J. Appl. Environ. Biol. Sci., 5(10)109-121, 2015

© 2015, TextRoad Publication

ISSN: 2090-4274

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and Biological Sciences

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*Corresponding Author: Aneela Shakir, Department of Mathematics, Abdul WaliKhan University, Mardan, Khyber

Pakhtunkhwa, Pakistan

Analysis of MHD and Thermally Conducting Unsteady thin film Flow in a

Porous Medium

Aneela Shakir1 , Taza Gul2, S.Islam2

1Department of Mathematics, ISPaR/ Bacha Khan University, Khyber Pakhtunkhwa, Pakistan 2Department of Mathematics, Abdul WaliKhan University, Mardan, Khyber Pakhtunkhwa, Pakistan

Received: March 12, 2015

Accepted: August 30, 2015

ABSTRACT

This article deals with Magneto Hydro Dynamic (MHD) thin film unsteady second grade fluid through a porous

medium in the existence of heat.Heat transfer flow in a porous medium has vital applications in biomechanics

like the purification of blood in kidneys and other fields.The governing nonlinear partial (PDE,s) for velocity

and temperature profiles have been obtained. These nonlinear (PDE, s) is solved analytically by using the

Adomian Decomposition technique (ADM) and Optimal Homotopy Asymptotic technique (OHAM). The

comparisons of these two methods for velocity and temperature distribution have been analyzed graphically and

numerically.It has been found that both these solutions are well agreed.The effects of model parameters have

been discussed graphically as well as numerically.

KEYWORDS: Unsteady thin film flows, porous medium, second order fluid, MHD, Lifting, Drainage, ADM,

OHAM, Temperature distribution.

1. INTRODUCTION

The thin film flow of non-Newtonian fluids has been mostly studied in Engineering and industry. The flow of

heat transfer in a porous medium has various applications in engineering and in industry. Frequent studies of such

flows have been found in current years. Recently, out of several models of non-Newtonian behaviors demonstrated

by specific fluids, ample attention has been given to differential type fluids [1]. The second order fluid which

makes a subclass of these fluids of the differential type, has been exhaustively investigated in numerous types of

circumstances.An easy structure of second order, fluids received the special attention of researchers.Some

development and related work of the unsteady non-Newtonian thin film flow on moving and oscillating vertical

belt discussed by TazaGul et al. [2]. They show the compression between (ADM) and (OHAM) for both velocity

and temperature profiles. Fetecau & Zierep [3] determined the velocity field of second order fluid by using the

Fourier sine transformation. Ali et al. [4], the heat transfer investigation of unsteady flow a second order fluid on

vertical oscillating plates and closed form solution obtained by using Laplace transforms. Erdog and Imrak [5]

investigated the properties of unsteady one directional flows of second order fluid. They obtained the velocity field

for couette and peaceable flows. Farooq et al. [6], studied the thin film flow using Oldroyd 8-constant fluid model

on a vertically moving belt and solved the problem by using VIM and ADM.

The analysis of electrically conducting fluid in the with MHD has various importance in Engineering and

technology such as magneto-hydrodynamic generators, nuclear reactors and MHD pumps. In literature many

researchers have studied the unsteady flows of second order fluid in different structures which contain the

effects of MHD. Hang and Liao [7] determined the analytical solution of unsteady magneto-hydrodynamic flows

of non-Newtonian fluids using HAM and studied the effect of magnetic parameter.Siddiqui, et al. [8],

investigated unsteady flow electrically conducting second grade fluid.They solved the nonlinear partial

differential equation numerically and show the effect of model parameters graphically. Alam et al. [9], studied

the MHD thin film flow of Johnson Segalman fluid in vertical plate and solved the lift and drainage velocity by

using ADM. The properties of different numbers,and magnetic parameter have been discussed graphically.

Heat transfer analysis and magneto-hydrodynamics in different non-Newtonian flows in porous medium have

been taken a particular interest from researchers. Faisal et al. [10], determined the exact solutions of magneto-

hydrodynamics second order fluid in a porous medium for the velocity field. Fourier sine and Laplace

transformation technique is used for the solution of the velocity field. Zaman et al. [11] examined the Stokes first

problem of an unsteady MHD third grade fluid in a non-porous half space in the presence of hall current. Hay

obtained Analytical solution by using HAM and studied the effect of the dimensionless parameter on the velocity

profile numerically.Rahman [12] investigated numerically the results of magneto-hydrodynamic on thin films of

unsteady micropolar fluid over a porous medium in three different geometries. Using shooting method the velocity

profile has been solved numerically and shows the result graphically. Gul et al. [13] studied the heat transmission

investigation and MHD thin film flow. The constitutive equations of the third order fluid are solved analytically by

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Shakir et al.,2015

using the Adomian decomposition technique and compared the result with OHAM. Imran et al. [14] discussed the

exact solution of MHD flow of a second order fluid over a porous medium by using Laplace transforms. Samiulhaq

et al. [15] determined the exact solution of MHD second order unsteady fluid through porous media using Laplace

transforms.The result of the velocity field and skin friction are shown graphically. Mahmood and Khan [16]

analytically studied the solution of thin film flow of a non-Newtonian third order fluid through a porous medium

over an inclined plane by using the HAM. Hussain et al. [17] discussed the MHD flows of the second grade fluid in

a porous medium and find the exact solution.The effect of various parameters is shown graphically. Hamza et al.

[18] studied the heat transmission to the MHD oscillatory flow through porous media.They obtained the velocity

field as well as the temperature field analytically.They discussed the effect of different parameters on velocity field

,the frequency of the oscillation and Peclet number. Makinde and Maone [19] investigated the MHDoscillatating

flow and heat transmission with a porous medium.The solutions obtained for velocity and temperature profiles is

of the close form. Stokes first problem has been investigated by Tan and Masuoka [20] in a porous half-space for a

second grade fluid with heated flat plate and obtained the exact solution.

Aiyesmi et al. [21] discussed heat transmission analysis of unsteady MHD thin film flow of a third order

fluid down an inclined plane and obtained equations for velocity and temperature distribution.

The aim of the current research is to analyze heat transfer and the influence of magneto-hydrodynamic

unsteady thin films flow of second grade fluid through a porous medium on vertical belt. (ADM) is being

employed to find the velocity and temperature profile.Adomian and Rach [22, 23] presented the ADM for the

approximate solutions for linear and nonlinear boundary value problems.ADM had been used by Wazwaz [24,

25] for reliable treatment of Bratu-type and propose an algorithm for nonlinear boundary value problem.ADM

has been discussed by Hasan [26] for solving some linear and non- linear boundary value problems. A complete

parametric study has been carried out to show the effect of physical parameter like stock number, magnetic

parameter, second grade perameters, Brinkman and Prandtl numbers and porosity on both velocity and

temperature distribution.

2. Formulation of lifting problem Suppose,the flow of second order fluid under the effect of Magneto Hydro Dynamic on a porous vertical

belt. The belt is oscillating as well as moving upward with uniform velocity .V During its upward motion, the

belt carries along a thin film of second order fluid having constant thickness δ .Constant magnetic field

( )00,B ,0=B is acting transversely on the belt as mentioned in [2]. 2

00, ( , ), 0B v x tσ × = − J B is the

Lorentz force per unit volume, whereσ is the electrical conductivity and the electric field is not considered. We

select x-axis and y-axis perpendicular and parallel to the belt respectively. Furthermore, we assumed that the

flow is unsteady, laminar and pressure gradient is constant. The Darcy resistance [27-30] for second-grade fluid

in the case of porous medium is given by

1 .v

rt k

ξµ α

∂ = − + ∂ (1)

Here k and ξ show permeability and porosity of the porous medium respectively.

The velocity field and temperature distribution are:

{ }= ; 0,v(x,t),0 Θ = Θ(x,t).v (2)

Using the boundary conditions

cos ,at 0 and 0, at ,(x,t)

(x,t) V V t x xx

ω δ∂

= + = = =∂

vv (3)

0 1( , ) , at 0 and ( , ) , at .x t x x t x δΘ = = Θ = =Θ Θ (4)

ω is the used as the frequency of the oscillating belt andΘ is temperature.

Using the above assumption, continuity equation is identically satisfied where the

Navier Stokes equation and heat equation are reduced to

2

0

vv + ,yxT B r

t xρ ρ σ∂ ∂

= − −∂ ∂

g (5)

2

2

v.p yxc k T

t xxρ

∂Θ Θ ∂ ∂= + ∂ ∂ ∂ (6)

The componentyxT of the Cauchy stress tensor T is obtained as

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J. Appl. Environ. Biol. Sci., 5(10)109-121, 2015

1

v v.xy yxT T

y t yµ α

∂ ∂ ∂= + = ∂ ∂ ∂

(7)

Inserting Eq. (7) in Eq. (5), we get

2 22

1 02 2

v v vv + ,B r

t x t xρ µ α ρ σ

∂ ∂ ∂ ∂= + − − ∂ ∂ ∂ ∂

g (8)

Eq. (6) becomes 22

12

v v v.pc k

t x t x xxρ µ α

∂Θ Θ ∂ ∂ ∂ ∂= + + ∂ ∂ ∂ ∂ ∂ ∂ (9)

Introducing the following non-dimensional physical quantities

(10)

Here M shows magnetic parameter,α represent non-dimensional variable, Φ is the porosity parameter, ,t rS B

and rP the Stock ,Brinkman and Prandtl numbers respectively .

Applying dimensionless variables on Eq. (8) and (9) and dropping bars to obtain the velocity and temperature

fields as

2 2

2 2,t

v v v vv Mv S

t x t x tα α

∂ ∂ ∂ ∂ ∂ = + −Φ −Φ − − ∂ ∂ ∂ ∂ ∂ (11)

2 22

2.

r r

v v vP B

t x t x xxα

∂Θ Θ ∂ ∂ ∂ ∂= + + ∂ ∂ ∂ ∂ ∂ ∂ (12)

(3) and (4) are reduced as

1 cos ,at 0 and 0, at 1,v(x,t)

v(x,t) t x xx

ω∂

= + = = =∂

(13)

( , ) 0, at 0 and ( , ) 1, at 1.x t x x t xΘ = = Θ = = (14)

Figure 1. Lift Geometry Figure 2. Drainage Geometry

Basic Idea of ADM:

The ADMtechnique is used to decompose the unknown function ( , )v x t into its components as

0( , ) ( , ).nn

v x t v x t∞

==∑ (15)

To understand the basic concept of ADM, we suppose the nonlinear PDEs in an operating system as

( , ) ( , ) ( , ) ( , ) ( , ),t x

L v x t L v x t Rv x t Nv x t g x t+ + + = (16)

111

( )

2 2 2 2

01

2 2

2

0

1 0 1 0

v, , , , , , , ,

, .

p

t r

r

cBx t gv x t M S P

V V k k

VB

k

µσ δαµ δ ρ δ ξα

δ ρδ ρδ µ µ

µ

= = = = = = Φ = =

Θ−Θ= Θ =

Θ −Θ Θ −Θ

Shakir et al.,2015

( , ) ( , ) ( , ) ( , ) ( , ).x tL v x t g x t L v x t Rv x t Nv x t= − − − (17)

Here 2

2

xLx ∂

∂= and

tLt ∂

∂= , ),( txg shows source term, while the remaining linear term is ),( txRu and

the non-linear term is ),( txNu .

Applying1

xL− on Eq. (17)

1 1 1 1 1( , ) ( , ) ( , ) ( , ) ( , ),x x x x t x xL L v x t L g x t L L v x t L Rv x t L Nv x t− − − − −= − − − (18)

1 1 1( , ) ( , ) ( , ) ( , ) ( , ),x t x xv x t K x t L L v x t L Rv x t L Nv x t− − −= − − − (19)

Here ( , )K x t function denotes the terms arising from ),(1 txgLx

−by applying dxdxLx ∫∫=−

(.)1

. Since ADM

defines the ( , ) and ( , )v x t Nv x t , by infinite series as

( )0 0

( , ) ( , ) and N ( , ) ,n nn nv x t v x t v x t A

∞ ∞

= == =∑ ∑ (20)

Where nA is adomian polynomial and Eq. (19), becomes

1 1

0 0 0( , ) ( , ) ( , ) ,n x n x nn n n

v x t K x t L R v x t L A∞ ∞ ∞− −

= = == − −∑ ∑ ∑ (21)

The components form of Eq. (21) is

( )1

0 1 2 0 1 2

1

0 1 2

( , ) ( , ) ( , ) ..... ( , ) ( , ) ( , ) ( , ) ...

( ...),

x

x

v x t v x t v x t K x t L R v x t v x t v x t

L A A A

−

−

+ + + = − + + +

− + + + (22)

By comparison, both sides of Eq. (22) we get zero, first , second order components of the velocity profile

0 ( , ) ( , ),v x t K x t=

1 1

1 0 0( , ) [ ( , )] [ ],x xv x t L R v x t L A− −= − −

1 1

2 1 1( , ) [ ( , )] [ ],x xv x t L R v x t L A− −= − −

1 1

3 2 2( , ) [ ( , )] [ ],x xv x t L R v x t L A− −= − −

and so on . (23)

2.1. Lift velocity problem and ADM solution We apply ADM method on Eq. (11) and (12). Consider a nonlinear partial differential Eq. (11), (12) in operator

form

2

2( , ) ( , ) ,x

v vL v x t K x t v Mv

t y tα α

∂ ∂ ∂ = − +Φ +Φ + ∂ ∂ ∂ (24)

2 2

( , ) .x r r

v v vL x t P B

t x t x xα

∂Θ ∂ ∂ ∂ Θ = − + ∂ ∂ ∂ ∂ ∂ (25)

Applying the inverse operator on Eq. (24) and (25) we get

21 1 1 1 1

2( , ) (x, t) ,x x x x x x

v vL L v x t K L L v L ML v

t x tα α− − − − − ∂ ∂ ∂ = − +Φ +Φ + ∂ ∂ ∂

(26)

2 21 1 1( , ) .x x r x r x

v v vL L x t P L B L

t x t x xα− − −

∂Θ ∂ ∂ ∂ Θ = − + ∂ ∂ ∂ ∂ ∂ (27)

For the solution of velocity and temperature profile in series form is

0 0

(x, t) (x, t) and (x, t) (x, t).n n

n n

v v∞ ∞

= =

= Θ =Θ∑ ∑ (28)

From Eq. (35) and (36) we can write the Adomian polynomials as 22 2

20 0 0

, , .n n n

n n n

v v v vA E F

t x x x t x

∞ ∞ ∞

= = =

∂ ∂ ∂ ∂ ∂ = = = ∂ ∂ ∂ ∂ ∂ ∂ ∑ ∑ ∑ (29)

The velocity and temperature distribution in components form as

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J. Appl. Environ. Biol. Sci., 5(10)109-121, 2015

[ ][ ]

10 1 20 1 2

10 1

( , ) ( , ) ( , ) ... ( , ) ...

( , ) ( , ) ... ,

x

x

x t x t x t K x t Lv v v A A A

ML x t x tv v

α −

−

+ + + = − + + +

+ + + (30)

[ ][ ]

10 1 20 1 2

0 1 2

( , ) ( , ) ( , ) ... ( , ) ...

... .

r xx t x t x t h x t B L E E E

F F Fα

− + + + = − + + + +Θ Θ Θ

+ + + (31)

Comparing both sides of Eqs. (30), and (31), we get the velocity and temperature components. we take it only

up to second order

( ) ( )0 , ,v x t K x t= , (32)

( ) ( )0 , , ,x t h x tθ = (33)

( ) [ ] [ ] [ ]1 1 1 101 0 0 0, ,x x x x

vv x t ML v L v L aL A

tα− − − −∂ = +Φ +Φ − ∂

(34)

( ) 1 1 101 0 0, E ,r x r x r xx t P L B L B L F

t

θθ − − −∂ = − − ∂

(35)

( ) [ ] [ ] [ ]1 1 1 112 1 1 1, ,x x x x

vv x t ML v L v L aL A

tα− − − −∂ = +Φ +Φ − ∂

(36)

( ) 1 1 112 1 1, E .r x r x r xx t P L B L B L F

t

θθ − − −∂ = − − ∂

(37)

Solution of Eq. (32)- (37) by applying the boundary conditions from Eq. (13) and (14) are

( ) [ ] [ ] 2

0 , 1 os 1 cos ,2 2

t tS Sv x t c t t x xω ω = + − + + +

(38)

( )0 , ,x t xθ = (39)

( ) ( ) ( )

( ) ( ) ( )

( )

2

1

2

2

2

2 2, cos sin 1 cos

3 2 2 3 2 24 24

cos cos sin 1 cos sin 12 2 2 3 2 2

1cos

3 2 12 12

t t

t t

MS St t tv x t M x

t t t t tM x

MS StM

ω ω ω ωα

ω ω ω ω ω ωα α

ω

Φ = + Φ − −Φ + +

+ +Φ − − Φ + + Φ

Φ − −Φ − − ( )3 4 ,

24

tMSx M x

+ +Φ

(40)

( ) [

[

4 3 2

1

3 4 22

2, 2cos 2 cos sin cos cos

2 2 2 3 2 6 2

sin 2 cos sin 2cos cos2 24 2 2 2 2 2

cos sin2 2

r t

tt t

t t t t tx t B S

St t t t tS x S

t tS

ω ω ω ω ωα ωαω

ω ω ω ω ω ωααω

ω ω

Θ = − + −

+ + − + −

22 22 2

24

cos cos sin8 2 3 2 2 6

.12

t tt t t

t

S St t tx S S x

Sx

ω ωα ω ω + + − +

+

(41)

The solution of the second component of velocity and temperature distribution is too large. So derivations are

given up to first order while, graphical solutions are given up to second order.

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Shakir et al.,2015

2.2. The OHAM solution of lifting problem

Solve the given problem in Eqs. (11) and (12), we use OHAM for finding the velocity components for

different order problems, we obtain, the 0th, 1st and 2ndcomponent problems are 2

0 0

2

( , ): ,t

v x tp S

x

∂=

∂ (42)

20 0

2

( , ): 0,

x tp

x

∂ Θ=

∂ (43)

2 2 221 0 0 0 0 01

1 0 02 2 2 2

( , ): ,t t

v v v v vv x tp S c S Mv v

x t t x t x xα α

∂ ∂ ∂ ∂ ∂∂ ∂= − − + +Φ + + Φ + − + ∂ ∂ ∂ ∂ ∂ ∂ ∂

(44)

( )2 2 22

1 0 0 0 0 013 3 32 2

( , ): 1 .r r

v v vx tp P c B c c

x t x x t x xα

∂Θ ∂ ∂ ∂ ∂ Θ∂ Θ = − − + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ (45)

The solution of zero and the first components of velocity as well as temperature field is given. While the

solution of second component of both profile are too large, so given only graphically.

( ) [ ] [ ] 2

0 , 1 os 1 cos ,2 2

t tS Sv x t c t t x xω ω = + − + + +

(46)

( )0 , ,x t xθ = (47)

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

2

1 1

2 2

2

2 2, cos cos sin 1

3 2 3 2 2 24

cos sin 1 cos cos2 2 2 3 2

cos sin 13 2 2 12

t

t

St t tv x t c M M x

t t t M tM M x M

St tM

ω ω ω ωα

ω ω ω ωα

ω ω ωα

= +Φ − + Φ − +Φ

+ + Φ − +Φ + +Φ

− + Φ − +Φ ( )3 4 ,

24

tMSx M x

− +Φ

(48)

( ) [3 4 2

1 3

4 3 22

, 2 cos sin 2cos cos2 2 2 3 2 6

cos sin 2cos 2 cos sin cos2 2 24 2 2 2 2

cos sin2 2 2

t tr

tt

t

S St t t tx t B c

St t t t t tx S

S t t

ωαω ω ω ωαω

ω ω ω ω ω ωαω

ωα ω ω

Θ = − + +

− + − +

−

22 22 2

24

cos cos sin8 2 3 2 2 6

.12

t t tt

t

S S St t tx S x

Sx

ωαω ω ω + − − +

+

(49)

The values of iC for the lift velocity components are

1 20.9548774554,C 0.0008152351.C = − = −

The values of iC for the lift temperature distribution are

1 2 3 40.0143829706,C 0.0150995207,C 0.9345978812,C 0.0043072243.C = − = = − = −

3 Description of drainage problem

In the second problem, the porous belt is only oscillating. We assumed that thin film draining the vertical

belt due to gravity as mentioned in [2]. Therefore, the stock is taken positive in Eq. (11). Furthermore, an

assumption of this problem is the same as of the above problem.

After non-dimensional sing the boundary conditions for only oscillating belt become as

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J. Appl. Environ. Biol. Sci., 5(10)109-121, 2015

cos ,at 0 and 0, at 1.v(x,t)

v(x,t) t x xx

ω∂

= = = =∂

(50)

3.1 ADM solution of the Drainage problem:

The model for both problems is same; but the there is one difference in drainage problem which is t that

the belt is oscillating. Due to gravity force the stock number is mentioned positively .Write Eq. (11) and (12) in

standard form of Adomian Decomposition Method. The Adomian Polynomials for drainage and lift problem are

same.

Consuming the boundary conditions (50)

( ) [ ] [ ] 2

0 , os cos ,2 2

t tS Sv x t c t t x xω ω = − − −

(51)

( )0 , ,x t xθ = (52)

( ) ( ) ( ) [ [ ]

( ) [ ]( ) [ [ ]( ) [ ]( )

( )

1

2

3 4

1, sin 1 cos cos

3 2 3 2 24 24

1sin 1 sin 1 os

2 6 2

,2 24

t t

t

t t

MS St tv x t M x t

MSM t x t c t M

S Sx M x

ω ω ωα ω

ωω α ω ω α ω

Φ = + Φ − +Φ − − +

+Φ − − Φ + + Φ − −Φ +

Φ + − −Φ

(53)

( ) [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[ ] [ ]

2

1

22

2 23 4

1 1, cos 2 sin 2 cos sin

4 4 4 6 12 24

1 1sin 2 cos 2 cos sin

4 4 4 2 4 8 6

sin 2 cos .3 8 12

t t tr

t t t

t t t

S S Sx t B t t t t x

S S St t t t x

S S St t x x

ωαωαω ω ω ω

ωαωα ωαω ω ω ω

ω ω

Θ = + − − + +

+ − − + − − +

− + −

(54)

3.2 The OHAM solution of drainage problem

By applying the standard form of OHAM on a drainage problem, we construct 0th, 1st and 2nd component

problem as 2

0 0

2

( , ): ,t

v x tp S

x

∂= −

∂ (55)

20 0

2

( , ): 0,

x tp

x

∂ Θ=

∂ (56)

2 2 221 0 0 0 0 01

1 0 02 2 2 2

( , ): ,t t

v v v v vv x tp S c S Mv v

x t t x t x xα α

∂ ∂ ∂ ∂ ∂∂ ∂= + − −Φ − − Φ + − Φ + ∂ ∂ ∂ ∂ ∂ ∂ ∂

(57)

( )2 2 22

1 0 0 0 0 013 3 32 2

( , ): 1 .r r

v v vx tp P c B c c

x t x x t x xα

∂Θ ∂ ∂ ∂ ∂ Θ∂ Θ = − + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ (58)

The solution is given up to first components, while the solution of the second component is large, so it shown

just graphically.

( ) [ ] [ ]0 2

0: , os cos ,2 2

t tS Sp v x t c t t x xω ω = − − −

(59)

( )0 , ,x t xθ = (60)

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Shakir et al.,2015

( ) [ ]( ) [ ]( ) [ ]

( ) [ ]( ) [ ]( ) [ ]( )

( )

1

1

2

3 4

1 1: , cos sin 1 cos

3 3 24 24 2

1sin 1 os sin 1

2 6 2

,2 24

t t

t

t t

MS Sp v x t t M t x t

MSM t x c t M t

S Sx M x

ωω ω α ω

ωω α ω ω ω α

Φ = +Φ − + Φ + + −

+Φ + + Φ + +Φ − + Φ +

Φ + + +Φ

(61)

( ) [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [

[ ] [ ]

2

1

22

22 3 4

1 1, cos cos 2 sin 2 sin

6 4 4 12 24 4

1 11 sin 2 cos 2 2 cos sin

4 2 6

sin cos .12

t t tr

tt t

tt t

S S Sx t B t t t t x

St t S t S t x

St S t S x x

ωαωαω ω ω ω

ωα ω ω ω ωα ω ωα

ω ω

Θ = − + − − −

+ − + − + + −

− + +

(62)

The values of iC for the lift velocity components are

1 20.9581265072,C 0.0006733414.C = − = −

The values of ic for the lift temperature distribution are

1 2 3 40.0275769521,C 0.0287710716,C 0.9292482971,C 0.0050987101.C = − = = − = −

Figs.3,4: Comparison graphs of OHAM and ADM methods for

0.2, 0.02, 0.5, 0.5, 0.4, 10, 0.8, 1.5.t r rM S t P Bω α= = = = Φ = = = =

Figs. 5, 6: Comparison of ADM and OHAM methods in case of drainage for “V” (on left) and for

“T”distribution (on right) when

0.3, 0.02, 0.9, 0.5, 0.4, 10, 0.8, 1.5.t r rM S t P Bω α= = = = Φ = = = =

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J. Appl. Environ. Biol. Sci., 5(10)109-121, 2015

Figure 7, 8: Lift velocity distribution of fluid (on the left) and temperature distribution of fluid (on the right) at

different time level when 0.2, 0.02, 0.5, 0.6, 0.4, 10, 0.8, 1.5.t r rM S t P Bω α= = = = Φ = = = =

Figs 9, 10: Lift velocity distribution of fluid (on left) and drainage casesof fluid (on right) at various values of

0.2, 0.5, 0.5, 0.4, 0.02, 0.8, 1.5.t r rM S P Bω α= = = Φ = = = =

Figs 11, 12: Lift temperature distribution of fluid (on left) and drainage temperature distribution of fluid (on

right) when 0.2, 0.02, 0.5, 0.5, 0.3, 0.6, 4.t r rM S P Bω α= = = = Φ = = =

Figure 13, 14: Effect of � on the Lift velocity profile (on left) and drainage velocity field (on right) by taking

0.2, 0.02, 0.5, 0.7, 10, 0.8, 1.5.t r rS t P Bω α= = = Φ = = = =

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Figure 15, 16: Effect of the porosity � on the Lift velocity profile (on left) and drainage velocity profile (on

right) by taking 0.2, 0.02, 0.7, 0.8, 0.5, 0.8, 1.5.t r rM S x P Bω α= = = = = = =

Figure 17, 18: Effects of the tS on the Lift velocity fields (on left) and drainage velocity profile (on right) by

taking 0.3, 0.02, 0.5, 0.6, 0.3, 0.8, 1.5.r rM x P Bω α= = = Φ = = = =

Figure 19, 20: Effect of rB on theLift temperature field (on left) and drainage velocity profile (on right) by

taking 0.4, 0.2, 0.05, 0.5, 0.5, 0.4, 2.t rx M S Pω α= = = = = Φ = =

Figure 21, 22: Effect of the rP on the Lift temperature field (on left) and drainage velocity profile (on right) by

taking 0.3, 0.2, 0.02, 0.5, 0.5, 0.4, 1.5.t rx M S Bω α= = = = = Φ = =

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4. RESULTS AND DISCUSSION

Magneto-hydrodynamic unsteady (thin film) flows of second order fluid on a porous vertical oscillating

and moving belt have been discussed for velocity and temperature fields. Fig, 1 & 2 shows the geometry of the

problems respectively.The solutions of the velocity as well as temperature distribution are studied graphically

using ADM and OHAM.The graphical comparisons of these methods are shown in Fig 3-6 at various values of

physical parameters , , , , .t r rM S B and Pω Φ It is clear from Tables 1-4 and from Figs 3-6 that these two

methods are will agreed.The effects of some physical dimensionless parameters discussed graphically in figs 7-

22 like magnetic parameter M , porosity parameter Φ , frequency parameter�, stock number tS ,Brinkman

number rB and Prandtl number rP on velocity and temperature distribution. The effect of different time levels for

velocity distrubationareobserved in Figs 7-10 while the influence of various time level on the temperature field

specified in Fig 11-1 2 for both lift and drainage cases.

At the solid boundary due to the no-slip boundary condition,the flow along with the belt renders oscillation

in the same phase and the amplitude.Decresing the amplitude of velocity quickly increase with the distance from

the belt.Whileflow of the second order fluid in the whole flow domain oscillates with the driving belt

movement.

Figs 13 & 14 show the influence of magnetic parameteron the lift and drainage velocity field.The increase

in M (magnetic parameter) increases the lift velocity while an increase in M decreases the drainage motion

because the action of the magnetic field have to control the shearing of a narrow layer near the boundary.Figs

15-16 show the influence of porosity on first and second problem.In both lifting and drainage problems, the

velocity field decreases with increase in porosity parameters. Generally,thin film flows for the viscus fluid

comparatively thinner than that for the said fluid.

The influence of tS , on lifting and the drainage fluid motion is discussed in Fig 17-18. The fluid motion

increases with the increase in stock number and this is happening in lift velocity while in drainage the fluid

motion decreases. Figs 19-20 show the influence ofrB in the form of temperature distribution for lift as well as

drainage problems respectively. When Brinkman number increases, then temperature distribution also increases

for both lift and drainage cases. The effect of Prandtl number rP for both problems are shown in Figs 21-22.

5. Conclusion:

We have studied the heat analysis of unsteady MHD thin film flow of a second order fluid over porous

medium on a oscillating and moving vertical belt. The belt is oscillating and translating in lifting problem, but in

case of drainage problem the belt is only oscillating. From the modeling we get nonlinear PDE,s. Analytical

solution is obtained by using ADM and OHAM for velocity and temperature fields. Compare the result

numerically and graphically. We plot the effect of the physical dimensionless parameters like porosity parameter

Φ , frequency parameterω , stock numbertS , Brinkman number

rB and Prandtl numberrP for velocity and

temperature distributions.

Generally, a far thicker boundary layer is demonstrated by the non-Newtonian fluid, denoting a superior

and more visible viscosity than the viscose fluids. An increase in the second order model material parameterα ,

helps more thickening in the boundary layer.

Conflict of interests: The authors of this article do not have any conflict of interests.

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