Applied Mathematical Sciences, Vol. 9, 2015, no. 66, 3291 - 3302

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2015.54316

Solution of Micropolar Fluid Flow through Porous

Channels – A Differential Transform Approach

P. Vimala

Department of Mathematics

Anna University Chennai – 600025, India

P. Blessie Omega

Department of Mathematics

Anna University Chennai – 600025, India

Copyright © 2014 P. Vimala and P. Blessie Omega. This article is distributed under the

Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Abstract

The steady laminar two-dimensional flow of a micropolar fluid through a porous

channel with variable permeability is studied. Various cases of injection at one

wall and suction at the other wall with different velocities and injection at both

walls or suction at both walls are considered. A semi-analytical Differential

Transform Method is used to solve the problem. Comparison of the results with

the corresponding Newtonian case is carried out and the efficiency of the DTM is

analysed.

Keywords: Micropolar Fluid, Differential Tranform Method, Porous Channel

1 Introduction

The flow through channels and circular pipes with permeable walls has

fundamental importance in biomedical and engineering industries. Two

dimensional steady laminar flow of an incompressible Newtonian fluid through

uniform porous walls with same or different permeability have been investigated

by several researchers. Berman [3] has investigated the effects of wall porosity on

the two dimensional steady incompressible laminar flow of a fluid in a channel with rectangular cross-section. The governing third order ordinary differential equa-

3292 P. Vimala and P. Blessie Omega

tion has been solved by a perturbation technique for slow flows through the

porous walls with uniform injection or uniform suction at both walls. Yuan [15]

has analyzed the investigation of laminar flow in channels with porous walls.

Here, an exact solution has been obtained and the results have been extended to

the case of moderate to high- suction or injection velocity. Terrill and Shrestha

[11] have studied the flow through parallel uniform porous walls of different

permeability for small Reynolds numbers of the flow. The problem has been

solved using perturbation technique and a numerical method.

Non-Newtonian fluids play a major role in many industries and machineries.

Microfluid is one such fluid. Eringen [6] has developed the theory of microfluids,

dealing with a class of fluids, which exhibit certain microscopic effects, arising

from local structures and micromotion of fluid elements. These fluids can support

stress moments and body couples and are influenced by the spin inertia. Nearly

after a decade, Eringen [7] introduced the theory of micropolar fluid, which is a

subclass of microfluids. Micropolar fluids are those fluids which are consisting of

randomly oriented particles suspended in a viscous medium. In the above work,

the theory of micropolar fluids has been formulated and the constitutive laws for

fluids with microstructures have been derived. This theory has provided a

mathematical model for the non-Newtonian behavior which could be observed in

certain fluids such as polymers, colloidal suspensions, animal blood, crystals and

so on. Kelson et al. [8] have determined the influence of the material constants of

the micropolar fluids on the flow with high mass transfer through the channel

walls for uniform injection or suction at the walls. Xin-hui SI et. al. [13] have

analyzed the flow of micropolar fluid between two orthogonally moving porous

disks. Review for microcontinuum fluid mechanics has been presented by Ariman

et al. [1]. Xin-yi SI et. al. [14] have given a homotopy analysis solution for

micropolar fluid flow through porous channel with expanding or contracting walls

of different permeability.

A semi-analytical approach namely Differential Transform Method (DTM)

has been used by several researchers for solving linear and non-linear problems

involving ordinary differential equations or partial differential equations

[2,4,5,9,10&16]. In an earlier work, the authors [12] have given a DTM approach

of solving the problem of two dimensional laminar flow of an incompressible

Newtonian fluid through parallel porous walls with variable permeability

considering injection at one wall and suction at the other wall with different

velocities and injection at both walls or suction at both walls. The present

investigation focuses on the use of a micropolar fluid model in such a geometry

using DTM.

Solution of micropolar fluid flow through porous channels 3293

2 Mathematical Formulation

Consider a steady two dimensional laminar flow of a micropolar fluid through

a channel of height ‘h’ having porous walls of different permeability (Figure 1).

The flow is driven by a constant pressure gradient with uniform entrance velocity

U and is superimposed by the injection at either of the walls and suction at the

other wall. The fluid is assumed to have different normal velocities 1

V and 2

V

at the lower and upper walls respectively. Different types of flows arise in a

porous channel depending on the signs and magnitudes of 1

V & 2

V . All these

flows fall under two distinct categories: (i) 1 2

V V and (ii)2 1

V V .

Figure 1 Flow Geometry

The velocity components u and v are taken to be in the direction of x and y-axes

respectively. The fluid is either injected or ejected with a constant velocity1

V at

the lower wall and a constant velocity 2

V at the upper wall. The governing

equations of the steady laminar flow of an incompressible micropolar fluid in the

absence of body forces are the continuity equation and the momentum equations

[8], given by

1

0u v

x h

(2.1)

2 2

2 2 2

1 1u v u p u u Nu

x h x x h h

(2.2)

2 2

2 2 2

1 1v v v p v v Nu

x h h x h x

(2.3)

2 2

2 2 2

1 12 sN v N u v N N

u Nx h j h x j x h

(2.4)

where /y h , p is the pressure, the kinematic viscosity, the fluid density,

N the microrotation whose direction of rotation is in the xy- plane and the material

parameters s and ,,j are microinertia density, coupling coefficient and spin

gradient viscosity respectively. The boundary conditions are given by

1V

y=0

y=h

h

h

h

h

h

h

h

h

h

2V

y

x

3294 P. Vimala and P. Blessie Omega

1 2

( ,0) 0, ( ,1) 0, ( ,0) , ( ,1)V Vu x u x v x v x (2.5)

0)1,(,0)0,( xNxN . (2.6)

2.1 Case (i): 1 2

V V

A similarity transform [10, 19] of the form

11 1( , ) ( / ) ( )Vx Uh x f (2.7)

1 1 1( , ) ( / ) ( / ) ( ) /VN x U x h g h (2.8)

is used, where U is the entrance velocity and2 11 / 1V V , and hence the

velocity components in terms of stream function become

11 1/ / ( ) /Vu y Uh x f h (2.9)

1 1/ ( )Vv x f . (2.10)

Therefore (2.2)-(2.4) respectively become

2 21 1

1 1 1 1 1 12/ / ( ) /

g V pU V x h h f f f f

h h x

(2.11)

1

1

22 1 1 1 1

1 2/

V VV

g f f ph f

h h h

(2.12)

2 2

1 1 1 1 1 1 1 12s Vg h f h g jh f g f g (2.13)

The expression for /p in Eq.(2.12) is a function of only and thus

2

0p

x

. (2.14)

Hence (2.11)-(2.13) reduce to

2

1 1 1 1 1 1 1 1( )f C g R f f f A (2.15)

)()2( 111131121 fggfCgfCg (2.16)

where 1 1

2

1 1 2 3/ ( ), / ( ), / , /s sV VR h C C h C j h and A1

is an arbitrary constant of integration. The boundary conditions (2.5) and (2.6)

become

1 1 1 1 1(0) 0, (0) 1, (1) 1 , (1) 0f f f f (2.17)

1 1(0) 0, (1) 0.g g (2.18)

For the case of suction at one wall and injection at the other wall with the upper

wall velocity being less than the lower wall velocity, it is required that

01 1 and 01 R . For suction at both walls, ,01 R 12 1 and for

injection at both walls, 12,0 11 R .

Solution of micropolar fluid flow through porous channels 3295

2.2 Case (ii): 2 1

V V

In this case the equations of the similarity transformation are

22 2( , ) ( / ) ( )Vx Uh x f (2.19)

22 2( , ) ( / ) ( / ) ( ) /VN x U x h g h (2.20)

where 1 22 /1 V V .

The velocity components in terms of stream function in this case, are

22 2/ / ( ) /Vu y Uh x f h (2.21)

2 2/ ( )Vv x f (2.22)

Using (2.21) and (2.22), (2.2) - (2.4) become

22 22 2 2 2 2 22 2

/ /g V p

U V x h f f f fh h h x

(2.23)

2 2

2

2 2 2 222 2

V VV g f f pf

h h h h

(2.24)

2

2 2

2 2 2 2 2 2 22s Vg h f h g jh f g f g (2.25)

(2.24) shows that /p is a function of only and using (2.14) in (2.23)-(2.25),

222

2

22212 )( AfffRgCf (2.26)

2 2 2 2 4 2 2 2 2( 2 ) ( )g C f g C f g g f (2.27)

where 2

2 2 1 2 4 2/ ( ), / ( ), / , /s sR V h C C h C jV h and 2A

is an arbitrary constant of integration.

The boundary conditions from (2.5) and (2.6) become

,0)1(,1)1(,0)0(,1)0( 22222 ffff (2.28)

0)1(,0)0( 22 gg (2.29)

For the case of suction at one wall and injection at the other wall with lower wall

velocity being less than the upper wall velocity, 10 2 and 02 R . For

suction at both walls, ,02 R 21 2 and for injection at both walls,

21,0 22 R .

3 Solution by DTM

The differential transform of a function Xf and its inverse transform are

defined as

0

1(k) ( ( )) ( )

k!

k

k

X X

dF f X f X

dX

D

(3.1)

3296 P. Vimala and P. Blessie Omega

0

0

( ) (k)( )k

k

f X F X X

(3.2)

For implementation purpose, )(xf is expressed by a finite series such that

0

0

( ) ( )n

k

k

Xf X F k X

. (3.3)

Usually, the values of ‘n’ are decided by the convergence of the series

coefficients.

3.1 Case (i) 21 VV

In the present problem, taking differential transform about 0 , (2.15) and (2.16)

are transformed into

)(

)2()()2)(1(

)1()1()1()1(

)1()1()3()3)(2)(1(

1

111110

1

111110

1

1

111

1

1kA

kkFkFkkkk

kkFkFkkk

R

kGkCkFkkk

k

k

k

k

(3.4)

k

k

k

k

kkGkFkkkGkFkkC

kGkFkkCkGkk

0 011111111113

1121

1 1

)()1()1()1()()1(

)(2)2()2)(1()2()2)(1(

(3.5)

and the first two boundary conditions from (2.17) and the first condition from

(2.18) are transformed into the first three conditions in the following:

1 1 1(0) 1, (1) 0, (0) 0F F G & 1111 )1(,)2( dGbF (3.6)

Using (3.6) in (3.4) and (3.5), the values of )(1 kF and )(1 kG are obtained

iteratively and the other three boundary conditions are written as

1

0

111 1)(or1)1(

n

k

kFf (3.7)

1

0

11 0)1()1(or0)1(n

k

kFkf and (3.8)

n

k

kGg0

11 0)(or0)1( (3.9)

Solution of micropolar fluid flow through porous channels 3297

For n=8, solving (3.7), (3.8) and (3.9), the values of 1b , 1d and 1A are obtained

and using inverse differential transform (3.3), the solutions for )(gand)( 11 f are

obtained. Here, the skin-friction is defined as

2 2

2 ( / )

/ 2 hU

wall wallf

uc

U

(3.10)

where /wall

u is computed by using (2.9).

3.2 Case (ii) 2 1

V V

In this case (2.26) and (2.27) can be transformed into

1

1

2 1 2

1 1 2 1 2 1

0

2 2

1 1 2 1 2 1

0

( 1)( 2)( 3) ( 3) ( 1) ( 1)

( 1)( 1) ( 1) ( 1)

( )

( 1)( 2) ( ) ( 2)

k

k

k

k

k k k F k C k G k

k k k F k F k k

R A k

k k k k F k F k k

(3.11)

1 1

2 2 2 2

4 1 2 1 2 1 1 2 1 2 1

0 0

( 1)( 2) ( 2) ( 1)( 2) ( 2) 2 ( )

( 1) ( ) ( 1) ( 1) ( 1) ( )k k

k k

k k G k C k k F k G k

C k k F k G k k k F k G k k

(3.12)

and the first two boundary conditions from (2.28) and the first condition from

(2.29) are transformed into the first three conditions in the following:

2 2 2 2(0) 1 , (1) 0, (0) 0F F G & 2222 )1(,)0( dGbF . (3.13)

Using (3.13) in (3.11) and (3.12), the values of )(2 kF and )(2 kG are obtained

iteratively and the other three boundary conditions are written as

n

k

kFf0

22 1)(or1)1( (3.14)

1

0

22 0)1()1(or0)1(n

k

kFkf and (3.15)

n

k

kGg0

22 0)(or0)1( (3.16)

3298 P. Vimala and P. Blessie Omega

For n=8, solving (3.14), (3.15) and (3.16), the values of 2b , 2

d and 2A are

obtained and using inverse differential transform (3.3), the solutions for

)(gand)( 22 f are obtained. In this case also, the skin-friction is given by

(3.10). Here /wall

u is computed using (2.21).

4 Results and Discussion

In case (i), 21 VV the lower wall velocity is greater than or equal to the upper

wall velocity in magnitude and in case (ii), 12 VV the upper wall velocity is

greater than or equal to the lower wall velocity in magnitude. 0&0 21 VV ,

correspond to injection at the lower wall and suction at the upper wall.

0&0 21 VV , correspond to injection at both walls. 0&0 21 VV ,

correspond to suction at both walls. 0&0 21 VV , correspond to suction at

lower wall and injection at the upper wall. In case (i),

)/(11 hVR ,2 11 / 1V V , )(1 f represents the normal velocity

component and )(g1 represents the microrotation. Similarly, in case (ii),

)/(22 hVR ,1 22 /1 V V , )(2 f represents the normal velocity

component and )(g2 represents the microrotation. If 1 0R or 2 0R , the

problem in the respective cases reduce to the channel flow with impermeable

walls. If1 2

V V , then 1 22, 2 in the respective cases. This reduces the

problem to the symmetrical injection and suction cases. If1 2

V V , then

1 20, 0 . This represents suction at one wall and equal injection at the other

wall. The axial flow is identical to the flow through impermeable flat plates, but

there is also a constant flow in the y-direction.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f 1 ()

R1= 10,

1= -0.8

R1= 1,

1= -0.8

R1= 1,

1= -0.2

R1= 10,

1= -0.2

C1=1.7,C

2=1.8,C

3=1.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

f 1 ' ()

R1=10,

1= - 0.8

R1=10,

1= - 0.2

R1=1,

1= - 0.8

R1=1,

1= - 0.2

C1=1.7,C

2=1.8,C

3=1.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

g1 () R

1= 1,

1= - 0.2

R1= 10,

1= - 0.8

R1= 1,

1= - 0.8

R1= 10,

1= - 0.2

C1=1.7,C

2=1.8,C

3=1.9

Figure 2 Normal Velocity Figure 3 Axial Velocity Figure 4 Microrotation

Figure 2 shows that the normal velocity increases with Reynolds number for

8.01 . Further increase in normal velocity is observed with an increase in the

Solution of micropolar fluid flow through porous channels 3299

value of 1 . In Figure 3 injection velocity at the lower wall is greater than the

suction velocity at the upper wall and the axial velocity profile becomes

asymmetric with respect to the middle of the channel and it is pushed towards the

upper wall. As 1R is decreased for fixed values of 1 the profile becomes more

asymmetric. For fixed values of 1R , as 1 is increased, the profile becomes less

asymmetric. It is observed that the velocity increases as the value of 1R increases

up to a certain value of (near lower wall). After that the velocity decreases due

to suction at one wall and injection at the other wall.

The behavior of microrotation for different values of 1 and 1R is shown in

figure 4. The shear stress at the two walls tends to rotate the fluid in opposite

directions because of which the microrotation has opposite signs near the walls. In

this case, as the lower wall injection velocity is greater than the upper wall suction

velocity, microrotation in the fluid near the lower wall increases with 1 for

fixed 1R whereas that near the upper wall decreases with an increase in 1 for

a fixed 1R . For a fixed 1 , with an increase in 1R , the microrotation decreases and

this decrease is significant in the middle of the channel and negligible near the

upper wall, whereas the trend is reversed with negligible difference near the lower

wall.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

f 2 ()

R2=10,

2 = 0.8

R2=1,

2 = 0.8

R2=1,

2 = 0.2

R2=10,

2 = 0.2

C1=1.7,C

2=1.8,C

4=1.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

f 2 ' ()

R2=10,

2=0.8

R2=1,

2=0.8

R2=1,

2=0.2

R2=10,

2=0.2

C1=1.7,C

2=1.8,C

4=1.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

g2 ()

R2=1,

2= 0.8

R2=10,

2= 0.8

R2=1,

2= 0.2

R2=10,

2= 0.2

C1=1.7,C

2=1.8,C

4=1.9

Figure 5 Normal Velocity Figure 6 Axial Velocity Figure 7 Microrotation

Figure 5 shows that the normal velocity decreases with increase in Reynolds

number for 2.02 . Further decrease in normal velocity is observed with an

increase in 2 . In figure 6 suction velocity at the upper wall is greater than the

injection velocity at the lower wall and the axial velocity profile becomes

asymmetric with respect to the middle of the channel and it is pushed towards the

upper wall. As 2R is increased for fixed values of 2 the profile becomes

more asymmetric. For fixed values of 2R , as 2 is increased, the profile

becomes less asymmetric. It is observed that the velocity decreases as the value of

2R increases up to a certain value of (near lower wall). After that the velocity

increases due to suction at one wall and injection at the other wall.

3300 P. Vimala and P. Blessie Omega

The behavior of microrotation for various values of 2 and 2R is shown in

figure 7. The shear stress at the two walls tends to rotate the fluid in opposite

directions because of which the microrotation has opposite signs near the walls. In

this case, as the lower wall injection velocity is less than the upper wall suction

velocity, microrotation in the fluid near the upper wall decreases with 2 for

fixed 2R whereas that near the lower wall increases with an increase in 2 for

a fixed 2R . For a fixed 2 , with an increase in 2R , the micro rotation decreases

and this decrease is significant in the middle of the channel and negligible near the

upper wall, whereas the trend is reversed with negligible difference near the

lower wall.

In tables 1 and 2, the values of skin friction 1 (0)f , 1 (1)f , 2 (0)f and 2 (1)f at the

two walls in their respective cases are presented with their corresponding

Newtonian results. The results obtained from the present theory with C1 = 0 show

good agreement with the Newtonian results. When n=17, the Newtonian results

and the present results are found to be exactly the same. This validates the

solution by the present method.

Table 1 Skin Friction – A Comparison with Newtonian Result (Case i)

1 1,R 1 (0)f

1 (1)f

Newtonian

(n=17)

Present Result with C1=0 Newtonian

(n=17)

Present Result with C1=0

n=8 n=11 n=14 n=17 n=8 n=11 n=14 n=17

1 , -0.2 –1.0362 –1.0351 –1.0362 –1.0362 –1.0362 1.3921 1.397 1.3921 1.3921 1.3921

1 , -0.8 –4.3415 –4.3222 –4.3426 –4.3415 –4.3415 5.2593 5.3334 5.2532 5.2591 5.2593

10 , -0.8 –2.6848 –2.766 –2.616 –2.7627 –2.6848 9.1636 9.387 11.232 8.1328 9.1636

Table 2 Skin Friction – A Comparison with Newtonian Result (Case ii)

2 2,R

2 (0)f 2 (1)f

Newtonian

(n=17)

Present Result with C1 = 0 Newtonian

(n=17) Present Result with C1 = 0

n=8 n=11 n=14 n=17 n=8 n=11 n=14 n=17

1 , 0.2 1.036 1.0369 1.036 1.036 1.036 –1.4001 –1.3961 –1.4001 –1.4001 –1.4001

10 , 0.2 0.4891 0.5592 0.5139 0.4968 0.4905 –4.3644 –2.8171 –3.5305 –4.0047 –4.259

1 , 0.8 4.3558 4.3556 4.354 4.3556 4.3558 –5.3610 –5.3546 –5.3708 –5.3628 –5.3612

5 Conclusion

A semi-analytical solution by Differential Transform Method (DTM) for the two

dimensional steady laminar flow of an incompressible micropolar fluid between

parallel porous walls with variable permeability has been used. This method of

solution overcomes the errors and extensive computing involved in the usual

numerical procedures with discretization techniques. Also, DTM does not require

exertion of any small parameters as in perturbation methods. Solving the coupled

Solution of micropolar fluid flow through porous channels 3301

set of non-linear partial differential equations is generally difficult. Hence, the

problem has been reduced to a system of non-linear ordinary differential

equations using Berman’s similarity transformation and the resulting problem has

been solved using DTM with ease. This semi-analytical DTM provides a simple

iterative method of solution for the problem. On the whole, this paper portrays the

efficiency of the DTM for solving such problems governed by highly non-linear

coupled set of partial differential equations by reducing to a system of ordinary

differential equations. However, it can also be solved using DTM in two

dimensions without reducing to ordinary differential equations. There is also

scope for solving the above problem using a multi-step DTM which gives a better

approximation than DTM.

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Received: December 15, 2014; Published: April 20, 2015