Finite-difference analysis of MHD stokes problem for a vertical infinite plate in a dissipative fluid with constant heat flux

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Voi.13(3),155-163, 1986. Printed in the USA. Copyright (c) 1986 Pergamon Journals Ltd.

FINITE-DIFFERENCE ANALYSIS OF MHD STOKES PROBLEM FOR A VERTICAL INFINITE PLATE IN A DISSIPATIVE FLUID WITH CONSTANT HEAT FLUX V.M.Soundalgekar and S.B.Hiremath 7/lO,Vivekanand Housing Society, Saraswat Colony Dombivali (E) 421201, India

(Received 10 January 1986; accepted for print 5 May 1986)

Introduction Stokes ~ ] first studied the unsteady flow of a viscous incompressible fluid past an infinite horizontal plate moving impulsively in its own plane.Stewartson[2,3]presented the analytical solution of unsteady flow past an impulsively moving semi-infinite plate.The numerical solution of Stewartson's problem was presented by Hall[4].The flow of a compressible _gas past an infinite vertical plate was studied by Illingworth[51who presented approximate solution.Elliott 6 studied the flo~ ~ast an infinite vertical plate with time-dependent motion. In Ref.[6]only mathematical solution was presented and the corresponding physical situation was not discussed.In case of motion past a vertical plate,the presence of free-convection currents due to temperature difference between the plate temperature T~ and T~ must be considered.IIence such a study,on taking into account the free-convection currents, of flow past an impulsively started infinite vertical plate was recently presented by Soundalgekar[7~who gave an exact solution by using Laplace-transform technique.Another situation which is also of interest in technology is the study of flow past an impulsively started vertical plate receiving heat at constant rate. Exact solution to the ~roblem considered in ~ef.~7]with constant heat flux was presented by Soundalgekar and l~atil[8].In both these papers,the effects of viscous dissipative heat were neglected.On tsd~ing into account the viscous dissipative heat,the oroblem is governed by coupled nonlinear equations and hence exact or approximate solutions could not be obtained.So in case of both isothermal and constant heat flux,these nonlinear coupled equations were solved by explicit finite-difference method by Soundalgekar, Bhat and Mohiuddin [9,10]. Rossow[ll]first presented the exact solution of MHD Stokes problem on negleting the induced magnetic field. The corresoonding I~ID flow past a vertical plate was presented by Soundalgekar, Gupta and A r a n ~ e [ 1 2 ] . I n thes~, last two pap~:r,~,-~l,~cous dissipative heat ~Jas assumed to be neglected.On taking into account 155

156

V.M. SOUNDALGEKAR and S.B. HIREMATH

viscous dissipative heat but neglecting Joule dissipative heat, the coupled nonlinear equations governing the MHD flow past an impulsively started infinite plate were studied by finite-difference method by Soundalgekar and Hiremath[13].The plate was assumed to be isothermal.The MHD flow past an impulsively started infinite plate in the presence of constant heat flux and viscous dissipative heat has not been studied as yet.Hence the motivation to study this problem. In Sec.2,the mathematical analysis has been presented and in Sec.3,the conclusions are set out. Analysis Here

the MHD flow of an electrically

viscous

fluid past an impulsively

te under transverse magnetic

conducting incompressible

started infinite

vertical pla-

field is considered. The x-axis is

taken along the plate and the y-axis is taken normal to the plate, Initially,the

temperature of the fluid and the plate is assumed

to be the same for all y.At time t'>O,the plate is being heated by supplying heat at constant rate and simultaneously,the

plate

is assumed to get an impulsive motion in the upward direction. Applying the usual Boussinesq's an infinite

approximation,the

vertical plate is now governed by the following

tem of coupled nonlinear

equations in non-dimensional

~u ~2u ~Y - ~y2 + GO - Mu p ~0 t-

The initial

~.G~D flow past

u(y,t)

:

(i)

~2e ~u 2 a y2 + PE ( ~ )

and boundary

form

sys-

(2)

conditions are

u(o,t)

= O, @(y,t) = 0 30 = o, y ~ = - i

u(~,t)

= O, O(~,t)

= 0

t~O

(3) t~O

The nondimensional quantities are defined as follows : ' °2 ' u = u/U o, t = tU /~ , y = YUo/~ , O = ( T ' - T ~ ) / ( q ' ~ / k U o)

G =~g~(q'~/kUo)/Uo 2 , P =.~Cp/k

, ~ = Uo2/Cp(q'~/kU o)

!

Here u is the velocity, U o the velocity of the plate,t' the time~ Cp the specific heat at constant pre~sure, k the thermal conductivity,

the kinematic

viscosity,

T' the temperature

MHD STOKES PROBLEM

157

of the fluid,T'cothe temperature of the fluid far away from the plate,g the acceleration due to g r a v i t y , ~ t h e

coefficient of vo-

lume expansion,q' the heat flux per unit a r e a , ~ t h e

coefficient

of viscosity, o-the scalar electrical conductivity,B o the constant magnetic field, e the non-dimensional temperature,P the Prand t l number,E the Eckert number and M is the magnetic parameter. Equations (1) and (2) are coupled and nonlinear equations and their analytical solutions are not possible to obtain.Hence, we solve them by finite-difference method.The mesh-system is shown on Fig. 1.

X

-1

0

1 2

40 41

FIG.I. M E S H

SYSTEM

We can write the explicit finite-difference equations (1) and (2) as follows :

u(i+i,j)

u(i,j+l) - u(i,j)

: G~(i,j)

+

-2u(i,j)+u(i-l,j) (Ay) z

- Mu(i,j)

~(i,j+l) - @(i,j) P

~t

(5)

~(i+l,j) - 2~(i,j) + @(i-l,j) :

(AY) 2

+ 2

PEI u(i+l'J) u ( i '-J ) - ~ Z 1 y

(6)

The index i refers to y and j to t.AY is taken as O.1.We can write the initial conditions from (3) as u(o,o)

= l, e ( o , o )

= o

(7)

This means that due to sudden velocity given to the plate, the

158

V.M.

velocity

at y = 0 c h a n g e s

at t < 0 . T h e adual

initial

c h a n g e in the

of c o n s t a n t h e a t conditions

from

discontinuously

(]),the

that

temperature

due

of the p l a t e

: 0

boundary

for all i

the b o u n d a r y

@(i,j)

: i

for all

form

(6

lated j=O.

grid-point

(i0)

(-l,j)

from the s e c o n d The b o u n d a r y

the v e l o c i t y

0.?i,

compared our

ence

rw>d.Hencc

of thes~ ~ re±" c~aiua-

(6).~ut e(-l,O)

is

calcu-

time-ster;

t ~ k e n as y : 4 and h e n c e j.From

(9),we

calculate :i,~O,

compute ~(i,j+l),

j = lO00,i.e,

in

earlier

i : O,L~O.This

t = i.

on a c o m p u t e r

for P = 0.~,

i[ = ~ , 4 . T h e

order

the a c c u r a c y

of the r e s u l t s , w e

have

the a n a l y t i c a l

solution obtained

for

cent. To

0.0006,O.000~

at the h y p o t h -

at p o i n t s on the

computations

(l<~)

±~ = 0 . 0 1 , O . 0 2 , to check

solutions

scheme,we

follov/i-

:-i

for the i n i t i a l

at y = o ~ i s

(6),we

with

time-step ~t

and 1 for the entire

and the a g r e e m e n t

than i per

the

- 8(-1,j)

the v a l u e s

: 0 for all

tiil

these

L = 0 at t = 0 . 2 , 0 . 9 i = 1,~O

takes

to take

time-axis.

Z3 Y

(i0)

and t e m p e r a t u r e s

c a r r i e d out

en as O . O 0 1 . 1 n

have

end of a t i m e - s t e p , v i z . , u ( i , j + l ) , i

is r e p e a t e d

C = 2,6,

or

side of

e q u a t i o n of

: O, @ ( 4 l , j )

at the

-i

and we ta/~e a v e r a g e

time-steps. Similarly,from

',Te have

(9)

side of the p l a t e on

for {J give

condition

terms of velocities

procedure

j

8(©,j) =

on the r i g h t h a n d

we set u(41,j)

tsdtcs the

,on the p l a t e , w e

- 8(-l,j)

equations in

tini[ @ ( O , j + ! )

(8)

:

2~),

etical

initial

(>0)

c o n d i t i o n on ~ at the p l a t e

finite-difference

two

zero.The

:

a p o i n t i = -i on the left h a n d

These

is a gr-

to the p r e s e n c <

c o n d i t i o n on u at y = 0

N o w to c o m p u t e ~ at i : O , f r o m

ng

is t a k e n

there

o

are

u(O,j)

Then

at i from its v a l u e

c o n d i t i o n on ~ i n d i c a t e s

: O, @ ( i , O )

follov~ing form

and S.B. HIREMATH

flux and h e n c e ~ ( © , 0 )

for y>O

u(i,O) Also

SOUNDALGEKAR

the c o n v e r g e n c e the

solutions

and an i n s i g n i f i c a n t

~w c o n c l u d e

and c o n v e r g e n t .

The v e l o c i t y

profiles

are

s h o w n on

of the

i.u.

e r r o r of l e s s finite-differ-

for small v a l u e s of At =

c h a n g e in the v a l u e s

that o u r e x p l i c i t

me is stable

chos-

of y - v a l u e s ,

was g o o d v&ith a m a x i m u m

test

repeated

range

was

was obse-

finite-difference

Figs.2

sche-

and ~ for P : ;9.5 an<~

MHD STOKES PROBLEM

159

0.71 respectively.We observe from these two figures that the velocity decreases due to the application of the magnetic field.An increase in G or E leads to an increase in the velocity.On Figs. 4 and 5,the temperature profiles are shown. We observe from these figures that due to the application of the magnetic field,the temperature increases.An increase in G or E also leads to an increase in the temperature.From the velocity field,we now calculate the skin-friction. It is given by ~g = -

~u i ~Y IY = 0

(il)

l

whereT=~/~U~.The

numerical values o f ~ a r e

calculated by numeric~

al differentiation using Newton's 5-point interpolation formula by taking 5 points.These values are entered in Table I and Fig.6. We observe from this table and Fig. 6 that due to the application of the magnetic field,there is a rise in the value of the skin-friction.An increase in G or E leads to a fall in the value of the skin-friction under the action of the magnetic

field.~g increases

with increasing the Prandtl number. We now calculate the rate of heat transfer at the plate.lt is given in terms of the Nusselt number as Nu

=-

1 ~--~

d8 I dy y=0

=

1 ~

d9 (Vdy

-

(12)

r

The numerical values of Nu are calculated and are entered in Table I and also shown in Fig. 7.We observe from this table and Fig.7 that due to the application of the magnetic field,there is a fall in the value of the Nusselt number as compared to non-magnetic case.Greater viscous dissipative heat or an increase in G leads to an increase in the value of Nu but the Nusselt number decreases due to increasing the strength of the magnetic field. Conclusions 1.Due to the application of the magnetic field,there is a decrease in the velocity and an increase in temperature,skin-friction and the Nusselt number.2.Greater viscous dissipative heat or an increase in G leads to a fall in the skin-friction but a rise in the Nusselt number.3.An increase in P leads to a rise in the value of the skin-friction and the Nusselt number.4.Greater viscous dissipative heat or an increase in G leads to a rise in

160

V.M.

SOUNDALGEKAR

and S.B. HIREMATH

the velocity,temperature. References l.G.G.Stokes,Camb.Phil.Trans.iX,8(1851) 2.K.Stewartson,Quart.Mech. Appi.Math.4,182(1951) 3.K.Stewartson,quart.Mech. Appl.Math.26,i43(1973) 4.M.G.Hail,Proc.Roy. Soc.(London),310A,401(1969) 5.C.R.lllingworth,Proc. Camb.Phil.Soc.,46,60j(1950) 6.L.Elliott,Zeit.Angew.Math. Mech.,49,647(1969) 7.V.M.Soundalgekar,J.Heat Transfer(Tr.ASME)99C,499(1977) 8.V.M.Soundalgekar and M.R.Patil,Astrophysics and Space gci.,70, 179(1980) 9.V.M.Soundalgekar,J.P.Shat and M.Mohiuddin,lntl.j.~ingng. Sci.,17, i283(1979) i0.V.M.Soundalgekar,J.P.Bhat and M.Mohiuddin,Latin American J. Heat Mass Tr.,9,(1985) II.V.J.Rossow,NASA Report No 1358(1958) 12.V.M.Soundalgekar,S.K.Gupta and R.N.Aranke,J.Nuciear Engng Des. 51,405(1978) lj.V.M.Soundalgekar and S.}~.Hiremath,Magnetohydrodynamics(USSR) (To be published) Table I

Values of "g and

q~ ~

MQ

~

/t

o.~ 2"> d.bl 2 2 2 4 4 0 0

2 0.02 40.01 4 0.02 2 0.0! 2 0.02 20.O1 2 0.02

0.712 2 2 2 4

2 0.01 2 0.02 4 0.0i 4 0.02 20.O1

4 2 0.02

0 20.01 0 2 0.02

Nu

Nu

1.0

o.2

t".4290 1.4270 1.1259 1.1220

o.1378 0.1341 -1.1487 -1.1556 0.9091 0.9053 -i.0726 -1.0774

i~300 1.4232 1.4303 1.4238 1.4294 1.4220 1.4306 1.4243

1.O O. 897'2 O. 6322 0.8944 O. 6309 0.8976 o. 6922 0.8952 o. 63lo 0.8967 o. 6317 0.8933 0.6300 0.8977 o. 6323 0.8954 0.6311

1.4967 1.4944 1.2612 1.2568 1.9254 1.9232 1.O120 1.0097

0.9255 0.4088 0.9219 0.4041 0.3677 -0.6066 0.3611 -0.6145 1.5083 1.1288 1.5048 1.1243 0.1669 -0.7062 0.1632 0.7114

1.7096 1.6983 1.71oo 1.6991 1.7086 1.6963 1.7106 1.7003

i.O710 1.O662 1.o715 1.0673 1.0700 1.0642 1.0719 1.0680

O.2

O.5

0.7708 0.7677 0.0584 0.0527 1.8620 1.3734 1.8600 1.3703 0.9394-0.0157 0.9373-0.0191

b.~

O. 7541 0. 7517 0. 7545 0.7525 0.7532 0.7501 O. 7546 0.7527

0

0.2

0.z,

U

0.6

0.8

1'°II

Y

2

0.2 0,2 0.2 0.2 0.5 0.2

t 0.2

FIG. 2. VELOCITY PROFILES

1

II HI IV V Vl VI'[

I

3

P: 0.5

2 2 2 4 2 0

M 2 2 2 4 2 2 2

G 2 0.01 0.02 0,01 0.01 0.01 0.01

E 0

0.2

0.4

U

0.6

0.8

1.o

2

y

3

P= 0.71

M 2 2 2 2 4 2 0

FIG. 3. VELOCITY PROFILES

1

1; I 0.2 fT 0.2 ]]'I 0,2 IV 0.2 V 0.2 VI 0.5 V~ 0.2

G 2 2 2 4 2 2 2

E 0 0.01 0.02 0.01 0.01 0.01 0.01

! 4.

~ o

r.~

O

0

0,2

0.4

1

Y

2

3

G 2 4 2 2 2 2 2

FIG. 4. TEMPERATURE PROFILES

1

P:0.5

M 2 2 2 2 2 0 2

~J---

E 0 0.01 0.01 0.01 0.02 0.01 0.01

0

0.2

0.4

O

0.6

t I 0.2 IV 0.2 II 0.2 V 0.2 flT 0.2 VII 0.2 V[ 0.5

0,6

8

0.8

0.~

2

y

G 2 4 2 2 2 2 2

3

P : 0.71

M 2 2 2 4 2 0 2

FIG 5 TEMPERATURE PROFILES

1

t I 0.2 IV 0.2 II 0.2 V 0.2 ffT 0.2 VE 0,2 VI 0.5

E 0 0.01 0.01 0.01 0'02 0.01 0.01

~-]

t~

i-I

Z t~

r.~ 0

~o

0

.

FIG. 6

0"5

t

0.71

=0-5

I "0

THE SKIN- FRICTION

E 0"01 I \ 0-02 II 0.01111 0.01 IV

0.2

IMG 0./-, 2 2 2 2 2 Z, 2

0"8

"12

1"2

1.(

2-1

I

III

I!

0"4

Nu

0

1-0

1"I

Ill

I

1,0

II

II

I

III

III

THE NUSSELT NUMBER

I

0"5 t

0.01

I

FIG. 7

I

0.02 II

0-01

0-2

~

2 2

2

2

2

M

= 0.71-

P=0"5

O

l,.o

O

"O

i'x/