Cmjv02i01p0153

J. Comp. & Math. Sci. Vol.2 (1), 153-158 (2011)

Heat Transfer in Magnetohydrodynamics MADAN LAL and SARVESH KUMAR Department of Applied Maths, M.J.P. Rohilkhand University, Bareilly 243 006 (India) ABSTRACT In the present paper, the effects of transversely applied magnetic field on three dimensional flow and heat transfer in presence of viscous dissipative heat has been discussed and the components of the plate skin friction and the rate of heat transfer are obtained. It is found that the rate of heat transfer and skin friction both decreases in presence of applied magnetic field.

Key words: MHD Flow, Incompressible Fluid, Skin Friction, Heat Transfer, Hartmann Number.

1. INTRODUCTION Two and three-dimensional MHD stagnation point flow was studied by several authors such as Rossow8, Meyer6, Mcllroy and Neuringer5. The heat transfer aspect of such a flow for constant wall temperature and in the presence of injection was studied

by Sparrow et al.10. Mahapatra and Gupta4 studied the heat transfer in the steady two dimensional stagnation point flow of an incompressible fluid over a stretched sheet with a velocity proportional to the distance from the stagnation point. The boundary layer flow for an incompressible viscous fluid near a stagnation point at a heated stretching sheet placed in a porous medium, using Lie-group method is analysed by Boutros1. Jing ZHU3 found the analytical solution to a stagnation point flow and heat transfer over a stretching sheet using Homotopy analysis, Gersten and Gross2 have studied the effect of slightly sinusoidal transverse suction velocity distribution leads to a cross flow and hence to a three-

dimensional flow over the surface. Singh et al.9 further extended this idea by applying the transverse sinusoidal suction velocity distribution fluctuating with time at the plate. In both these studies the heat transfer aspect has been investigated ignoring the heat due to viscous dissipation.

Recently Pal7 studied the effect of thermal radiation on heat and mass transfer stagnation point flow towards a stretching sheet.

Hence, the object of this chapter is to analyse the effects of transversely applied magnetic field on the three dimensional flow and heat transfer in the presence of viscous dissipative heat when the plate is subjected to a slightly sinusoidal transverse suction velocity distribution.

2. FORMULATION AND SOLUTION OF THE PROBLEM Here, we have considered the steady and laminar flow of viscous, incompressible and finitely conducting fluid with constant

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

Madan Lal, et al., J. Comp. & Math. Sci. Vol.2 (1), 153-158 (2011)

154

Type equation here.

Where v0 is the free stream velocity, U is the basic steady suction velocity, L is the wave length of the weak superimposed suction velocity distribution and the other physical quantities have their usual meaning.

If the amplitude ∈ of the suction velocity is very small. Then we can assume the main flow velocity u(x, y) in the neighbourhood of the plate as

∇. q

(2.1)

u( x , y) = ∑ ε r u r ( x, y )

(2.2)

(2.7) and the similar expression hold for other variables v, w, p and θ.

If ∈ = 0, then the equations (2.1) to (2.6) become

physical properties. Let the magnetic field r

B is applied perpendicular to the infinite porous flate plate which lies horizontally in x-y plane. If (u, v, w) are the nondimensional velocity components in x, y and z directions respectively. Then, the flow is governed by the following non-dimensional equations: =

∂p 1 2 − + ∇ u ∂x R

=

Du

=

Dv

−

Dw

=

Dθ

=

Where

r q r ∇

=

D

≡

∞

0

2

∂p 1 2 M + ∇ v− v ∂y R R (2.3)

1 2 M2 ∇ w− w R R 1 E ∇2θ + φ RPr R

≡

,

u

u0 (2.5)

u0

∂ ∂ +ν ∂x ∂y

2

2 2

u = −α(1+ ε cosπy), u = −α ,

v = w = 0,

θ =1

v = 0, w = 1, θ = 0, p = Π

(2.6) Consider the following non-dimensional quantity : x=

x' , L

p=

y=

p' , ρv 0 2

y' , L

dp 1 d u0 du 0 =− 0 + dx dx R dx 2 2

u0

The boundary conditions of the problem in the non-dimensional form are x → ∞,

(2.8) 2

u=

θ=

u' , U

v=

T'− Π ' Tw' − Π '

v' , v0

w=

w' v0

(2.9)

2

dv 0 1 d v 0 M = − v0 dx R dx 2 R 2

. q ²

2 (

x = 0,

du 0 =0 dx

(2.4)

( u, v, w )

r =0

(2.10)

2

dw 0 1 d w 0 M = − w0 dx R dx 2 R

(2.11)

dθ 0 E dw0 2 1 d 2θ 0 u0 = + dx RPr dx 2 R dx

(2.12) with corresponding boundary conditions x=0 ,

x→∞ ,

u 0 = −α ,

u0 =−α ,

v0 =0,

v0 = w 0 = 0 , θ0 = 1

w0 =1,

p0 =Π,

θ=0

(2.13) The solutions of equations (2.8) to (2.12) under the boundary conditions (2.13) are

u0 = −α ,

v0 = 0 ,

p 0 = Π (2.14)

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

Madan Lal, et al., J. Comp. & Math. Sci. Vol.2 (1), 153-158 (2011) ∞

w 0 = ∑ ( −1)

j−1

j=1

( a1 x) j j!

(2.15)

∂v1 ∂p1 1 2 M2 âˆ’Îą =− + ∇ v1 − v1 ∂x ∂y R R (2.19) u1

∂w 0 M2 ∂w 1 w1 − Îą 1 = ∇ 2 w1 − R R ∂x ∂x

u1

∂θ 0 ∂θ 1 2 E ∂w 0 ∂w1 âˆ’Îą 1 = ∇ θ1 + 2 ∂x ∂x RPr R ∂x ∂x

ஶ

θ଴ 1 ŕ­¨ 1 ଵ ௼ ŕŻ? ŕŻ?ŕ­€ŕŹ´

ଵ 2 ଵ ŕŻ?

x j!

aଵ

E1 =

a1 EPr 2 ( 2a 1 − ÎąRPr )

ଶ

R ÎąR

(2.16)

(2.21) The corresponding boundary conditions become

ŕ°­

4M ଶ మ

x = 0,

which represents the solution of twodimensional problem of flow past a porous flate plate with constant suction and transversely applied magnetic field.

If Îľ ≠0 , then using equation (2.7) in equations (2.1) to (2.6) and equating the like powers of ∈ and neglecting those of ∈2, we get following equations as the coefficient of âˆˆď€ :

∂u1 ∂v1 + =0 ∂x ∂y

∂u ∂p 1 âˆ’Îą 1 = 1 + ∇ 2 u1 ∂x ∂x R

(2.20)

ŕ­¨

wherein ଵ ଶ

155

(2.17)

x → ∞,

u1 = âˆ’Îą cos Ď€y, u1 = 0 ,

v1 = w1 = 0,

v1 = w 1 = 0 ,

θ1 = 0

p1 = 0 ,

θ1 = 0

(2.22) The above partial differential equations describes the three-dimensional flow whose solutions may be considered as :

u1 ( x, y)

=

v1 ( x, y)

=

w 1 ( x, y )

=

p1 ( x, y) θ1 ( x, y)

= =

u11 ( x) cos πy 1 1 u11 ( x) sin πy π

w 11 ( x ) cos πy

(2.23)

p11 ( x) cos πy θ11 ( x) cos πy

(2.18)

Putting above values in equations (2.18) to (2.21) and using the boundary conditions (2.22), we get

Îąa 3a 4 ∞ u1 = ( −1) j a a x cos Ď€y ∑

( a 3 − a 4 ) j= 0 j!

v1 =

∞ Îąa 3a 4 xj ( −1) j ( a 3j − a 4j ) sin Ď€y ∑ Ď€ ( a 3 − a 4 ) j= 0 j!

w1 =

ÎąR a3 − a4

∞

∑ j= 0

ŕą

( −1) j C1 (a1 + a 4 + b1 ) j − C 2 {b a a cos Ď€y !

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

Madan Lal, et al., J. Comp. & Math. Sci. Vol.2 (1), 153-158 (2011)

156

p1 = θ1 =

+

∞ αa 3a 4

ౠ ( −1) j a a R M a a (a αR M a ! cos πy ∑ 2 π R ( a 3 − a 4 ) j= 0

LM l N

αRPr ∞ 2a 2a ∑ ( −1) j A 1 a 3 E1 + EC1 (a1 + a 4 ) (2a1 + a 4 + b 2 ) − A 1 a 4 E1 + EC2 (a1 + a 3 ) b 2j − (2a1 + a 3 ) j ( a 3 − a 4 ) j= 0 1 2

q

l

qm

αRPr a 4 j αRPr a 3 b 2 − ( a 3 + αRPr ) j − (1 + E1 ) b 2j − ( a 4 + αRPr ) j A3 A4

m

r

m

rOPQ

r

rOPQ

2a1b1E(C2 − C1 ) j xj b 2 − (a1 + b1 ) j cos πy A5 j! where 1/ 2 1 a 2 = αR − (αR ) 2 + 4 M 2 2 1 a 3 = a1 + (a12 + 4π 2 )1/ 2 2 1 a 4 = a 2 + (a 22 + 4 π 2 )1/ 2 2 1/ 2 1 b1 = αR + (αR) 2 + 4( π 2 + M 2 ) 2 1/ 2 1 b 2 = αRPr + (αRPr ) 2 + 4 π 2 2 A1 = ( 2a1 + a 4 ) 2 − αRPr ( 2a1 + a 4 ) − π 2 +

m

m

r

m

r

m

r

A 2 = ( 2 a 1 + a 3 ) 2 − α RPr ( 2 a 1 + a 3 ) − π 2

A 3 = a 3 (a1 + αRPr ) A 4 = a 4 (a 2 + αRPr )

A 5 = ( a 1 + b1 ) 2 − α RPr ( a 1 + b 1 ) − π 2

C1 =

a3 a 1a 4 , C2 = a4 a 3 ( 3a1 − αR )

3. RESULTS AND DISCUSSION The expressions for the shear stress components in x-y directions in the nondimensional form are calculated as :

τx =

τ = − ε F2 (α , R , M ) sin πy π ' x

τy =

τ 'y ρ

=

a1 + ε 1 − F1 (α , R , M ) cos πy αR

where F1 (α, R, M) = 1 −

l

q

1 C1 (a1 + a 4 − b1 ) − C 2 (a1 + a 3 − b1 ) a3 − a4

l

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

q

(3.1)

Madan Lal, et al., J. Comp. & Math. Sci. Vol.2 (1), 153-158 (2011)

F2 (α , R , M ) = If M=0 reduces Gersten (αR,M) to:

a 3a 4 π

(3.2)

and α=1, then the above problem to similar which was studied by and Gross. The expressions for F1 and F2 αR,M) in this case reduces

π (λ + π) 2λ 2 F2 (1, R ,0 ) = λ F1 (1, R ,0) =

wherein

1 λ = R + ( R 2 + 4 π 2 )1/ 2 2 The skin friction factor F1(αR,M) has been shown against Reynolds number in Fig. 1, from which, it is clear that F1 decreases with the increase of R as well as α and approaches to zero as R → ∞ . It can also observed that an increase in Hartmann number M, leads to a significant decrease in the skin friction factor F1.

157

For different values of α, R and M, the numerical values of F2(α, R, M) have been listed in the following table which shows that F2 increases with the increase of M. Table for the function F2 (α α, R, M)

Table 1 : The function F2 (α α, R, M) given in equation (3.2) α

M/R

10-2

10-1

1

10

102

1.0

0

3.1466

3.1920

3.6811

10.9050

0.5

0.5

3.1428

3.1653

3.3993

6.5011

50.14

1.0

0.5

3.1478

3.1931

3.6806

10.8855

100.06

0.5

1.0

3.1428

3.1651

3.3968

6.4679

50.04

100.10

Now the rate of heat transfer at the surface of the plate in terms of Nusselt number, is given by Nu = 1 + E1 −

2 a 1 E1 + ε 1 − F3 (α, R, Pr , M, E) cos πy αRPr

l

q

where F3 (α , R , Pr , M , E ) = 1 −

−

1 2a1 a 3 E1 + EC1 ( a1 + a 4 ) ( b 2 − 2a1 − a 4 ) a 3 − a 4 A1

LM l N

q

2 a1 a 4 E1 + EC 2 (a1 + a 3 ) ( b 2 − 2a1 − a 3 ) A2

l

q

αRPr a 4 (1 + E 1 )( b 2 − αRPr − a 3 ) A3 αRPr a 3 − (1 + E 1 )( b 2 − αRPr − a 4 ) A4 +

+

2a1 b1E(C 2 − C1 ) ( b 2 − a1 − b1 ) A5

OP Q

(3.3)

In the absence of magnetic field and viscous dissipative heat (i.e.) for M = E = 0 and α = 1 the expression for F3 (α R,Pr, M, E) reduces to : F3 (1, R , Pr ,0,0) =

Fig. The function F1 (α α, R,M) defined by eqn. (3.1)

LM RS NT

1 λ πPr πPr2 R πPr λRPr λ − + + − −π λ−π π λ (1 + Pr ) λ (1 + Pr ) 1 + Pr π

UV W

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

OP Q

Madan Lal, et al., J. Comp. & Math. Sci. Vol.2 (1), 153-158 (2011)

158 wherein

1 R + ( R 2 + 4 π 2 )1/ 2 2 1 λ = RPr + ( RPr ) 2 + 4 π 2 2

λ=

m

1/ 2

r

The effect of Hartmann number M on the rate of heat transfer is comparatively less as compared to that on the skin friction. The effect of M at M = 10 and 102 can be pointed out graphically.

or Prandtl number Pr but increases with Eckert number E. It is noted that with the increase of M, F3 first increases slightly for small values of R but then decreases for large values of R. Remark : It is concluded that the skin friction factor F1(α, R, M) and the rate of heat transfer factor F3(α, R, Pr, M, E) both decreases in the presence of transversely applied magnetic field.

REFERENCES

α 0.5 0.5 1.0 0.5 1.0 0.5

M 0 10 10 102 10 10

Pr 0.71 0.71 0.71 0.71 0.71 7.00

E 0.01 0.01 0.01 0.01 0.05 0.01

I II III IV V VI

Fig. 2 The function F3 (α α,R, Pr, M,E) defined by eqn. (3.3)

The above figure shows that F3(α, R, Pr, M, E) decreases with suction parameter α

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3. Jing, ZHU. Applied Mathematics and Mechanics (English Edition), 30(4), 463 (2009).

4. Mahapatra, T.R. and Gupta, A.S., Heat and Mass Transfer, 38, 517 (2002).

5. Mcllroy, W. Jour. and Neuringer, J.L., Aeronau. Sci. 25, 194 (1958).

6. Meyer, R.C., Jour. Aeronau. Sci. 25, 561 (1958).

7. Pal, D., Meccanica, 44, 145 (2009).

8. Rossow, V.J., Jour. Aeronau. Sci. 25, 334 (1958).

9. Singh, P., Sharma, V.P. and Mishra, U.N. Int. Jour. Heat Mass Transfer, 21, 1117 (1978).

10. Sparrow, E.M. and Cess, R.D., Radiation heat transfer. Hemisphere, Washington, (1978).

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)