Journal of Modern Physics, 2011, 2, 1290-1304
doi:10.4236/jmp.2011.211160 Published Online November 2011 (http://www.SciRP.org/journal/jmp)
Effect of Magnetohydrodynamic on Thin Films of
Unsteady Micropolar Fluid through a Porous Medium
Gamal M. Abdel-Rahman
Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt
Received July 9, 2011; revised August 22, 2011; accepted September 3, 2011
Abstract
In This paper, we deal with the study of the effect of magnetohydrodynamic on thin films of unsteady mi-
cropolar fluid through a porous medium. These Thin films are considered for three different geometries. The
governing continuity, momentum and angular momentum equations are converted into a system of non-lin-
ear ordinary differential equations by means of similarity transformation. The resulting system of coupled
non-linear ordinary differential equations is solved numerically by using shooting method. A representative
set of numerical results in the three thin film flow problems for velocity and micro-rotation profiles are dis-
cussed and presented graphically. A comprehensive parametric study is carried out to show the effects of the
micropolar fluid parameters, magnetic field parameter, permeability parameter and etc. on the obtained solu-
tions.
Keywords: Unsteady Flow, MHD, Thin Film, Porous Medium, Micropolar Fluid
1. Introduction
Nowadays, the world has a great interest in the study of
non-Newtonian fluids from both fundamental and prac-
tical point of view. The understanding of physics in-
volved in the flows of such fluids can have immediate
effects on polymer processing, coating, ink-jet printing,
micro fluidics, geological flows in the earth mantle,
homodynamic, the flow of colloidal suspensions, liquid
crystals, additive suspensions, animal blood, turbulent
shear flows and many others. In view of this, a lot of
interest has been shown towards the study of non-New-
tonian flows and hence extensive literature regarding
analytic and numerical solutions is available on the topic.
It is also accepted now that in general, the governing
equations of non-Newtonian fluids are highly non-linear
and of higher order than the Navier-Stokes equations.
Because of the non-linearity and the inapplicability of
the superposition principle, the exact solutions are even
difficult to be obtained for the case of viscous fluids.
Such exact solutions further narrow down when non-
Newtonian fluids are taken into account. A lot of work
has been carried out regarding the analytic solutions for
flows of non-Newtonian fluids in cases where the in-
volved equations have been linearized and in cases
where the partial differential equations have been re-
duced into ordinary differential equations. Some recent
attempts in this direction have been made in the studies
[1-10].
Due to complexity of fluids there is not a single con-
stitutive equation which can describe the properties of all
non-Newtonian fluids. In view of this, several non-
Newtonian fl uid models have been pr op osed.
In micropolar fluids, rigid particles contained in a
small volume element and rotate about the center of the
volume element are described by the micro-rotation vec-
tor. This local rotation of the particles is in addition to
the usual rigid body motion of the entire volume ele-
ment.
In micropolar fluid theory, the laws of classical con-
tinuum mechanics are augmented with additional equa-
tions that account for conservation of microinertia mo-
ments and balance of first stress moments that arise due
to consideration of the microstructure in a material.
Amongst these, a micropolar fluid model is introduced
by Eringen [11-13]. This model includes the effects of
local rotary and couple stresses. Physically, some fluids
with additives, nemotogenic and smectogenic liquid
crystals, flow of colloidal fluids, suspension solutions,
blood, fluid with bar like elements may be represented
by the mathematical model underlying micropolar fluids.
The study of micropolar fluid mechanics has attracted

G. M. ABDEL-RAHMAN
1291
the attention of many researchers. A good list of refer-
ences on the published papers for this fluid can be found
in Eringen [14] and Ishak et al. [15]. Recently, Lok et al.
[16] analyzed the boundary layer flow of a micropolar
fluid near the forward stagnation point of a plane surface.
Nazar et al. [17] studied the stagnation point flow of a
micropolar fluid towards a stretching sheet. The problem
of the porous stretching sheet has been of great use in
engineering studies. Numerical study for micropolar flow
over a stretching sheet is presented by Moncef [18].
The work on the thin film flows of non-Newtonian
fluids under various configurations is relatively of recent
origin. Siddiqui et al. [19] discussed the thin film flows
of a third grade fluid down an inclined plane. In [20] they
examined the thin film flows of Sisko and Oldroyd-6
constant fluids on a moving belt. The thin film flow of a
fourth grade fluid down a vertical cylinder is also ana-
lyzed by Siddiqui et al. [21]. Sajid et al. [22] studied the
above mentioned on ex act solutions for thin film flows of
a micropolar fluid in the absence of a magnetic field, i.e
M = 0. The studied effect of MHD on thin films of a mi-
cropolar fluid in the absence of a porous medium, i.e S = 0
by Abdel-Rahman [23].
Hence, the objective of the present paper is the study
of the effect of magnetohydrodynamic on thin films of
unsteady micropolar fluid through a porous medium.
These Thin films are considered for three different ge-
ometries. A representative set of numerical results in the
three thin film flow problems for velocity and micro-
rotation profiles are discussed and presented graphically.
A comprehensive parametric study is carried out to show
the effects of the micropolar fluid parameters (K, m1, m2
and m3), magnetic field parameter, permeability parame-
ter and etc., which are also discussed.
2. Mathematical Analysis
The equation of continuity and the conservation equa-
tions of linear momentum and angular momentum for an
incompressible unsteady micropolar fluid, in the pres-
ence of magnetohydrodynamic through a porous medium,
by neglecting the body force and body couple, are:
0 V, (1)


2
*,
Dpx
Dt
k


 

Ω
VV
JB V
  
(2)


2.
D
jDt
xx xV




 
ΩΩ
ΩΩ
(3)
Subject to appropriate initial an d boundary conditions.
Here V, and B represent the velocity, micro-rotation
and total magnetic vectors respectively, ρ and j denote
the density and the gyration parameters of the fluid. p is
the pressure and μ, κ, α, β, k* and γ are the material con-
stants. If the velocity, micro-rotation and magnetic com-
ponents are (u, v, 0), (0, 0, N) and (0, 0, B0) respectively,
where B is the total magnetic field, so B = B0 + b, b is the
induced magnetic field.
Ω
In the low magnetic Reynolds number approximations,
in which the induced magnetic field b can be ignored, the
magnetic body force becomes; (see e.g. [24])
2
0
x
B  
J
BV (4)
In which

is the electrical conductivity of the fluid.
Consider the two-dimensional flow of unsteady, in-
compressible and micropolar fluid in absence of pressure
gradient [25] and applying the magnetic field B0 per-
pendicular to the velocity field through a porous medium.
Under the usual boundary layer approximation, the gov-
erning equations for this problem can be written as fol-
lowing:
Continuity equation :
0
uv
xy



 (5)
Momentum equation in the
x
direction is:
22
122
2
0*,
uuv uu
uv F
txy
x
y
B Nuu
yk






 
 
 
 
 



 

(6)
Momentum equation in the
y
direction is:
22
222
2
0*,
vvv vv
uv F
txy
x
y
B Nvv
xk


 
 
 
 

 
 



 

(7)
Angular momentum equation:
22
22
2.
NNN NN
uv
txyj xy
vu
N
jx





 
 

 

 




y
(8)
where, u and v are the respective velocity compo-
nents in
x

and
y

directions, N is the micro- rota-
tion or angular v elocity who se direction of ro tation in th e
x
y

plane,

is the kinematics viscosity and j, γ and κ
are the micro inertial per unit mass, spin gradient viscos-
ity and vortex viscosity, respectively. Here γ is assumed
to be given by [26];

1292 G. M. ABDEL-RAHMAN

2j  

(9)
In which μ is the dynamic viscosity and j is the refer-
ence length. As pointed out by Ahmadi [27].
Equation (9) is invoked to allow Equations (5)-(8) to
predict the correct behavior in the limiting case when
microstructure effects become negligible and in this case
micro-rotation is reduced to the angular velocity. The
micro-rotation N at the wall is related to the shear stress
at the wall by the relation:
Nn



 (10)
Where N

and


are micro-rotation and shear stress
at the wall and n is a constant .
(0 1)n0N
The case at n = 0 indicates that

, which repre-
sents concentrated particle flows, in which the micro-
elements, close to the wall surface are unable to rotate
[28]. This case corresponds to the strong concentration of
microelements [29]. The case at n = 0.5 indicates the
vanishing of the anti symmetric part of the stress tensor
and denotes weak concentration of microelements [27].
We now consider the thin film flows for both the cases.
3. Thin Film Flow down an Inclined Plane
In the presence of a magnetic field B through a porous
medium, we consider the thin film of an incompressible
micropolar fluid down an inclined plane. The ambient air
is assumed stationary so that the flow is caused by grav-
ity only. Also, the surface tension is assumed negligible
and the film thickness

is uniform, as shown in Fig-
ure 1. The velocity V, microrotation N and the magnetic
body force
x
J
B are:
 

2
11 0
,0,0,0,0,,
sin and,0,0
uy Ny
Fgx Buy 

 
 
 

VN
JB (11)
In view of above definition of velocity, Equations (5)
and (7) are satisfied automatically and Equations (6) and
(8) give:
,
x
u
g
1

,yv

Figure 1. The geometry of the problem.
2 20
1
2*
sin 0
B dudN gu
dy dy k
  

  
 u

 

 ,
(12)
2
22
2
dN du
N
jdy dy


  0


 
 
 . (13)
Subject to the boundary conditions
0,at 0
0at
du
uNny
dy
du Ny
dy


 
 
(14)
where 1
g
is the gravity.
We introduce the following dimensionless variables:
2
,and
y
yuuN

 
 .N
where j is equal to 2

. With the help of Equations (12)-
(14), we have:


1
10KuKNmMu Su
 

, (15)

12
2
KNKNu

 0



 , (16)
where 32
11
sinmg

  is the parameter of film
thickness

, K



is the parameter of micropolar
fluid, 22
0
MB

  is the magnetic parameter and
2*
kS   is the permeability parameter.
The boundary conditions (14) become:


 
00,0 0
110
uNnu
uN




(17)
4. Thin Film Flow on a Moving Belt
We consider presence of container with wide moving
belt passing through it, and contain thin films of a mi-
cropolar fluid through a porous medium affected by
magnetohydrodynamic. A the container. The belt moves
in the vertical direction with velocity 0 as shown in
Figure 2. The belt p ick s u p a thin film th ick ne ss
U

. The
fluid is drain ed down due to gravity. We assume a steady
laminar flow with uniform film thickness. The
x

axis
is taken normal to the belt and therefore the velocity V,
micro-rotation N and the magnetic body force
x
J
B are:



2
21 0
0,,0 ,,0,0 ,
and0, ,0
vx Nx
FgxB vx 

 


 

VN
JB

(18)
Note that Equations (5) and (6) are identically satisfied
while Equati on s (7) an d (8 ) give:

G. M. ABDEL-RAHMAN
1293
x
y
Belt

Fluid layer
Figure 2. Geometry of the flow of moving belt through a
micropolar fluid.
2 20
1
2*
0
B dvdN gvv
dx dx k
  
 




 
, (19)
2
22
2
dN dv
N
jdx dx


 

 

 0

. (20)
For the problem under consideration, the boundary
conditions are:
0_
,at
0at
dv
vUN nx
dx
dv Nx
dx

 
 
0
(21)
We introduce the following dimensionless variables:
00
,and .
xv
x
vNN
UU



Equations (19)-(21) are reduced to:

2
10KvKNmMv Sv
 
, (22)

12
2
KNKNv

 


 0
. (23)
where 2
21
mg U  0 is the parameter of film thick-
ness

,
The boundary conditions (21) become:



 
01, 00
110
vNnv
vN




(24)
5. Thin Film Flow down a Vertical Cylinder
The governing equatio ns here are [29]
0
uu w
rr z




, (25)
22
1222
1,
uu u
uw
trz
uuuu N
Frr z
rrz





 




 
 





(26)
22
222
2
0*
1
,
ww wwww
uwF
trzrr rz
B
NN ww
rr k




 
 
 
 
 
 

 






(27)
22
22
1
2.
NN NNNNN
uw
trzjrr rr
wu
N
jrz




2
z

  
 


  








(28)
where u and w are the velocity components in the r

and z

directions.
Consider a micropolar fluid falling on the outside sur-
face of an infinitely long vertical cylinder of radius R in
the presence of a magnetic field B and a porous medium
as shown in Figure 3. We also assume that, the uniform
magnetic field of intensity 0 acts in the radial direc-
tion and the effect of the induced magnetic field is negli-
gible, which is valid when the magnetic Reynolds num-
ber is small.
B
The flow is in the form of a thin, uniform axisymmet-
ric film of thickness

, in contact with stationary air.
The velocity and micro-rotation are


21
0,0, ,andwrN rFg




VN
z, w
r, u

B
0
R
Figure 3. Physical model and coordinate system.

1294 G. M. ABDEL-RAHMAN
Obviously, Equations (25) and (26) are satisfied identi-
cally and Equations (27) and (28) become:
2
2
2
0
1*
1
0,
dwdwdNN
rdrdrr dr
B
gww
k





 

  

 


(29)
2
22
120 dNdNNdw
N
jrdr jdr
dr r





 

 . (30)
With the appropriate boundary conditions are:
0, at
0at
dw
wNn r
dr
dw NrR
dr
R


 
(31)
Defining
2
,andrRwf N
RR


  
Equations (29)-(31) are read as:
 
3
1
0,
KffK
mMfSf


 
 
  
 (32)


22
1
2
KKf


 


 20
. (33)
where 33
31
mgR

 is the parameter of film thickness δ,
22
0
MBR

  is the magnetic parameter and
2
SRk  *
f
is the permeability parameter.
The boundary conditions (31) become:
 
 
10,1 1
0.
fn
fd d




 (34)
where 1dR

 .
6. Numerical Results and Discussions
The system of the non-linear ordinary differential equa-
tions together with the boundary conditions are solved
numerically by using Shooting method. Numerical re-
sults are presented for velocity and micro-rotation fields
in the three thin film flow problems of a micropolar fluid:
1) flow down an inclined plane, 2) flow on a moving belt
and 3) flow down a vertical cylinder, with the boundary
layer for different parameters of the problem including
micropolar fluid parameters (K, m1, m2 and m3) and the
magnetic parameter for all (n = 0.0 and n = 0.5).
For the three considered problems, Figures 4 and 5
show magnetic field effect on a) the velocity and b) the
micro-rotation profiles, it is noted that, the increase of
the magnetic parameter M leads to; at n = 0.0 and n = 0.5,
the velocity decreases and at n = 0.0, the micro-rotation
decreases in two cases 1) and 3) but it increases in case 2)
while at n = 0.5 the micro-rotation decreases.
Figures 6 and 7 show micropolar fluid parameter K
effect on a) the velocity and b) the micro-rotation pro-
files, it is noted that, the increase of the micropolar fluid
parameter K leads to; at n = 0.0 and n = 0.5 the velocity
decreases in two cases 1) and 3) but it increases in case 2)
and at n = 0.0, the micro-rotation decreases while at n =
0.5 it decreases in two cases 1) and 3) but it increases in
case 2). However, the velocity and the micro-rotation
profiles are greater for n = 1/2 when compared with the
case at n = 0.0. We have also prepared Figures 8 and 9
just to see the effects of film thickness δ on a) the veloc-
ity and b) the micro-rotation profiles. It is seen that, the
increase of the film thickness δ results in; at n = 0.0 and
n = 0.5, the velocity increases in two cases 1) and 3) but
it decreases in case 2) and at n = 0.0, the micro-rotation
increases in two cases 1) and 2) but it decreases in case
3), while at n = 0.5, the micro-rotation increases in two
cases 1) and 3) but it decreases in case 2). Furthermore,
this increase is enhanced when the value of n increases
from zero to 1/2.
Figures 10 and 11 show porous medium effect on a)
the velocity and b) the micro-rotation profiles , it is noted
that, the increase of the porous medium parameter S
leads to; at n = 0.0 and n = 0.5, the velocity decreases
while at n = 0.0 the micro-rotation decreases and at n =
0.5 the micro-rotation decreases in two cases 1) and 3)
but it increases in case 2).
7. Conclusions
In this paper, we have studied numerically the effect of
magnetohydrodynamic on thin films of unsteady mi-
cro p o l a r fluid through a porous medium. These Thin films
are considered for three different geometries, named: 1)
flow down an inclined plane, 2) flow on a moving belt
and 3) flow down a vertical cylinder. From the present
study we hav e found that:
Numerical results are presented for velocity and mi-
cro-rotation fields in the three thin film flow problems of
a micropolar fluid. The results are graphically presented
and the influence of micropolar fluid parameters, the
porous medium parameter and the magnetic parameter
are discussed for strong and weak concentrations of the
microelements. It is observed that, the rotation of the
microelements at the boundary increases the velocity

G. M. ABDEL-RAHMAN
1295
(1)
(2)
(a) (b)
(3)
Figure 4. (a) velocity and (b) micro-rotation profiles for different values of M for n = 0.0.

1296 G. M. ABDEL-RAHMAN
(1)
(2)
(a) (b)
(3)
Figure 5. (a) velocity and (b) micro-rotation profiles for different values of M for n = 0.5.

G. M. ABDEL-RAHMAN
1297
(1)
(2)
(a) (b)
(3)
Figure 6. (a) velocity and (b) micro-rotation profiles for different values of K for n = 0.0.

1298 G. M. ABDEL-RAHMAN
(1)
(2)
(a) (b)
(3)
Figure 7. (a) velocity and (b) micro-rotation profiles for different values of K for n = 0.5.

G. M. ABDEL-RAHMAN
1299
(1)
(2)
(a) (b)
(3)
Figure 8. (a) velocity and (b) micro-rotation profiles for different values of m1, m2 and m3 for n = 0.0.

1300 G. M. ABDEL-RAHMAN
(1)
(2)
(a) (b)
(3)
Figure 9. (a) velocity and (b) micro-rotation profiles for different values of m1, m2 and m3 for n = 0.5.

G. M. ABDEL-RAHMAN
1301
(1)
(2)
(a) (b)
(3)
Figure 10. (a) velocity and (b) micro-rotation profiles for different values of S for n = 0.0.

1302 G. M. ABDEL-RAHMAN
(1)
(2)
(a) (b)
(3)
Figure 11. (a) velocity and (b) micro-rotation profiles for different values of S for n = 0.5.

G. M. ABDEL-RAHMAN
1303
when comp ar ed with the cas e wh en th e r e is no ro ta tion at
the boundary.
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