Differential Transform Method for MHD Boundary-Layer Equations: Combination of the DTM and the Padé Approximant

Mohammad Mehdi Rashidi

Engineering Faculty of Bu-Ali Sina University Hamedan, Iran

e-mail: mm_rashidi@yahoo.com

Abstract—The purpose of this study is to implement a new analytical method (DTM-Padé technique which is a combination of the differential transform method and the Padé approximant) for solving magnetohydrodynamic boundary-layer flow. We will show that the DTM solutions only valid for small values of independent varible, therefore the DTM not applicable for solving MHD boundary-layer equations. Numerical comparisons between the DTM-Padé and the classical fourth-order Runge–Kutta reveal that the new technique is a promising tool for solving MHD boundary-layer equations.

Keywords-DTM-Padé; MHD; Boundary-layer

I. INTRODUCTION Magnetohydrodynamics (MHD) is the study of the

interaction of conducting fluids with electromagnetic phenomena. The flow of an electrically conducting fluid in the presence of a magnetic field is of importance in various areas of technology and engineering such as MHD power generation, MHD flow meters, MHD pumps, etc [1-6]. The viscous flow due to stretching boundary is important in extrusion processes where sheet material is pulled out of an orifice with increasing velocity. If the boundary velocity is linear with respect to a fixed point, exact solutions of the Navier–Stokes equations may be obtained [7-8].

The concept of the DTM was first proposed by Zhou [9], who solved linear and nonlinear problems in electrical circuit problems. Chen and Ho [10] developed this method for partial differential equations and Ayaz [11] applied it to the system of differential equations, the validity of the DTM is independent of whether or not there exist small parameters in the considered equation. The motivation of this letter is to extend the DTM to solve the MHD boundary-layer equations, for this purpose, we introduce a new analytical method (DTM-Padé). In this letter, DTM-Padé is employed to give series solution for MHD boundary-layer flow.

II. GOVERNING EQUATIONS Let us consider the MHD flow of an incompressible

viscous fluid over a stretching sheet at 0.y The fluid is electrically conducting under the influence of an applied magnetic field ( )B x normal to the stretching sheet. The governing equations are

0,u vx y

220

2 ( ) ,Bu v v

u v g T T ux y y

2 *

2

0

( )

1( ) ,

p

r

p p

T T T uu v T Tx y cy

Q qT T

c c y

where u and v are the velocity componenets in the x and y directions respectively, ,T , , pc and are the temperature, kinematic viscosity, fluid density, specific heat at constant pressure and thermal diffusivity, respectively.

,k , , 0B and g are the fluid thermal conductivity, volumetric expansion coefficient, electric conductivity, applied magnetic induction and the gravitational acceleration, respectively. rq and T are the thermal radiation and the free stream temperature.

By using Rosselant approximation, the radiative heat flux rq is given by

* 4

*

4 ,3r

Tq

yk

where * is the Stefan-Boltzmann constant and *k is the mean absorption coefficient. The obtained Taylor series expansion for 4 ,T neglecting higher order terms, is

4 3 44 3 ,T T T T

where the higher-order terms of the expansion are neglected. By employing (4) and (5), (3) becomes

2009 International Conference on Signal Processing Systems

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2 *

2

* 3 20

* 2

( )

16( ) .

3

p

p p

T T T uu v T Tx y cy

Q T TT T

c c k y

The boundary conditions corresponding to governing

equations are

0

( ,0) , ( ,0) ,( , ) 0, ( , ) ,( ,0) ( ) ,

w

w

u x a x v x vu x T x TT x T x T A x

where a is the stretching rate (a constant), ( )wT x is the wall temperature. Using the similarity variables

/ , ( ),

( ) ,w

t a y a x f tT T

tT T

where , ,y xu v (1)-(3) are trasformed to

2

2

( ) ( ) ( ) ( )( ) ( ) 0,

f t f t f t f t

M f t Gr t

1( ) ( ) ( ) (1 )

Pr( ) ( ) ( ) 0,

Rx

Nt f t t

f t t t

where the primes denote differentiation with respect to t and

0

* 3

2 *

*0

, Pr ,

( ) 16, ,

3

, ,

wR

xp p

M Ba

g T T TGr N

a x k k

Qxc c a

are the Hartmann number, Prandtl number, Grashof number, thermal radiation parameter and heat generation or absorption coefficients, respectively.

If 0 0,M f the solution for ( )f t can be written as

( ) 1 exp( ).f t t

We shall solve the nonlinear differential equations (9)-(10) using DTM, DTM-Padé and numerical method (fourth order Runge-Kutta).

III. THE DIFFERENTIAL TRANSFORM METHOD Transformation of the kth derivative of a function in one

variable is as follows

0

1 ( )( ) [ ] ,!

k

t tkd f t

F kk dt

and the inverse transformation is defined by

00

( ) ( ) ( ) ,k

k

f t F k t t

00

( ) ( ) ( ) ,i

k

k

f t F k t t

where ( )F k is the differential transform of ( ).f t Taking differential transform of (9)-(10), we obtain

0

2

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

( 1) ( 1) ( ) 0,

k

r

k k k F k

k r k r F r F k rr k r F r F k r

M k F k Gr k

0

1( 1) ( 2) ( 2)

Pr( 1) ( ) ( 1) (1 )( 1) ( 1) ( )

( ) 0,

R

kx

r

Nk k k

k r F r k rk r F k r r

kAfter finding the DTM solutions for (16)-(17), the Padé

approximant must be apllied. The Padé approximants, that often show superior performance over series approximations, provide a successful tool and promising scheme for identical applications.

IV. RESULTS AND DISCUSSION The system of transformed governing equations (16)-

(17) with transformed boundary conditions are solved analytically and numerically. Figs. 1-2 show the analytical solution of ( )f t obtained by the DTM-padé and the DTM for different values of wall mass transfer coefficient (such that 0 0f indicates wall suction and 0 0f corresponds to wall blowing conditions). We can see that the DTM solutions is only valid for small values of independent variable ( t ). In the Figs. 3-4, the comparison between the DTM-padé and the numerical solutions is presented, the results obtained by the DTM-padé have good agreement with the numerical results for all values of .t

Figs. 5-6 show the analytical solution of ( )t obtained by the DTM-padé and the DTM for different values of wall mass transfer coefficient, the DTM solutions, can’t satisfy infinity boundary condition ( ( ) 0 ), on the other hand, for the DTM-padé solutions, ( ) 0 , see Figs. 7-8.

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The influence of the presence of a heat source ( 0)or a heat sink ( 0) in the boundary layer on the velocity and temperature fields is presented in Figs. 9 and 10, respectively. The presence of a heat source in the boundary layer generates energy which causes the temperature of the fluid to increase. This increase in temperature produces an increase in the flow field due to the buoyancy effect.

V. CONCLUSION

In this paper, a new analytical method (the DTM-padé)is used to find analytical solutions of magnetohydrodynamics boundary-layer equations. The DTM combined with padé approximants are also shown to be a promising tool in solving two point boundary value problem consisting of systems of nonlinear differential equations. Without using padé approximation, the analytical solution obtained by the DTM, can’t satisfy infinity boundary conditions ( ( ) 0, ( ) 0f ).

REFERENCES

[1] T. Hayat, C. Fetecau, M. Sajid, “Analytic solution for MHD Transient rotating flow of a second grade fluid in a porous space,” Nonlinear Analysis: Real World Applications, vol. 9, pp. 1619–1627, 2008.

[2] T. Hayat, T. Javedb, M. Sajid, “Analytic solution for MHD rotating flow of a second grade fluid over a shrinking surface,” Physics Letters A, vol.372, pp.3264–3273, 2008.

[3] M.M. Abdelkhalek, “Heat and mass transfer in MHD flow by perturbation technique,” Computational Materials Science, vol. 43,pp. 384–391, 2008.

[4] M.S. Abel, M.M. Nandeppanavar, “Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/sink,” Communications in Nonlinear Science and Numerical Simulation, vol.14, pp. 2120–2131, 2009.

[5] A. Ishak, R. Nazar, I. Pop, “MHD boundary-layer flow of a micropolar fluid past a wedge with constant wall heat flux,”Communications in Nonlinear Science and Numerical Simulation,vol. 14, pp. 109–118, 2009.

[6] K.V. Prasad, D. Pal, P.S. Datti, “MHD power-law fluid flow and heat transfer over a non-isothermal stretching sheet,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, pp. 2178–2189, 2009.

[7] C.Y. Wang, “Analysis of viscous flow due to a stretching sheet with surface slip and suction,” Nonlinear Analysis: Real World Applications, vol. 10, pp. 375–380, 2009.

[8] C.Y. Wang, “Exact solutions of the steady-state Navier–Stokes equations,” Annual Review of Fluid Mechanics, vol. 23, pp. 159–177,1992.

[9] J.K. Zhou, “Differential Transformation and Its Applications for Electrical Circuits,” Huazhong University Press, Wuhan, China, 1986 (in Chinese).

[10] C.K. Chen, S.H. Ho, “Solving partial differential equations by two dimensional differential transform method,” Applied Mathematics and Computation, vol. 106, pp. 171–179, 1999.

[11] F. Ayaz, “Solutions of the systems of differential equations by differential transform method,” Applied Mathematics and Computation, vol. 147, pp. 547–567, 2004.

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t

f(t)

0 1 2 3 4-0.5

0

0.5

1

1.5

2

2.5

DTM-Padé, f0=0.5

DTM, f0=0.5

DTM-Padé, f0=-0.5

DTM, f0=-0.5DTM, f0=0

DTM-Padé, f0=0

t

f(t)

0 1 2 3 4 5-0.5

0

0.5

1

1.5

2

2.5

3

DTM-Padé, f0=0.5

DTM, f0=0.5

DTM-Padé, f0=-0.5

DTM, f0=-0.5DTM, f0=0

DTM-Padé, f0=0

Figure 1. The analytical solution of ( )f t obtained by the DTM-padé andthe DTM for

0, 0,Pr 0.71, 1, 0, 0, 0.R R xM N Gr N

Figure 2. The analytical solution of ( )f t obtained by the DTM-padé andthe DTM for

0, 5,Pr 0.71, 1, 0, 0, 0.R R xM N Gr N

t

f(t)

0 0.1 0.2 0.3 0.4 0.5-0.5

-0.25

0

0.25

0.5

0.75

DTM-Padé, f0=0.5

Numerical, f0=0.5

DTM-Padé, f0=0DTM-Padé, f0=-0.5

Numerical, f0=-0.5Numerical, f0=0

t

f(t)

0 0.1 0.2 0.3 0.4 0.5-0.5

-0.25

0

0.25

0.5

0.75

DTM-Padé, f0=0.5

Numerical, f0=0.5

DTM-Padé, f0=0DTM-Padé, f0=-0.5

Numerical, f0=-0.5Numerical, f0=0

Figure 3. Comparison of the solution obtained by the DTM-Padé and the numerical solution for

5, 0,Pr 0.71, 1, 0, 0, 0.R R xM N Gr N

Figure 4. Comparison of the solution obtained by the DTM-Padé and the numerical solution for

5, 5,Pr 0.71, 1, 0, 0, 0.R R xM N Gr N

t

(t)

0 1 2 3 4

0

0.5

1

DTM-Padé, f0=0.5

DTM, f0=0.5

DTM-Padé, f0=-0.5

DTM, f0=-0.5DTM, f0=0

DTM-Padé, f0=0

t

(t)

0 2 4 6 8-0.25

0

0.25

0.5

0.75

1

DTM-Padé, f0=0.5

DTM, f0=0.5

DTM-Padé, f0=-0.5

DTM, f0=-0.5DTM, f0=0

DTM-Padé, f0=0

Figure 5. The analytical solution of ( )t obtained by the DTM-padé andthe DTM for

0, 0,Pr 0.71, 1, 0, 0, 0.R R xM N Gr N

Figure 6. The analytical solution of ( )t obtained by the DTM-padé andthe DTM for

0, 5,Pr 0.71, 1, 0, 0, 0.R R xM N Gr N

873

t

(t)

0 5 10 15 200

0.25

0.5

0.75

1

DTM-Padé, f0=0.5

DTM-Padé, f0=-0.5DTM-Padé, f0=0

t

(t)

0 5 10 15 200

0.25

0.5

0.75

1

DTM-Padé, f0=0.5

DTM-Padé, f0=-0.5DTM-Padé, f0=0

Figure 7. The analytical solution of ( )t obtained by the DTM-padé for5, 0,Pr 0.71, 1, 0, 0, 0.R R xM N Gr N

Figure 8. The analytical solution of ( )t obtained by the DTM-padé for5, 5,Pr 0.71, 1, 0, 0, 0.R R xM N Gr N

t

f(t)

0 1 2 3 40

0.5

1

1.5

2

2.5

DTM-Padé, Gr=5, =1

DTM-Padé, Gr=5, =-1DTM-Padé, Gr=5, =0

t

(t)

0 0.5 1 1.5 2 2.50

0.25

0.5

0.75

1

DTM-Padé, Gr=5, =1

DTM-Padé, Gr=5, =-1DTM-Padé, Gr=5, =0

Figure 9. The analytical solution of ( )f t obtained by the DTM-padé for

05, 0, 0, 0,Pr 0.71, 0.R xGr f M N

Figure 10. The analytical solution of ( )t obtained by the DTM-padé for

05, 0, 0, 0,Pr 0.71, 0.R xGr f M N

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