaltic wall can be represented by:
0(x0) =h+"cos[2
x0]; (8)
and the dimensionless variables dened by:
x=x0
; y =y0
h;  =0
h; (9)
where,
(x) = 1 +cos[2x]: (10)
The boundary conditions associated electric potential as following,
 = 1 at y=(x);@
@y= 0 at y= 0: (11)
Under the long wavelength and low Reynolds number, the linearlized Poisson-Boltzmann equation
can be solved to obtain the EDL potential distribution as
 =cosh(my)
cosh(m); (12)
wherem=h
is the ratio between the half-channel height and the Debye length.
Thus, we get the charge density distribution as following:
e=m2
h2cosh(my)
cosh(m): (13)
The eect of parameters on potential distribution and the charge density distribution are shown by
Figs. 2a and 2b.

4
(a) potential distribution for dierent values of
electro-osmotic parameter mwhen=0.2.
(b) charge density distribution for dierent val-
ues of electro-osmotic parameter mwhen=0.2.
Figure 2
2.3 Flow analysis
The continuity and Navier-Stokes equations which governs the
ow under the action of externally
applied electric and magnetic elds, in the moving coordinates given as:
@u0
@x0+@v0
@y0= 0; (14)

(u0+C)@u0
@x0+v0@u0
@y0
=@p0
@x0+ 2@
@x0
(y0)@u0
@x0
+@
@y0
(y0)@v0
@x0+@u0
@y0
B2
0(u0+C) +eEx;(15)
and

(u0+C)@v0
@x0+v0@v0
@y0
=@p0
@y0+ 2@
@y0
(y0)@v0
@y0
+@
@x0
(y0)@v0
@x0+@u0
@y0
; (16)
in whichdenotes the electrical conductivity of the
ow.
The slip and symmetry boundary conditions on the planes y0= 0 andy0=0(x0) can be
expressed as follows:
Symmetry boundary condition:@u0
@y0= 0; aty0= 0;
Slip boundary condition: u0=C+b@u0
@y0aty0=0(x0):(17)
in whichbis the slip length, B0is the externally applied constant magnetic eld in the transverse
direction to the
ow and Exis the axial component of applied electric eld.
The dimensionless variables dened by:
x=x0
; y =y0
h; u =u0
C; v =
Chv0; = 0
Ch;
p=h2
0Cp0;  =0
h;  (y) =(y0)
0;  =h
;
M=r
0B0h Re=Ch
0;  ="
h:(18)
will be used; where \ " is the amplitude ratio, \ " is the wave number, \ M" Hartmann number and
\Re" is the Reynolds number. Equation of motion and boundary conditions in the dimensionless
form become:
@u
@x+@v
@y= 0; (19)

5
Re
(u+ 1)@u
@x+v@u
@y
=@p
@x+22@
@x
(y)@u
@x
+@
@y
(y)
2@v
@x+@u
@y
M2(u+ 1) +eh2
0CEx;(20)
and
Re3
(u+ 1)@v
@x+v@v
@y
=@p
@y+ 22@
@y
(y)@v
@y
+2@
@x
(y)
2@v
@x+@u
@y
: (21)
Also, the conguration of the peristaltic wall can be represented by:
(x) = 1 +cos[2x]; (22)
Under the long wavelength and low Reynolds number and use Eq(13), it follows from Eqs. (19)(21)
that the appropriate equations describing the
ow are:
@p
@x=@
@y
(y)@u
@y
M2(u+ 1) +m2cosh(my)
cosh(m(x)); (23)
and
@p
@y= 0: (24)
It is noted from Eq. (24) that pis not a function of y.
The dimensionless boundary conditions are:
@u
@y= 0; aty= 0
and
u=1 +1(@u
@y) aty=(x);(25)
in which1=b=his dened as the slip parameter, =uHS=CanduHS=(Ex)=(0) is
the Helmholtz-Smoluchowski or maximum electro-osmotic velocity. On dening the dimensionless
time-mean
ows andFrespectively in the xed and wave frame as[28]
=Q
ChandF=q
Ch; (26)
now we nds that
=F+ 1; (27)
where
F=Z
0udy: (28)
We note that represents the dimensionless form of the surface of the peristaltic wall.
As referred to in the previous papers ( [29], [30] ), the viscosity can be taken as a function of
yonly. For our analysis, we consider the following form of .
(y) =ey=1y+2y2for1 (29)
whereis Reynolds model viscosity parameter.
2.4 Perturbation analysis
The Reynolds model viscosity parameter is considered to be very small. In order to solve Eq.(23)
with the boundary conditions(25), we consider the perturbation expansion by writing
f=f0+f1+2f2+


; (30)
in whichfcan be replaced by u, ,Forp.

6
Now by using Eq. (30) in Eqs. ( 23, 25) and collecting the coecients of like power of , Firstly,
we gets the zeroth-order equation as:
@2u0
@y2M2u0=@p0
@x+M2m2cosh(my)
cosh(m(x))(31)
the corresponding boundary conditions
@u0
@y= 0; aty= 0
and
u0=1 +1(@u0
@y) aty=(x):(32)
Secondly, the rst-order perturbation equation can be written as:
@2u1
@y2M2u1=@p1
@x+y@2u0
@y2+@u0
@y; (33)
Combined with the corresponding boundary conditions
@u1
@y= 0; aty= 0
and
u1=1(@u1
@y) at y=(x):(34)
Finally, the second-order perturbation equation can be written as:
@2u2
@y2M2u2=@p2
@x+y@2u1
@y2+@u1
@yy2@2u0
@y22y@u0
@y; (35)
Combined with the corresponding boundary conditions
@u2
@y= 0; aty= 0
and
u2=1(@u2
@y) at y=(x):(36)
Now, by integrate Eq.(31) and use corresponding boundary conditions (32), we get the expression
of the zero order of velocity prole as follows:
u0(x;y) =Bcosh(my) +C1cosh(My)1
M2@p0
@x
1; (37)
by using Eqs. (37), (30) and (28) the zero order of axial pressure gradient written in the form
@p0
@x=M3
Asinh(M(x))M(x)BLsinh(M(x))
MBsinh(m(x))
m+F0+(x)
; (38)
also, by using denition of the stream function , we get:
0(x;y) =1
m(Asinh(M(x))M(x))
(Asinh(My)My)(F0mBsinh(m(x)))
sinh(M(x))(my(A+BL)ABsinh(my)) +(x)(m(A
+BL) sinh(My)BMsinh(my))!
;
(39)

7
where,
C1=@p0
@xA
M2BL
; A =1
cosh(M(x))1Msinh(M(x));
L=A
cosh(m(x))1msinh(m(x))
; B =m2sech(m(x))
m2M2;(40)
Similarly, we obtain the rst perturbation( using the solution it we get from zero perturbation )
and the second perturbation ( using the solution it we get from zero and rst perturbation ) of
velocity, pressure gradient and the stream function from Eqs.(33 – 36) as follows:
u1(x;y) =
1
8M3!
2Mcosh(My)
A@p0
@xyBLM2y+ 4C2M2
+ sinh(My)
M2
2y2
A@p0
@x
BLM2
+BL+ 8C3M
A@p0
@x
+1
(m2M2)2
8BmM3
my(mM)
(m+M) cosh(my)(m2+M2) sinh(my)
8M@p1
@x!
;
(41)
u2(x;y) =
1
96M4!
3 cosh(My)
M2
y2

A@p0
@x
M2y24
+ 8C3M3
+BL
M4y4
4M2y23
+ 4M(C32M(C2y+ 4C4))
+ 3A@p0
@x
+ 2Msinh(My)
y
M2
A@p0
@x
y2BL
M2y2+ 9
+ 12C3M
+ 9A@p0
@x
+ 6C2M2
2M2y21
+ 48C5M3
+1
m2M24
96BmM6
m
y2
m2M22
+ 4m2+ 8M2
cosh(my) + 2y
2m4
+m2M2+M4
sinh(my)
96M2@p2
@x!
(42)
where,
C2=
1
4M3
m2M22
(1Msinh(M(x))cosh(M(x)))!
sinh(M(x))
BM2
L
m2
M22
+ 8m2M2
A@p0
@x
m2M22
+M(m+M)(x)
(mM)2(m+M)
BLM2
A@p0
@x
1M2(x)1
cosh(M(x))M((x)31) sinh(M(x))
+ 4B1m3M2(M
m) sinh(m(x)) + 4Bm2M2(mM) cosh(m(x))
+ 4BmM3
21mM2(cosh(m(x))
cosh(M(x)))
m2+M2
sinh(m(x))
4M@p1
@x
m2M22!
;
and
C3=B
L
M3m2M
2+ 16m2M4
A
@p0
@x
m2M2
2
8M3(m2M2)2
(43)

8
C4=
1
96
M3m2M4
(1Msinh(M(x))cosh(M(x)))!
M
m2M24
(x)
6M
cosh(M(x))
1
A@p0
@xM2(BL+ 12C3M)
+ 4C2M2
+ 6 sinh(M(x))
3A@p0
@x
3BLM2+ 4M3(C331C2M)
+M(x)
6Msinh(M(x))
1
A@p0
@xM2(BL+ 4C3M)
+ 4C2M2
+ 12 cosh(M(x))
A@p0
@x+BLM2+ 2M3(C31C2M)
+M(x)
BLM2
A@p0
@x
31M2(x)2
sinh(M(x)) +M(1413(x)) cosh(M(x))
3(mM)4
(m+M)4
M
sinh(M(x))
3A1@p0
@x3B1LM2+ 8C2M2+ 41C3M3
+ 32M@p2
@x
+ cosh(M(x))
3A@p0
@x3BLM2+ 4C3M3
96BmM6sinh(m(x))
21M2
5m2
+M2
+ (mM)(m+M)(x)
2
2m2+M2
+1m2(mM)(m+M)(x)
+ 96Bm2
M6cosh(m(x))
m2M22
(x)2+ 4
m2+ 2M2
+ 21
m4+m2M22M4
(x)!
;
and
C5=C2
8M:
(44)
Thus, you can get:
u(x;y) =u0(x;y) +u1(x;y) +2u2(x;y);
(x;y) = 0(x;y) + 1(x;y) +2 2(x;y);
@p
@x=@p0
@x+@p1
@x+2@p2
@x:(45)
While, the pressure rise
pand the friction force F(at the wall) for a channel of length L,
in their non-dimensional forms, are given by

p=Z1
0(@p
@x)dx (46)
F=Z1
0(@p
@x)dx (47)
The integrals in Eqs. (46) and (47) are not integrable in closed form, they are evaluated numerically
using a digital computer.
3 Results and discussion
This section deals with the numerical and graphical implication of the present problem. Three
subsections are considered. In the rst subsection the eects of various parameters on the velocity
prole are investigated. The eects of the problem parameters on the pumping characteristics are
discussed in the second subsection. The trapping phenomena is illustrated in the last one.
3.1 Velocity characteristics
In this section, the electro-magnetohydrodynamic
ow characteristics are sketched out by the vari-
ation of non-dimensional
ow velocity ( u), for varying dierent physical parameters involved in
the present problem. With an aim to examine the variation of dierent quantities of interest, we
consider the following values of the parameters: Hartmann number M= 0;2;3;4; slip parameter

9
1= 0;0:1;0:2;0:3; electro-osmotic parameter m= 10;20;30;50; Helmholtz-Smoluchowski
or maximum electro-osmotic velocity =0.1, 0.2, 0.3 ,0.4; Reynolds model viscosity parameter
= 0;0:1;0:3;0:5; amplitude ratio parameter = 0:1;0:3;0:5;0:7; time averaged mean
ow rate
=1.
It has been observed from our present model that the axial velocity deserve to provide main
characteristics of
ow behavior in the micro-channel for micro-
uidic applications. Figs.(3) represent
the variation of dierent parameters on the axial velocity. The eect of electro-osmotic parameter
mon the axial velocity is shown in Fig.(3a). It shows that the velocity distribution depends slightly
on the value of mthroughout the channel. In the vicinity of the channel wall the magnitude of
the axial velocity decreases as the value of the parameter mincreases at the channel center. Since
mis the ratio of the height of the channel and Debye thickness , it indicates that an increase in
the height of the channel reduce the axial velocity of the
uid. Fig.(3b) shows that the magnetic
eld resists the motion of the
uid. This is because of the magnetic eld applied perpendicular
to the direction of axial velocity and there arises electromotive force (emf), which has a tendency
to slow down the
uid motion. It is observed from Fig.(3b) that the axial velocity decreases with
Hartmann number at the central part of the channel, while the trend is reversed near the channel
wall. From Fig. (3d), we observe that the velocity decreases in the central region and increases near
the channel walls with increasing slip parameter 1. The slip condition at the wall of the micro-
channel helps to drive the
uid faster and the dominating
uid
ows adjacent to the channel wall
due to which velocity increases. Figs(3e, 3c and 3f) illustrate the variation of the axial velocity with
the amplitude ratio parameter , the maximum electro-osmotic velocity parameter and Reynolds
model viscosity parameter respectively, we can see that all of these parameters increases the
velocity enhanced near the wall channel and reduced at the center of the channel.
3.2 Pumping characteristics
This subsection describes the in
uences of various emerging parameters of our analysis on the
pressure rise per wavelength
pand the friction force F. The eect of these parameters are
shown by Figs. 4 – 5.
Figs. 4 illustrate the change of the pressure rise
pversus the time averaged mean
ow rate
for various values of the parameters. It is shown in this gures, that there is an inversely linear
relation between
pand, i.e. the pressure rise decreases with increasing the
ow rate. Also,
the eect of increasing m,,,andMis to increase the pressure rise whereas it decreases as 1
increases.
The friction force Fare plotted in gures 5, it is observed that there is a directly linear relation
betweenFandi.e., the friction force Fincreases with increasing the
ow rate. Also the eect
of increasing m,,,andMis to decrease Fwhere it increases as 1increases, it is also clear
that the friction force has the opposite behavior compared to the pressure rise behavior.
3.3 Streamlines and
uid trapping
The phenomenon of trapping, whereby a bolus (dened as a volume of
uid bounded by a closed
streamlines in the wave frame) is transported at the wave speed. Fig.(6) illustrates the streamline
graphs for dierent values of the electro-osmotic parameter mwith a given xed set of the other
parameters. It is observed that the trapped bolus decreases in size as the electro-osmotic parameter
mincreases. Figs.(8 and 9) depicts the variation of 1andrespectively on trapping. It is
observed that the trapped bolus decreases in size as 1andincreases.The eect of Hartmann
number parameter Mon the trapping is illustrated in Fig.(7). It is evident that the trapped bolus
appears when Mis small but the important observation is that as Mis increases the trapped
bolus disappears. Fig.(10) illustrates the streamline graphs for dierent values of Reynolds model
viscosity parameter with a given xed set of the other parameters. It is observed that the trapped
bolus decreases in size as Reynolds model viscosity parameter increases.

10
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3: longitude velocity distribution for dierent values of physical parameters
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4: Variation of pressure rise per wavelength
pfor dierent values of physical parameters
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5: Variation of friction force Ffor dierent values of physical parameters

11
(a)m= 10
(b)m= 20
(c)m= 30
(d)m= 40
Figure 6: Streamlines for dierent values of mand other parameters are ( = 0:2,= 0:1= 0:1,
M= 2,1=0:1,= 1 )
(a)M= 0.0
(b)M= 2
(c)M= 4
(d)M= 6
Figure 7: Streamlines for dierent values of Mand other parameters are ( = 0:1,= 0:2,
1=0:1,m= 10,= 0:1,=1)
(a)1= -0.3
(b)1= -0.2
(c)1= -0.1
(d)0= 0
Figure 8: Streamlines for dierent values of 1and other parameters are ( = 0:2,= 0:1= 0:1,
M= 2,m= 10,= 1 )
(a)= 0.1
(b)= 0.2
(c)= 0.3
(d)= 0.4
Figure 9: Streamlines for dierent values of and other parameters are ( = 0:1,= 0:2,
1=0:1,m= 10,M= 2,=1)

12
(a)= 0.0
(b)= 0.1
(c)= 0.3
(d)= 0.5
Figure 10: Streamlines for dierent values of and other parameters are ( M= 2,= 0:2,
1=0:1,m= 10,= 0:1,=1)

13
References
[1] D. Burgreen & F. R. Nakache, J. Phys. Chem. , 68, (1964), pp. 1084-1091.
[2] C. P. Tso & K. Sundaravadivelu, J. Phys. D. Appl. Phys. , 34, (2001), 3522.
[3] Y. Q. Zu & Y. Y. Yan, J. Bionic Engg. ,3, (2006), pp. 179-186.
[4] M. Loughran , S.W. Tsai , K. Yokoyama & I. Karube, Current Applied Physics , 3, (2003), pp.
495 – 499.
[5] R. F. Probstein, 2nd ed.; New York (Wiley and Sons). , (1994).
[6] S. Chakraborty, J. Phys. D: Appl. Phys. , 39, (2006), pp. 5356 – 5363.
[7] J. C.Burns & T. Parkes, J. Fluid Mech. , 29, (1967), pp. 731 – 743.
[8] A. H. Shapiro, M. Y. Jarin & S.L. Weinberg, J. Fluid Mech. ,37, (1969), pp. 799 – 825.
[9] M. Y. Jarin, Int. J. Engng Sci. , 11, (1973), pp. 681 – 699.
[10] E. F. Elshehawey, Kh. S. Mekheimer, J. Phys. D: Appl. Phys. , 27, (1994), pp. 1163 – 1170.
[11] T. F. Zien& S. A. Ostrach, J. Biomech. , 3, (1970), pp. 63 – 75.
[12] S. N. Elsoud, S. F. Kaldas & Kh .S. Mekheimer, J. Biomath , 13, (1998), pp. 417 – 425.
[13] Kh. S. Mekheimer, E. F. El Shehawey& A. M. Elaw, Int. J. Theor. Phys. , 37, (1998), pp. 2895
– 2920.
[14] T. D. Brown& T. K. Hung, J. Fluid Mech. , 83,(1977), pp. 249 – 272.
[15] Kh. S. Mekheimer, Appl. Math. Comput. , 153, (2004), pp. 763 – 777.
[16] Kh. S. Mekheimer & Y. Abd Elmaboud, Physics Lett. A , 372, (2008), pp. 1657 – 1665.
[17] Kh. S. Mekheimer, Phys Let A , 372, (2008), PP. 4271- 4278.
[18] Y.Abd elmaboud, Kh. S. Mekheimer & A. Abdellatif, J. Heat Transfer , 135, 4, (2013), 044502.
[19] A. Sinha & G.C. Shit, Journal of Magnetism and Magnetic Materials, 378(2015), pp. 143 – 151.
[20] M. Y. Jarin & A. H. Shapiro , Annual Review of Fluid Mechanics 3, (1971), pp. 13 – 37.
[21] T. D. Brown & T. K. Hung, J. Fluid Mech. , 83,(1977), pp. 249 – 272.
[22] S. Takabatake, K. Ayukawa & A. Mori, J. Fluid Mech. , 193,(1988), pp. 267 – 283.
[23] S. Takabatake & K. Ayukawa, J. Fluid Mech. , 122,(1982), pp. 439 – 465.
[24] V.P. Srivastava & M. Saxena, Rheol. Acta. , 34, (1995), pp. 406 – 414.
[25] Kh. S. Mekheimer, K. A. Hemada, K. R. Raslan, R. E. Abo-Elkhair & A. M. A. Moawad,
Gen. Math. Notes , 20,(2014), pp. 22 – 49.
[26] L. M. Srivastava &V. P. Srivastava, J. Biomech. ,17, (1984), pp. 821 – 829.
[27] T. W. Latham, M.Sc. Thesis, MIT, Cambridge, MA, 1966.
[28] Kh. S. Mekheimer, S. Z. A. Husseny & Y. Abd elmaboud, Numer. Methods Partial Di. Equat.
, 26, (2010), pp. 747 – 770.
[29] A. M. EI. Misery, A. EI. Hakeem, A. EI. Naby & A. H. EI. Nagar, J. Phys. Soc. Jpn. , 72,
(2003), pp. 89 – 93.
[30] A. EI. Hakeem, A. EI. Naby, A. M. EI. Misery & I. I. EI. Shamy, J. Phys. A: Math. Gen. , 36,
(2003), pp. 8535 – 8547.