Hall and Magnetic Field Effects on Peristaltic Transport
through a Porous Medium of a Jeffery Fluid in a Channel
Soliman R. El Koumy, El Sayed I. Barakat, Sara I. Abdelsalam*
Abstract In this article, peristaltic motion induced by a sinusoidal traveling wave in the
flexible walls of a two-dimensional channel occupied by an incompressible, viscous and
electrically-conducting Jeffery fluid through a porous medium in the presence of a constant
magnetic field, has been investigated taking into consideration the Hall effect. The problem
is formulated using a perturbation expansion ( to the second order ) in terms of the small
amplitude ratio. The effects of pertinent flow parameters are discussed graphically. This work can
be considered as mathematical modeling to the case of gall bladder with stones.
Keywords Peristaltic motion; Jeffery fluid; Flexible walls; Hall effect; Porous medium
1. Introduction
The peristaltic pumping is a form of material transport that occurs when a progressive
wave of area contraction or expansion propagates along the length of a distensible tube
containing some material. This kind of fluid transport occurs in many biological systems;
in particular, a peristaltic mechanism may be involved in swallowing of food through the
esophagus, urine transport from kidney to bladder through the ureter. The study of the
mechanism of peristalsis, in both mechanical and physiological situations, has recently
become the object of scientific research. Since the first investigation of Latham [10],
several theoretical and experimental trials have been investigated to understand peristaltic
action in different situations. A review of much of the early literature is presented in an
article by Jaffrin and Shapiro [9]. A summary of most of the experimental and theoretical
investigations reported, with details of the geometry, fluid Reynolds number, wavelength
parameter, wave amplitude parameter and wave shape has been given by Srivastava and
Srivastava [13]. Peristaltic flow through a porous medium is presented by El-Shehawey et
al. [4]. Peristaltic motion of a generalized Newtonian fluid under the effect of a transverse
magnetic field is studied by El-Shehawey et al. [5]. However, some progress has also
been made in the field of non-Newtonian fluid mechanics. For recent contributions, we
refer to some interesting studies in the references [3, 8, 11, 12].
The Hall effect is important when the Hall parameter, which is the ratio between the
electron-cyclotron frequency and the electron-atom-collision frequency, is high. This
happens when the magnetic field is high or when the collision frequency is low [14]. The
aim of this article is to study MHD peristaltic channel flow of a Jeffery fluid through a
porous medium bounded by two flexible plates. A very strong magnetic field is imposed
on the flow so as to take Hall effects into consideration. Modified Darcy's law has been
used in the flow modeling. We formulate the problem in Section 2. In Section 3, we
discuss the perturbed systems. We solve the problem in Section 4. The numerical results
and discussion as well as the conclusions are presented in Sections 5 and 6, respectively.
Soliman R. Elkoumy . El Sayed I. Barakat
Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt
* Sara I. Abdelsalam
Basic Science Department, Faculty of Engineering, The British University in Egypt, Al-Shorouk
City, Misr-Suez Desert Road, P.O. Box 43, Cairo 11837, Egypt
Tel.: +(202) 2689-0000
Fax: +(202) 2687-5889 / 97
e-mail: sara.abdelsalam@bue.edu.eg
2. Formulation of the Problem
Consider a two-dimensional infinite channel of uniform thickness 2 h , and filled with an
incompressible, viscous and electrically conducting Jeffery fluid. We introduce Cartesian
coordinate system with the x  axis along the center line of the channel and y  axis
normal to it. A constant magnetic field B = (0, 0, Bo) is applied on the flow. The walls of
the channel are compliant on which imposed traveling sinusoidal waves of small
amplitude. We neglect the induced magnetic field under the assumption that the magnetic
Reynolds number is small. The governing equations representing this model are:
V
(2.1)
[
 ( V.) V] =  P + . S  J  B  R ,
t
 . V =0,
(2.2)

J   [E  V  B ] 
JB ,
(2.3)
e ne
(2.4)
  H  J ,  E  0 ,  . B  0 .
where  , t,  , p,  ,  , e, and ne are the density of fluid, time, kinematic viscosity,
pressure, dynamic viscosity, electrical conductivity, electric charge and number density of
electrons, respectively. V , J , B , E , and H are the velocity vector, electric current
density, magnetic induction vector, intensity of the electric field, and magnetic field
strength, respectively.
From Eq. (2.4) for the current density J = (Jx, Jy, Jz), we obtain from the relation
. J  0 that Jz = constant. Hence, we consider that the channel is non-conducting and
therefore Jz = 0 at the channel.
For Jeffery fluid, the constitutive equation for extra stress tensor S is
(1 


1
) S   (1 
t


2
t
)A 1
,
(2.5)
in which  1 is the relaxation time,  2 is the retardation time, and the first RivlinEricksen tensor A 1 is defined by
(2.6)
A 1  ( grad V)  ( grad V)T .
On the basis of Jeffery fluid model [15], the following expression of Darcy's resistance
has been suggested:
(1 


1
)R  

(1 


2
)V
,
(2.7)
t
K
t
where  (0    1) and K (positive value) are the porosity (constant) and permeability
of the porous medium, respectively. In the absence of an externally applied electric field
and with negligible effects of polarization of the ionized gas, we assume that the electric
field vector equals zero. i.e. E = 0. The x  and y  components of Eq. (2.1) are:
 B o2
u
u 
P S xx S xy
 u

u





(u  m1 )  R x , (2.8)
x
y 
x
x
y
(1  m12 )
 t
 B o2

 
P S yx S yy
 

u





(  m1u )  R y , (2.9)
x
y 
y
x
y
(1  m12 )
 t
where R x and R y are the x  and y  components of R and S xx , S xy and S yy are
to be computed from Eq. (2.5).
From Eqs. (2.2), (2.5), (2.7), (2.8) and (2.9) we obtain:


   u
u
u 
1
  P
  2
 1   1  t    t  u x   y      1   1  t  x    1   2  t   u







2
 Bo

 
 
 

1


(
u

m

)

1


1
1
2



u ,
t 
K 
t 
 (1  m12 ) 
(2.10)


   

 
1
  P
  2
 1   1  t   t  u x   y      1   1  t  y    1   2  t   







2
 Bo
 
 
 


1   1  (  m1u ) 
1   2  , (2.11)

2 
t 
K 
t 
 (1  m1 ) 
where u and  are the velocity components in the direction of increasing x and y ,
 Bo
) is the Hall parameter.
respectively , and m 1 ( 
ene
The fluid is subjected to boundary conditions imposed by the symmetric motion of the
flexible walls. Let the vertical displacements of the upper and lower walls be  and 
respectively, where [6]
 (x , t )  a cos(
2
(x  ct )) ,

(2.12)
y
 ( x , t )  a cos (
2

( x  ct ))
h

a
x
c
a
Bo
z
z
Fig. 1 Geometry of the problem
a is the amplitude,  is the wave length, and c is the wave speed. The horizontal
displacement will be assumed zero. The boundary conditions are
u (x ,  d   , t )  0 ,
 ( x , d   , t )  
(2.13)
 ( x , t )
,
(2.14)
t
Equation (2.2) allows the use of the stream function  ( x , y , t ) in terms of which
u

,
 

.
y
x
After eliminating p and dropping the stars, Eqs. (2.10) and (2.11) become
 1       ( 2 )    2   2     1      4


1
y
x
x
y 
2


t    t
t 






1  1

 (1  m ) 
t
 B 02
2
1
 2
 
  2
    K 1   2 t    ,



(2.15)
(2.16)
2
in which  denotes the Laplacian operator and subscripts of 
differentiation with respect to x and y .
The equation of motion of the compliant wall is [2]:
 2


4
2
m

d

B

T
 K   p  p0 .
 t 2
4
2
t
x
x


indicate partial
(2.17)
where m , d , B , T , K , and p 0 are the plate mass per unit area, the wall damping
coefficient, the flexural rigidity of the plate, the longitudinal tension per unit width, the
spring stiffness, and the pressure on the outside surface of the wall, respectively.
Assuming p 0  0 and the channel walls inextensible so that only their lateral motions
normal to the undeformed positions occur. The boundary conditions are then
 y  0 and  x 
 ( x , t )
t
at y   h  
(2.18)
Continuity of stresses requires that p must be the same as that which acts on the fluid on
y   h   at the interfaces of both the walls and the fluid. Employing the
x  momentum equation, we obtain:


     2

 4
 2
1


m

d

B

T
 K 
1

 
2
4
2
 t  x   t
t
x
x


 

 
 
   1   2   2 y   1   1  ( yt   y  xy   x  yy )
t 
t 

 

 B 02 

1  1
2 
 (1  m1 ) 
t
 


 ( y  m1 x ) 
1   2
K 
t


 y


.

(2.19)
Introducing
u



, u*  ,*  , * 
, *  ,
h
h
c
c
ch
h
ct
m
p
dh
B
*
*
t *  , m* 
, p* 
,
d

,
B

,
h

 h
 c2
 h 2
x*
T *
x
, y*
Th
y
, K* 
2
Kh
3
,  1
*


Dropping the stars we obtain
2
c
h
 1,  2 
*
c
h
 2,
1
K
*

 h
.
Kc
 1       (2 )  2  2   M 2   1   


1
y
x
x
y 
1
1
t   t
t



1 
  2
1
 
 1   2
    1   2  4 ,

K 
t 
R
t 
   cos  ( x  t ) ,
 y  0 and  x    sin  (x  t ) at y  1  


1   1  t


 
 x

 2


(2.20)
(2.21)
(2.22)
  2 d  B  4 T  2 K 
 m t 2  R t  R x 4  R 2 x 2  R 2  



1
  2
 
1 
 
2
1   2    y  M  1 1   1  ( y  m1 x )  1   2  y
R
t 
t 
K 
t 


 1   1


t

 ( yt   y  xy  x  yy )

at y  1  
(2.23)
where  
a
h
is the amplitude ratio,  
2 h

is the wave number, R  ch is Reynolds
h   c B 0 is Hartman number, and  1 
number, M 

1
1  m1
2
.
3. Perturbed Systems
Assuming the amplitude ratio  of the wave to be small, we obtain the solution for the
p
stream function as a power series in terms of  , by expanding  and
in the form [7]:
x
   o   1   2 2  ...
p
p
)o   (
p
)1   (
p
) 2  ...
(3.2)
x
x
x
x
The first term at the right-hand side of Eq. (3.2) corresponds to an imposed pressure
gradient which is considered as a constant. The higher-order terms correspond to the
peristaltic motion. Introducing Eqs. (3.1) and (3.2) into (2.20), (2.22) and (2.23), and
collecting the terms of like powers of  , we obtain three sets of coupled linear
differential equations with their corresponding boundary conditions in  0 ,  1 and  2
for the first three powers of  .
(
)(
(3.1)
2
3.1. System of Order Zero
1

  2 2
  
1   2     0  1   1  (
R
t 
t  t


 
1

 M 2  1 1   1  2 0  1   2
t 
K 
t

 2 0   0 y  2 0 x  0 x  2 0 y )
 2
 0 ,

(3.3)
 0 y ( 1)  0,
(3.4)
 0 x ( 1)  0,
(3.5)

1
  2
 
1   2    0 y (1)  1   1  ( 0 yt (1)  0 y (1) 0 yx (1)
R
t 
t 



 0x (1) 0 yy (1))  M 2  1 1   1  ( 0 y (1)  m1 0x (1))
t 

1
 
 1   2  0 y (1)  0 ,
K
t 
(3.6)
3.2. System of Order One
1

 
  
1   2   2 2 1   1   1  (  21   0 y  21x  1y  2 0x

R
t 
 t  t


  2
1 
  2
2
2
2
 0x   1y   1x   0 y )  M  1  1   1    1   1   2    1 ,
t 
K 
t 

 1 y ( 1)   0 yy ( 1) cos  ( x  t )  0 ,
(3.7)
(3.8)
 1x ( 1)   1xy ( 1) cos  ( x  t )   sin  ( x  t ) ,


1   1
t


(3.9)
d 2
B 5
 3
  m  sin  (x  t )   cos  (x  t )  2  sin  (x  t )
R

R


T 3
 sin (x  t )  K2  sin  (x  t )  1 1   2 
2
t
 R
R
R


 (1yxx (1)  1yyy (1)

 cos  ( x  t ) 0 yyxx ( 1)  cos  ( x  t ) 0 yyyy ( 1))  M  1 (1   1

2
t
)( 1y ( 1)
 cos  ( x  t ) 0 yy (1)  m1 1x (1)  m1 cos  ( x  t ) 0 xy ( 1))

 

1   2  ( 1y (1)  cos  ( x  t ) 0 yy ( 1))  1   1

K 
t 
t


 
 cos  (x  t ) 0 yyt (1))  1   1  ( 0 y ( 1) 1xy ( 1)
t 


1 

 ( 1yt (1)

 cos  ( x  t ) 0 y ( 1) 0 xyy ( 1)   1y ( 1) 0 xy ( 1)  cos  ( x  t ) 0 yy ( 1) 0 yx ( 1))


 ( 0 x ( 1) 1 yy ( 1)  cos  ( x  t ) 0 x ( 1) 0 yyy ( 1)
t 

 1x ( 1) 0 yy ( 1)  cos  ( x  t ) 0 xy ( 1) 0 yy ( 1)) ,
 1   1
(3.10)
3.3. System of Order Two
1
 
1   2   2 2 2

R
t 

 1   1

  
2
2
2
(   2   0 y   2x   1y   1x

 t  t
 2 y   0x  0x   2 y 1x  1y  2x   0 y )
2

2
M  1 1   1
2

2
 
2
1 

2
  2  1   2

t 
K 
t
 2
 2,

(3.11)
2
 2 y ( 1)  cos  ( x  t ) 1 yy ( 1)  1 2 cos  ( x  t ) 0 yyy ( 1)  0,
(3.12)
2
 2 x ( 1)  cos  ( x  t ) 1xy ( 1)  1 2 cos  ( x  t ) 0 xyy ( 1)
(3.13)
0
 0,
1
 
1   2  ( 2 yxx ( 1)   2 yyy ( 1)  cos  ( x  t )( 1yyxx ( 1)   1yyyy ( 1))

R
t 
 1 2 cos 2  ( x  t )( 0 yyyxx ( 1)   0 yyyyy ( 1))  M 2  1 (1   1

t
)( 2 y ( 1)
 cos  (x  t )1yy (1)  1 2 cos 2  (x  t ) 0 yyy (1)  m1 2x (1)
 

1   2  ( 2 y ( 1)
2
K 
t 

 
 cos  (x  t ) 1yy (1)  1 2 cos 2  ( x  t ) 0 yyy ( 1))  1   1  ( 2 yt ( 1)
t 

 m1 cos  ( x  t ) 1xy ( 1) 
m1
cos 2  ( x  t ) 0 xyy ( 1)) 
1
 cos  (x  t )1yyt (1)  1 2 cos 2  (x  t ) 0 yyyt (1))

 

t 
 1   1
 ( 0 y (1) 2 yx (1)  cos  (x  t ) 0 y (1) 1xyy (1)
 1 2 cos2  (x  t ) 0 y (1) 0xyyy (1) 1y (1)1yx (1)
 cos  ( x  t ) 1y ( 1) 0 xyy ( 1)   2 y ( 1) 0 yx ( 1)
 cos  (x  t ) 0 yy ( 1) 1yx ( 1)  cos 2  (x  t ) 0 yy ( 1) 0 xyy ( 1)
 cos  ( x  t ) 1yy ( 1) 0 yx ( 1)  1 2 cos 2  (x  t ) 0 yyy ( 1) 0 yx ( 1))


 1   1
 
 ( 0 x (1) 2 yy (1)  cos  ( x  t ) 0 x (1) 1yyy ( 1)
t 
 1 2 cos 2  ( x  t ) 0 x (1) 0 yyyy ( 1)   1x ( 1) 1yy ( 1)
 cos  ( x  t ) 1x ( 1) 0 yyy ( 1)   2 x ( 1) 0 yy ( 1)  cos  ( x  t ) 0 xy ( 1) 1yy ( 1)
 cos 2  (x  t ) 0 xy ( 1) 0 yyy ( 1)  cos  (x  t ) 1xy (1) 0 yy (1)
 1 2 cos2  ( x  t ) 0 xyy (1) 0 yy (1)) ,
(3.14)
4. Method of Solution
We note that the first set of differential equations in  0 subject to the steady parallel
flow and transverse symmetry assumption for a constant pressure gradient in the
x  direction, yields the following classical Poiseuille flow:
2K 0 
sinh y 
(4.1)
0 (y ) 
y 
 C1 ,
2 

 cosh  
RN 
K
0
 R / 2
 dp / dx 
,
0
in which N 2  M 2  1  1 K ,   RN 2 and C 1 is an arbitrary constant.
The second and third sets of differential equations in  1 and  2 with their
corresponding boundary conditions are satisfied by
1
 i ( x t )
*
 i ( x t )
 1 ( x , y , t )  ( 1 ( y ) e
 1 ( y )e
),
(4.2)
2
1
2 i  ( x t )
*
2 i  ( x t )
 2 ( x , y , t )  (20 ( y )  22 ( y ) e
 22 ( y ) e
),
(4.3)
2
where the asterisk denotes the complex conjugate. Substituting equations (4.2) and (4.3)
into the differential equations and their corresponding boundary conditions in  1 and
 2 leads to the following set of differential equations:
 d2 
K0
cosh y
R 
  1  i  1 
2
)  RM 2  1  
 2   i  R  2i  2 (1 
  (  ) 
N
cosh 
K 

  1  i  2 
 dy
 d2

 2 cosh y  1  i  1 
  2   2  1 ( y )  2 i  K 0 2

 1 ( y ) ,
N cosh   1  i  2 
 dy

 1/ (1) 
2K 0
RN
2
 tanh  ,

(4.5)
 1  i  1 
R  /
2
  (  )  1 (1)
K 
 1  i  2 
1/// (1)   i  R  RM 2  1  


  i  RM 2  1m1

2i 
(4.4)
K 0
N
2
K 0 3
  1  i  1  

(

1)

2
tanh 
 1
RN 2
  1  i  2  
tanh   
2M 2  1
 
////
20
K0 
2
N
 1  i  1 
 1  i  1 
K0 
tanh   R  
2

2
KN
 1  i  2 
 1  i  2 
tanh  
i 2 2
( R m  i  Rd   4B   2T  K ) ,
R2

iR
2
K 0 2 ,
1  //
*//


(

1)


(

1)

1
 RN 2
21
1/// (1)  
2


(4.7)
[1 ( y ) 1*// ( y )  1* ( y )1// ( y )] /  RN
20/ (1) 
//
20
(y ) ,
2
(4.8)
(4.9)
1
1// (1)  1*// ( 1) 
1//// ( 1)  1*//// ( 1) 



2
2 
K 0 4
RN 2
i  m1
2
 RN
20/ (1)  K 0  2 
2
RN
2
2
1// (1)  1*// (1) 


iR
1// ( 1)  1*// ( 1) 
RM 2  1 1/ (1)  1*/ ( 1) 

2 
iR
 1 (1)  *1// (1)   *1 ( 1)  1// (1) 
2
i  K 0 2
1 (1)  1* (1)  ,

2


N

(4.6)
(4.10)
2
 d2
 1  2i  1  
2 d
2 R
2
 2  4   2  4   2 i R  RM  1 
  22 ( y )
K
 1  2i  2  
 dy
  dy
4i K 0  1  2i  1   cosh y   d 2
2

1
  2  4  22 ( y )
2  1  2i  
cosh    dy
N 
2 

2
4i K 0 cosh y  1  2i  1 


 ( y )
cosh   1  2i  2  22
N2



i R  1  2i 1   /
1 ( y) 1// ( y) 1 ( y) 1/// ( y)  ,



2  1  2i 
(4.11)
2
/ (1) 
22
K 0 2
2RN 2
1 //
 (1) ,
2 1

 1  2 i  1   /
2R /
 (1)
 22 (1) 
K 22
 1  2 i  2  
222/// (1)  4  2  iR 

 1  2 i  1  /
 8 i  K 0
tanh 
 22 (1)   
N2

 1  2 i  2 
 2RM 2  1 
 1  2 i  1 
////
 4 i  RM 2  1 m1  
 22 (1) 1 (1)
1

2
i


2 
  1  2 i  1   //
R //
 i R 
    1 (1)  1 (1)
K
  1  2 i  2  
 RM
2
 1  2 i 1  //
 1  2 i 1  /
/
 1 (1)  i R 
 1 (1)1 (1)
 1  2 i 2 
 1  2 i 2 
1 
 1  2 i 1 
2 i K 0  2  1  2 i 1 
//

(

1)

(

1)

 1

 1 (1)
1
N2
 1  2 i 2 
 1  2 i 2 
i R 
(4.12)
 1  2 i  1  /
K02  2 K 

(

1)

 1
1 

KN 2 
R 
 1  2 i  2 
 m1 i RM 2  1 

M 2  1K 0  2  1  2 i   1 
N

,
 1 2i  2 
2
(5.13)
where the prime indicates the derivative with respect to y .
Thus, we obtained a set of differential equations together with the corresponding
boundary conditions which are sufficient to determine the solution of the problem up to
the second order in  . But these equations are fourth-order ordinary differential
equations with variable coefficients and the resulting problem is not an eigenvalue
problem since all the boundary conditions are not homogeneous. Therefore, we restrict
our investigation to the case of free-pumping. Physically, this means that the fluid is
stationary if there are no peristaltic waves. In this case we put (p / x )0  0 which
means K 0  0 , and we are able to obtain a simple analytical solution in a closed form [1,
6, 8]. Under this assumption, the solution of Eqs. (4.4)-(4.6) is
1 ( y )  L1 sinh  y  L2 cosh  y  L3 sinh  y  L4 cosh  y ,
(4.14)
where
 1 i  1 

,
p 2 q1  q 2 p1  1  i   2 
R q2
 1 i  1 
L3 

,
q 2 p1  p 2 q1  1  i   2 
R  p2
L4 
(4.15)
(4.16)
 sinh  ,
L
 sinh  4
 cosh 
L1  
L3 ,
 cosh 
L2  
  ( 
2
2
R
K
(4.17)
(4.18)
1 i  1 
 1  i    ,
2 

)  ( RM  1  i  R ) 
(4.19)
 1 i  1 
,
1 i  2 
(4.20)
2
1  i  RM  1m1 
2
p1  (  3   2 ) cosh  , q1 
in which
p2 
 1

 1

sinh  coth    1 cosh  ,
cosh  tanh   1 sinh  , q 2  (    ) sinh  ,
3
2
Next, in the expansion of  2 , we are interested only in the terms 20 ( y ) as our aim is
to determine the mean flow only. Thus the solution of Eqs. (4.8)-(4.10), under the
assumption K 0  0 , takes the following form:
/
20/ ( y )  F ( y ) 

F (1) sinh  (1  y )  F ( 1) sinh  (1  y )
sinh 2
D1 sinh (1  y )  D 2 sinh (1  y )
sinh 2
And the peristaltic mean flow is obtained as
 c 0 1 

cosh y 
,
cosh  
(4.21)
u
2
2
2 
F (1) sinh (1  y )  F ( 1) sinh (1  y )
F ( y ) 
20/ ( y ) 
2 

sinh 2
D1 sinh (1  y )  D 2 sinh (1  y )
sinh 2
cosh y 
 c 0  1 
,
cosh  

(4.22)
where
/
D1  20 ( 1) 
1
2
{ [( L1  L1 ) sinh   ( L 2  L 2 ) cosh  ]   L 3 sinh 
*
2
*
2
  L 4 cosh    L3 sinh    L 4 cosh 
*2 *
2
/
D 2  20 ( 1) 
1
2
*
*2 *
*
,
(4.23)
{ [( L1  L1 ) sinh   ( L 2  L 2 ) cosh  ]   L 3 sinh 
*
2
*
2
  2 L 4 cosh    *2 L*3 sinh  *   *2 L*4 cosh  * ,
c0  (1 

2
F ( 1)
//
) D1  F ( 1) 
2

i R
D3 
D5

i R
(4.24)
i RM  1m1
2
D7 
D9 , (4.25)
2
2
2
2
2






1
1
*//
*
//
*2
2
*
*
D 3   [1 ( 1) 1 ( 1)  1 ( 1) 1 ( 1)]   (    )  L1L3 sinh  y sinh  y
2
2

 L1L 4 sinh  y cosh  y  L 2 L3 cosh  y sinh  y  L 2 L 4 cosh  y cosh  y 
*
*
*
*
*
*
(   )  L1L3 sinh  y sinh  y  L 2 L3 cosh  y sinh  y
2
2
*
*
 L1*L 4 sinh  y cosh  y  L*2 L 4 cosh  y cosh  y 
 (  *2   2 )  L3L*3 sinh  y sinh  * y  L3L*4 sinh  y cosh  * y

L*3L4 cosh  y sinh  * y L4L*4 cosh  y cosh  * y  ,
1
D 5   [ 1 ( 1)   1 ( 1)]  
////
*////
2

2
1
4
( L1  L1 ) sinh    ( L 2  L 2 ) cosh 
*
*4 *
*
4
  L3 sinh    L4 cosh    L3 sinh    L4 cosh 
4
1
4
D 7   [ 1 ( 1)   1 ( 1)]  
//
* //
2

2
1
2
*
*4 *
*
*2 *
2
1
D 9  [ 1/ ( 1)   1*/ ( 1)] 
2
 (L
2
1
1
*
,
(4.27)
( L1  L1 ) sinh    ( L 2  L 2 ) cosh 
*
2
  L3 sinh    L 4 cosh    L3 sinh    L 4 cosh 
2
(4.26)
*
*2 *
*
,
(4.28)
 L1 ) cosh    ( L 2  L 2 ) sinh 
*
*

  L3 cosh    L 4 sinh    *L*3 cosh  *   *L*4 sinh  * ,
(4.29)
F ( y )  s1 cosh(   ) y  s 2 sinh(   ) y  s 3 cosh(   ) y
 s 4 sinh(   * ) y  s 5 cosh(   ) y  s 6 sinh(   ) y
 s 7 cosh(   ) y  s 8 sinh(   ) y  s 9 cosh(    * ) y
*
*
*
 s10 sinh(    * ) y  s11 cosh(    * ) y  s12 sinh(    * ) y
,(4.30)
s1  
s3 
iR
 )
*2
2
4 (   )  
* 2
iR
(
*2
 )
4 (   )  
iR
4 (   )  
2
( 2   2 )
4 (   )  
2
iR
(
 )
iR
 )
2
4 (   )  
2
*
( L 3 L3  L 4 L 4 ), s12 
2
2
( 2   2 )
4 (   )  
2
(
2
4 (   )  
*2

/
20
(L*2 L3  L1*L 4 ),
2
* 2
(
*
( L 2 L3  L1 L 4 ),
2
 )
*2
 )
*
2
*
( L 4 L 3  L 3L 4 ),
2
4 (   )  
/
*
4 (   )  
iR
*
(L 2 L*3  L1L*4 ),
(   )
* 2
Unlike most of the other investigations,  2 0 (+1)
2
2
iR
2
* 2
iR
iR
*
*
( L1L 4  L 2 L3 ),
2
(  *2   2 )
4 (   )  
( L 3L3  L 4 L 4 ), s10  
*
2
* 2
(L1*L3  L*2 L 4 ), s 6  
2
*
 )
*2
4 (   )  
iR
s4 
(
* 2
2
4 (   )  
(
s2  
(L1*L3  L*2 L 4 ), s 8 
2
* 2
iR
s11 
*2
*2

*
L 2 L 4 ),
*
*
( L L  L 2 L 4 ),
2 1 3
( 2   2 )
iR
s9  
*
( L1L 3
2
2
* 2
s5  
s7 
(
*
2
*
( L 3 L 4  L 4 L 3 ).
(  1) in our investigation which
gives the prediction that the motion of fluid is nonsymmetric and which will be discussed
later.
Now the critical reflux condition is defined as one for which the mean-velocity u ( y ) is
zero at the center line y  0 [1, 8]. Therefore, according to Eq. (4.22), the critical reflux
condition is given by
 g 21  ( g 21 )2  4 g 22
Tc 
,
2
(4.68)
where
g 1  1  (h1  h2  (    )2 h 3 ) cosh(    ), g 2  (h1  h2  (    ) 2 h3 ) sinh(    ),
g 3  1  (h1  h2  (    ) 2 h3 ) cosh(    ), g 4  ( h1  h2  (    ) 2 h3 ) sinh(    ),
g 5  1  (h1  h2  (   ) 2 h3 ) cosh(   ), g 6  (h1  h2  (   ) 2 h3 ) sinh(   ),
g 7  1  (h1  h2  (   ) 2 h3 ) cosh(   ), g 8  (h1  h2  (   ) 2 h3 ) sinh(   ),
g 9  1  (h1  h2  (    * ) 2 h3 ) cosh(    * ), g 10  (h1  h2  (    * ) 2 h3 ) sinh(    * ),
g 11  1  (h1  h2  (    * ) 2 h3 ) cosh(    * ), g 12  ( h1  h2  (    * ) 2 h3 ) sinh(    * ),
g 13  g 1 e1  g 1 e 4  g 3 e 7  g 4 e10  g 5 e13  g 6 e16  g 7 e19  g 8 e 22  g 9 e 25
 g 10 e 28  g 11 e 31  g 12 e 34 , g 14  g 1 e 2  g 2 e 5  g 3 e 8  g 4 e11  g 5 e14  g 6 e17
 g 7 e 20  g 8 e 23  g 9 e 26  g 10 e 29  g 11 e 32  g 12 e 35 , g 15  g 1 e 3  g 2 e 6  g 3 e 9
 g 4 e12  g 5 e15  g 6 e18  g 7 e 21  g 8 e 24  g 9 e 27  g 10 e 30  g 11 e 33  g 12 e 36 ,
j
j
g 16   1  2 sinh  (h4  h2   2 h3 )  5  2 cosh  (h4  h2   2 h3 )
2


i 
2
2
2R 2
c 2 sinh  L6 (h4  h2   2 h3 ) 
i  *2
2R 2
i  2
2R 2
c 2* sinh  *L*6 (h4  h2   *2 h3 ) 
c1 cosh  L6 (h4  h2   2 h3 )
i  *2
2R 2
c1* cosh  *L*6 (h4  h2   *2 h3 )
j3

2
 (h6 cosh   h5 sinh  ) 
i 

2R
2
2R
j2
g 17  
2
i 
3

2
2R 2
2R
j4

2R 2
2
j6
i  *
2
j8
2
( *

2
2
*
*
2R
2
c1*L*6 (h5  * cosh  *  h6 sinh  * )
 2 cosh  (h4  h2   2 h3 )
i  3 2
2R 2
c1 cosh  (h4  h2   2 h3 )
i  3  *2
2R
2
c1* cosh  * (h4  h2   *2 h3 )
 (h6 sinh   h5 cosh  )
i  3
2R 2
c 2 (h5  sinh   h6 cosh  ) 
*
h5
g 18  
c1L 6 (h5  cosh   h6 sinh  )
2R 2
*
c 2 (h5  sinh   h6 cosh  ) 
i  3 *
2R
i 
c 2* sinh  * (h4  h2   *2 h3 ) 
2
i  3

*
 (h6 cosh   h5 sinh  ) 
2

*
c 2 sinh  (h4  h2   2 h3 ) 
i  3  *2

 (h6 sinh   h5 cosh  )
c 2 L 6 (h5  sinh   h6 cosh  ) 
* *
 2 sinh  (h4  h2   2 h3 ) 
2
2
c 2 L 6 (h5  sinh   h6 cosh  ) 
i  *

j7
*
c1 (h5  cosh   h6 sinh  )
i  3 *
2R
2
c1* (h5  * cosh  *  h6 sinh  * ) ,
  2 )(c 5 sinh  sinh  *  c17 sinh  cosh  *  c 23 cosh  sinh  *
c11 cosh  cosh  * )  ( 2   2 )(c 8 sinh  sinh   c 32 cosh  sinh 
c 20 sinh  cosh   c14 cosh  cosh  )  (  *2   2 )(c 26 sinh  sinh  *

c 35 sinh  cosh  *  c 38 cosh  sinh  * c 29 cosh  cosh  * ) ,
g 19  
h5
2
( *
2
  2 )(c 6 sinh  sinh  *  c18 sinh  cosh  *  c 24 cosh  sinh  *
c12 cosh  cosh  * )  ( 2   2 )(c 9 sinh  sinh   c 33 cosh  sinh 
c 21 sinh  cosh   c15 cosh  cosh  )  (  *2   2 )(c 27 sinh  sinh  *

c 36 sinh  cosh  *  c 39 cosh  sinh  * c 30 cosh  cosh  * ) ,
g 20  
h5

(  *2   2 )(c 7 sinh  sinh  *  c19 sinh  cosh  *  c 25 cosh  sinh  *
2
c13 cosh  cosh  * )  ( 2   2 )(c10 sinh  sinh   c 34 cosh  sinh 
c 22 sinh  cosh   c16 cosh  cosh  )  (  *2   2 )(c 28 sinh  sinh  *

c 37 sinh  cosh  *  c 40 cosh  sinh  * c 31 cosh  cosh  * ) ,
g 21 
g 14  g 17  g 19
g  g 16  g 18
, h1  
, g 22  13
g 15  g 20
g 15  g 20


sinh   1 
sinh 2
 2
h2  
sinh 
sinh 2
, h3 
1 
1 
1
 , h4 
2 
  cosh  
 sinh 2

cosh  
,
1
sinh   

2


 1  1 
 sinh 2

cosh  
sinh 2
1
,
i  R
2
h5 
1 
i  R

1 
 , h6 
2
 cosh  
1  2

2
2
4
1 
 M 1 m1 , L5   R m   B  K ,
cosh



i  R
(  *2   2 )
4
(   * )2   2
i  R
(  *2   2 )
4
(   * ) 2   2
L6  i  Rd  L5 , L*6  i  Rd  L5 , e1 
e2 
i  R
(  *2   2 )
4
(   * ) 2   2
i  R
(  *2   2 )
4
(   * ) 2   2
i  R
(  *2   2 )
4
(   * ) 2   2
e4 
e6 
e8 
iR
(  *2   2 )
4
(   * ) 2   2
e10 
e12 
(  *2   2 )
4
(   * ) 2   2
iR
(  *2   2 )
4
(   * ) 2   2
e18 
4
(   ) 2   2
i  R
( 2   2 )
4
(   ) 2   2
i  R
( 2   2 )
4
(   ) 2   2
e 22 
e 24 
e 26 
e 28 
e 30 
e 32 
e 34 
iR
( 2   2 )
iR
( 2   2 )
4 (   )2  2
iR
( 2   2 )
4 (   )2  2
(c 34  c 22 ) , e 25 
(    * )2  2
i  R
(  *2   2 )
4
(    * )2  2
i  R
(  *2   2 )
4
(    * )2  2
iR
(  *2   2 )
4 (    * )2  2
iR
(  *2   2 )
4 (    * )2  2
(  *2   2 )
4
(   * ) 2   2
(c18  c 2 4 ) ,
(c 5  c11 ) ,
(c 7  c13 ) ,
iR
(  *2   2 )
4
(   * ) 2   2
(c 24  c18 ) ,
i  R
( 2   2 )
4
(   ) 2   2
( 2   2 )
4
(   ) 2   2
(c 8  c14 ) ,
(c10  c16 ) ,
i  R
( 2   2 )
4
(   ) 2   2
(c 21  c 33 ) ,
i  R ( 2   2 )
(c 8  c14 ) ,
4 (   ) 2   2
iR
(c 32  c 20 ) , e 23 
4
iR
i  R
(c 22  c 34 ) , e19 
(  *2   2 )
(   * ) 2   2
(   * ) 2   2
(c 20  c 32 ) , e17 
i  R
4
4
(c 9  c15 ) , e 21 
4 (   )2   2
(  *2   2 )
(  *2   2 )
(c 9  c15 ) , e15 
(c 7  c13 ) ,
i  R
iR
(c 25  c19 ) , e13 
( 2   2 )
e 20 
(c19  c 25 ) , e 7 
(c 23  c17 ) , e11 
i  R
e16 
(c17  c 23 ) , e 5 
(c 6  c12 ) , e 9 
iR
e14 
(c 6  c12 ) , e 3 
(c 5  c11 ) ,
( 2   2 )
4 (   )2   2
iR
(c10  c16 ) ,
( 2   2 )
4 (   )2   2
(c 33  c 21 ) ,
i  R
(  *2   2 )
4
(    * )2  2
(c 27  c 30 ) , e 27 
(c 35  c 38 ) , e 29 
(c 37  c 40 ) , e 31 
(c 27  c 30 ) , e 33 
(c 38  c 35 ) , e 35 
(c 26  c 29 ) ,
i  R
(  *2   2 )
4
(    * )2  2
i  R
(  *2   2 )
4
(    * )2  2
iR
(  *2   2 )
4 (    * )2  2
iR
(  *2   2 )
4 (    * )2  2
iR
(  *2   2 )
4 (    * )2  2
(c 28  c 31 ) ,
(c 36  c 39 ) ,
(c 26  c 29 ) ,
(c 28  c 31 ) ,
(c 39  c 36 ) ,
iR
e 36 
(  *2   2 )
4 (   )  
* 2
2
 1 i  1 
R p2
,

 1  i   2  p 2 q1  q 2 p1
(c 40  c 37 ) , c1  
 cosh 
 sinh 
 1 i  1 
R q2
, c3  
, c4  
,

 cosh 
 sinh 
 1  i   2  q 2 p1  p 2 q1
c2  
c5 
c8 

c c c * L L* , c 6  
4 2 3 2 6 6
R


c17 
R
4
c 29 


c 38 

2

*
2L6 L6 ,
2
R
i
2

4
R
4

4
R
c c * L L* , c 39  
4 1 2 6 6
( c 2
i
R
i
2
2
i
R
2
R

R
c 2 c 2 (L6 
, c19 
L*6 ) , 28
c

4
4
c c * ( L 6  L*6 ) , c 40 
4 1 2
i
R
c3 L6  c 2* c3*L*6 ) , j 4 
c 4 L6  c1* c 4*L*6 ) , j 8 
R
4
6
R
4
6
R
R
4
c1* c 2 c 3 ,
4
c1 c 2* c 3* ,
6
R4
c1 c 4 c 2* ,
c1 c1* ,
6
R
4
c 2 c1* c 4* ,
c 2 c1* ,
c1 c 2* ,
3
(c 2
2
i3
R
3
i
R

6
c1 c1* c 4* ,
c 2 c 2* ,
6
R
4
6
c c * c * ( L 6  L*6 ) , c 34 
4 2 1 4
c c * ( L 6  L*6 ) , c 37 
4 2 1
R
R
4
4
4

6
c c * ( L 6  L*6 ) , c 31 
4 1 1
R
R
c1 c1* c 4 ,
4
6
c c c * ( L 6  L*6 ) , c 25 
4 1 4 2
*

R
4
c 4 L6  c1* c 4*L*6 ) , j 6 
(c1
L*6 )
c c * c * (L 6  L*6 ) , c 22 
4 1 2 3
c 3 L6  c 2* c 3*L*6 ) , j 2 
(c 2
( c1

c1 c 2 c 3 (L 6 
*
R
c c * c *L L* , c 33  
4 2 1 4 6 6
c c *L L* , c 36  
4 2 1 6 6

R
4
4
c 27  
2
R

2
R
j7  
c2 c
*
c c *L L* , c 30  
4 1 1 6 6
R
R
, c18  
c c * c * (L 6  L*6 ) , c16 
4 1 1 4
c 2 c 2* c 3* ,
4
6
c c * c * (L 6  L*6 ) , c13 
4 1 1 4
4
c c c * L L* , c 24  
4 1 4 2 6 6
R
j3  
j5 
c1 c 2 c
*
3 L6L6
R
4
R
2
4
R

6
4
R
c c * c *L L* , c 21  
4 1 2 3 6 6
R


2
R
c 32 
c 35 
*
2
c 26 
R
c c * c * L L* , c15  
4 1 1 4 6 6

c 23 
c c * c * (L 6  L*6 ) , c10 
4 2 2 3
c 2 c 3 c 2* ,
4
R
4
c c * c L L* , c12  
4 1 1 4 6 6
2
c 20 

2
R
c c c * ( L 6  L*6 ) , c 7 
4 2 3 2
2
R
6
4
R
c c * c *L L* , c 9  
4 2 2 2 6 6

c14 
j1 
2
R
c11 

2
2
2
i
R
c 3  c 2* c 3* ) ,
*
*
(c 2 c 3  c 2 c 3 ) ,
(c1
c 4  c1* c 4* ) ,
3
2
(c1
c 4  c1* c 4* ) .
5. Numerical results and discussion
In order to have an estimate of the quantitative effects of various parameters involved in
the results of the present analysis, the mean velocity at the boundaries of the channel, the
mean velocity perturbation function, the time-averaged mean axial velocity distribution
and the reversal flow are calculated for various values of these parameters when K 0  0 .
The constants D1 and D2 arising from the nonslip boundary condition of the axial velocity
on the wall are due to  20/ ( y) at the boundaries and are related to the mean velocity at the
boundaries of the channel by u(1)    2/ 0 ( y)   D1 , u(1)    2/ 0 ( y)   D 2 . This
2
2
2
2
2
2
2
2
shows that the nonslip condition applies to the wavy wall and not to the mean position of
the wall. The variations of D1 and D2 with  for various values of the magnetic
parameter M , Hall parameter m1 , wall damping d , permeability of porous medium K ,
relaxation time  1 , and retardation time  2 are plotted in Figs. 2 and 3. In Figs. 2a  5d,
the effects of the various parameters involved are discussed for D1. In Figs. 2a and 2b, it
is seen that D1 increases by increasing M . Figures 2c and 2d reveal that D1 decreases by
increasing m1 . In Figs. 2a  2c, it is noted that D1 decreases by increasing d , K and
 1 , whereas it increases by increasing  2 as seen in Fig. 2d. The effects of various
parameters on D2 are shown to be similar to that on D1 except for the Hall parameter.
Figures 3a, 3b, 3c, and 3d illustrate that M , m1 and  2 have an increasing effect on D2.
However, d , K and  1 have a decreasing effect on it. It is noted that damping may
cause the mean flow reversal at the walls, which is not possible in the elastic case.
Following Fung and Yih [7], we define the mean-velocity perturbation function G ( y ) as

200 
G ( y )=
F (y ) 
2 2 
 R 


F (1) sinh  (1  y )  sinh 2 1 

sinh 2
cosh y
cosh 


,
F ( 1)  sinh  (1  y ) 
sinh 2 
The mean velocity perturbation function is plotted with  for various values of M , m1 ,
d , K ,  1 , and  2 in Figs. 4a  4d. It is obvious from Figs. 4a and 4b that M has an
increasing effect on G ( y ) , while m1 has a decreasing effect on it as seen in Figs. 4c and
4d. It is noted that d , K , and  2 have a decreasing effect on G ( y ) , while  1 has an
increasing effect on it. It is also observed that G ( y ) is minimum near the center of the
channel. Unlike all other investigations, it is noticed that G ( y ) is nonsymmetric as
illustrated in Figs. 4b and 4d (for example; G (0.4)  2.07327 and G (0.4)   4.51921 for
the dot  dashed curve in Fig. 4b). In Figs. 5a  5d, the variations of mean velocity
distribution with y for various values of the concerned parameters are illustrated. Figures
5a and 4b depict that the possibility of flow reversal increases by an increase in M .
However, Figs. 5c and 5d depict that the possibility of flow reversal decreases by an
increase in m1 . It is noted that d , K , and  1 have a decreasing effect on the mean
velocity distribution whereas  2 has an increasing effect on it. Figures 6a  6d elucidate
the variations of critical value of T with  for various values of the indicated
parameters. It is obvious that Tc increases with the increase in M and decreases with the
increase in m1 . It is also observed that R and  2 have an increasing effect on Tc , while
K and  1 have a decreasing effect on it. We note that the critical value of T becomes
exceedingly high for small values of  as compared with the large ones. This can be
interpreted physically that at high wave length (or at low frequency) reversal flow may
occur for a very high rigidity of the boundary walls. Thus, when T is lower than Tc ,
there is no mean flow at the center of the channel [1].
It should be noted that the effects of some parameters were found to be similar to Hayat et
al. [8] and Abd Elnaby and Haroun [1]; so the figures were excluded to avoid any kind of
repetition. In such figures, it was shown that D1, D2, and possibility of flow reversal
increase with an increase in  2 , T , and K , and decrease with an increase in  1 .
Conversely, G ( y ) decreases with the increase in  2 , T , and K , and increases with the
increase in  1 . It was also illustrated that Tc decreases with an increase in  1 and
increases with an increase in  2 , and that K has a slight decreasing effect on Tc .
(zero damping d  0.0)
M  3.0
0
0.02
0.02
0.03
0.03
0.04
0.05
0.06
0.07
0
0.04
M  0.0
M  3.0
0.05
M  3.0
M  6.0
0.06
M  6.0
0.2
0.4
0.6
Wave number,
0.8
1
0
0.2
M  3.0
0.4
0.6
Wave number,
d
0.01
0.8
1
 2  0.0
0
K  0.2
m 1  1.0
c
m1  1.0
0.02
D1
0.04
D1
 1  1.0
M  0.0
0
0.02
b
0.01
D1
D1
0
aa
0.01
0.06
m1  0.0
m1  1.0
m1  2.0
0.08
0.1
0
0.2
0.03
m1  0.0
m1  1.0
m1  2.0
0.04
0.05
0.06
0.4
0.6
Wave number,
0.8
0
1
0.2
0.4
0.6
Wave number,
0.8
1
Fig. 2 The variation of D1 with wave number  for three different values of magnetic parameter M (panels
a and b) and Hall parameter m 1 (panels c and d). The other chosen parameters are m  0.01, B  2,
T  1, K  1, R  10, d  0.35, K  1.5,  1 0.8,  2  0.5, and m 1  1 (panel a);
m  0.01, B  2, T  1, K  1, R  10, d  0.2, K  1.5,  1 2,  2  0.5, and m 1  1 (panel b);
m  0.01, B  2, T  1, K  1, R  10, d  0.5, K  1,  1 0.8,  2  0.5, and M  2 (panel c);
m  0.01, B  2, T  1, K  1, R  10, d  0.2, K  1.5,  1 0.8,  2  0.6 , and M  2 (panel d)
(zero damping d  0.0)
M  3.0
0
a
0
b
0.01
0.02
D2
D2
0.02
0.04
0.06
0
 1  1.0
M  3.0
0.03
M  0.0
0.04
M  0.0
M  3.0
0.05
M  6.0
0.06
M  3.0
M  6.0
0.2
0.4
0.6
Wave number,
0.8
1
0
0.2
0.4
0.6
Wave number,
0.8
1
0
0.02
d
0.005
0.01
0.03
0.015
D2
D2
0
K  0.2
m 1  0.5
c
0.01
0.04
0.06
0.07
0
0.2
m1  0.0
m1  0.5
m1  1.0
0.02
m1  0.0
m1  0.5
m1  1.0
0.05
0.025
0.03
0.4
0.6
Wave number,
0.8
1
0
0.2
 2  0.0
m1  0.5
0.4
0.6
Wave number,
0.8
1
0
5
10
a
The perturbationfunction, G y
The perturbationfunction, G y
Fig. 3 The variation of D2 with wave number  for three different values of magnetic parameter M (panels
a and b) and Hall parameter m 1 (panels c and d). The other chosen parameters are m  0.01, B  2,
T  1, K  1, R  10, d  0.35, K  1.5,  1 0.8,  2  0.5, and m 1  1 (panel a);
m  0.01, B  2, T  1, K  1, R  10, d  0.2, K  1.5,  1 2,  2  0.5, and m 1  1 (panel b);
m  0.01, B  2, T  1, K  1, R  10, d  0.5, K  1,  1 0.8,  2  0.5, and M  4 (panel c);
m  0.01, B  2, T  1, K  1, R  10, d  0.2, K  1.5,  1 0.8,  2  0.6 , and M  2 (panel d)
(zero damping d  0.0)
M  1.0
15
20
M  0.0
M  1.0
M  2.0
25
30
35
0
0.2
0.4
0.6
0.8
0
5
1  0.6
10
M  1.0
15
20
M  0.0
M  1.0
25
b
30
1
M  2.0
1
0.5
0
y
0
c
5
10
K  1.0
m1  1.0
15
m1  0.0
m1  1.0
m1  2.0
20
0
0.2
0.4
0.6
0.8
The perturbationfunction, G y
The perturbationfunction, G y
y
0.5
1
0
5
10
 2  0.0
15
20
1
d
1
m1  0.0
m1  1.0
m1  2.0
m1  1.0
0.5
y
0
y
0.5
1
Fig. 4 The variation of the mean velocity perturbation function G ( y ) for three different values of magnetic
parameter M (panels a and b) and Hall parameter m 1 (panels c and d). The other chosen parameters are
m  0.01, B  2, T  1, K  1, R  1, d  1, K  1.5,  1 0.8,  2  0.5,   0.5, and m 1  2
(panel a); m  0.01, B  2, T  1, K  1, R  1, d  0.5, K  1.5,  1 1.2,  2  0.5,   0.5,
and m 1  2 (panel b); m  0.01, B  2, T  1, K  1, R  1, d  0.5, K  1.5,  1 0.8,  2  0.5,
  0.5, and M  1 (panel c); m  0.01, B  2, T  1, K  1, R  1, d  0.5, K  1.5,  1 0.8,
 2  0.6,   0.5, and M  1 (panel d)
(zero damping d  0.0)
M  0.8
1
b
0.5
1
0.0006
M  0.8
0.5
M  0.4
M  0.8
M  1.2
y
y
0.5
0
 1  0.8
1
a
0
0.5
M  0.4
M  0.8
M  1.2
1
0.0004
0.0002
0
0.0002
T he mean velocity distribution, u y
0.002
0.0015
0.001
0.0005
T he mean velocity distribution, u y
0
1
0.5
d
m 1  0.0
m 1  2.0
m 1  5.0
0.5
 2  0.0
0.5
m1  2.0
m 1  0.0
m 1  2.0
m 1  5.0
0
y
y
0
1
K  0.3
m 1  2.0
c
0.5
1
1
0.0025
0.001 0.0008 0.0006 0.0004 0.0002
The mean velocity distribution, u y
0.002 0.0015 0.001 0.0005
T he mean velocity distribution, u y
Fig. 5 The variation of the mean velocity distribution and reversal flow for three different values of magnetic
parameter M (panels a and b) and Hall parameter m 1 (panels c and d). The other chosen parameters are
m  0.01, B  2, T  1, K  1, R  1, d  0.2, K  1.5,  1 0.8,  2  0.5,   0.5,   0.15,
and m 1  0.5 (panel a); m  0.01, B  2, T  1, K  1, R  1, d  0.5, K  1.5,  1 1.2,
 2  0.5,   0.5,   0.15, and m 1  0.5 (panel b); m  0.01, B  2, T  1, K  1, R  1,
d  0.5, K  0.9,  1 0.8,  2  0.5,   0.5,   0.15, and M  0.5 (panel c); m  0.01, B  2,
T  1, K  1, R  1, d  0.5, K  1.5,  1 0.8,  2  0.4 ,   0.5,   0.15, and M  1 (panel d)
M  0.4
M  0.5
10000
R  4.0
8000
M  0.6
M  0.5
6000
4000
a
0.5
0.6
0.7
0.8
Wave number,
TCritical Values
8000
6000
K  1.2
m 1  1.0
4000
2000
0.9
 1  0.9
6000
M  0.5
m 1  0.0
m 1  1.0
m 1  2.0
0.7
0.8
Wave number,
0.9
1
M  0.6
b
0.5
0.6
0.7
0.8
Wave number,
8000
 2  0.1
6000
0.9
1
m 1  0.0
m 1  1.0
m 1  2.0
m 1  1.0
4000
2000
0.6
M  0.5
4000
1
c
0.5
8000
2000
TCritical Values
2000
M  0.4
10000
TCritical Values
TCritical Values
12000
d
0.5
0.6
0.7
0.8
Wave number,
0.9
1
Fig. 6 The variation of critical values of the wall tension T with wave number  for three different values
of magnetic parameter M (panels a and b) and Hall parameter m 1 (panels c and d). The other chosen
parameters are m  0.01, B  2, K  1, R  6, d  0.5, K  1.5,  1 0.8,  2  0.5, and
m 1  0.5 (panel a); m  0.01, B  2, K  1, R  5, d  0.5, K  1.5,  1 1.6,  2  0.5, and
m 1  0.5 (panel b); m  0.01, B  2, K  1, R  5, d  0.5, K  1.9,  1 0.8,  2  0.5, and
M  0.5 (panel c); m  0.01, B  2, K  1, R  5, d  0.5, K  1.5,  1 0.8,  2  0.5, and
M  0.5 (panel d)
It is noted that the results of the hydrodynamic viscous fluid filling the porous medium of
Hayat et al. [8] when M  0 can be obtained by choosing M  0 in our investigation.
Also, the results of the hydrodynamic viscous fluid through non-porous medium of Abd
Elnaby and Haroun [1] can be obtained by choosing    1   2  0 and M  0 in our
investigation.
6. Conclusions
In this study, the effects of physical parameters of interest are discussed for the constants
D1 and D2, G ( y ) , u ( y ) , and Tc . The obtained results can be outlined and summarized as
follows: The constants D1 and D2 increase with an increase in M and  2 , and decrease
with an increase in d , K , and  1 . However, they behave differently with the Hall
parameter m1 ; m1 has a decreasing effect on D1, while it has a increasing effect on D2.
The mean velocity perturbation function G ( y ) increases by increasing M and  1 , while
it decreases by increasing d , K ,  2 , and m1 . The flow reversal increases with an
increase in d , K ,  1 , and m1 , and decreases with an increase in M and  2 . The
critical value of T increases by increasing M , R , and  2 , and decreases by increasing
m1 , K and  1 . The nonsymmetry of fluid motion was predicted since the mean flow at
one boundary of the channel is not equal to that at the other boundary; accordingly, the
result of the mean-velocity perturbation function being nonsymmetric is obtained.
Moreover, Since the Hall effect cannot be taken into consideration unless there is a strong
magnetic field imposed on the flow, so when there is no magnetic field, Hall effect
vanishes.
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