FINITE DIFFERENCE HYDROMAGNETIC

A N A L Y S I S OF U N S T E A D Y

FREE-CONVECTION

FLOW WITH

CONSTANT HEAT FLUX A. K. S I N G H

Department of Mathematics, Banaras Hindu University, Varanasi, India and N. C. S A C H E T I

Department of Mathematics and Computing, College of Science, Sultan Qaboos University, Muscat, Sultanate of Oman

(Received 21 June, 1988) Abstract. A numerical solution for unsteady hydromagnetic free-convection currents of a viscous incompressible and electrically conducting fluid induced by a vertical moving infinite plate is considered for constant heat flux at the plate. Velocity and skin-friction have been worked out for various values of the parameters occuring into the problem. It is found that magnetic parameter has a retarding effect on the velocity of air and water, while skin-friction increases with it.

I. Introduction

Rayleigh's problem in magnetofluid mechanics has been studied by a number of authors. Rossow (1957, 1958), Yang (1958), Gupta (1960), and Soundalgekar (1965) discussed a similar problem in the case of an accelerated infinite flat plate, by assuming the magnetic lines of force to be fixed relative to the fluid and the plate, respectively. Kakutani (1958) and Ong and Nicholls (1959) have extended the problem in the case of an oscillating infinite plate with or without the induced magnetic field produced by the current, respectively. The flow past an infinite vertical plate, started impulsively from rest, plays an important role in many industrial applications. This is particularly important in the design of spaceship, solar energy collectors, etc. The flow past a vertical plate moving impulsively in its own plane was studied by Soundalgekar (1977). Georgantopoulos et al. (1979) and Raptis and Singh (1983) have presented this problem in the presence of a transverse magnetic field, when the magnetic field is fixed relative to the fluid and plate, respectively. The effect of constant heat flux at the plate on the flow past an impulsively started vertical plate has been presented by Soundalgekar and Patil (1980). The problem of free convection of an electrically conducting fluid past a plate under the influence of a magnetic field is important and useful partly for gaining basic understanding of such flows, and partly for possible application to geophysical, astrophysical, and aerodynamical problems. The aim of the present paper is to study the effect of an applied magnetic field on unsteady hydromagnetic flow induced by an infinite vertical moving plate, when the heat flux at the plate is constant. It is difficult to obtain the exact Astrophysics and Space Science 150 (1988) 303-308. 9 1988 by Kluwer Academic Publishers.

304

A. K. SINGH AND N. C. SACHETI

solution of the problem by the Laplace transform method. The method used, hence, is finite difference method. Computations have been carried out at various time level for different values of other parameters entering into the problem. Also skin-friction is obtained by numerical differentiation using Newton's interpolation formula. Numerical result shows that magnetic field has retarding influence on the flow field. 2. Formulation of the Problem

We consider the unsteady two-dimensional flow of an electrically conducting and incompressible viscous fluid near an infinite non-conducting vertical plate (or wall). On this plate an arbitrary point has been taken as the origin O of a Cartesian coordinate system; the axis Ox' is taken along the plate in the upward direction and the axis Oy' perpendicular to it. A uniform magnetic field B o is assumed to be applied in the y'-direction. Initially, the plate and the fluid are at rest and at the same temperature T ' . Suddenly the plate starts moving impulsively with velocity U in its own plane along the x'-axis and instantaneously heat is also supplied to the plate at the constant rate. The magnetic Reynolds number of the flow is taken to be small enough so that the induced magnetic field can be neglected in comparison with the applied magnetic field. All the physical quantities are functions of the space coordinate y' and time t' only as the plate is infinite in extent. Within the framework of these assumptions, the free-convection flow relevant to the problem for the case of magnetic field fixed to the fluid is governed by the set of equations OH r

(~2 u '

-

&,

v --+ ~y,2

OT' Ot'

k

(1)

gfl(T' - T ' ~ ) - aB2 u ' , p

~2T'

(2)

pCp ~y, 2 '

where u' is the velocity component in the x'-axis; T', the temperature of the fluid; k, the thermal conductivity; g, the acceleration due to gravity; fi, the volumetric coefficient of thermal expansion; v, the kinematic viscosity; p, the density; ~r, the electrical conductivity; and Co, the specific heat of the fluid at constant pressure. The initial and boundary conditions of the problem are for

t'
u'=O,

for

t' > O;

u' = U,

u' ~ O ,

T' ~ T "

T'=7~

forall

OT'/Oy' = - q / k

y'; at y' = O,

(3)

as y ' - - . o o ;

where q is the constant heat flux per unit area at the plate. Introducing the following non-dimensinal quantities y=y'U/v,

t=t'U2/v,

M = aB~v/pU 2,

u=u'/U,

G =gflv2q/kU 4,

e=pvCp/k,

(4)

UNSTEADY HYDROMAGNETIC FREE-CONVECTION FLOW

305

in Equations (1)-(3), we have (~U

at

aEu -

GT

8T at

+

ay 2

-

(5)

Mu,

1 O2T p ~y2'

(6)

with the initial and boundary conditions t_<0;

u=0,

T=0

forall

t>0;

u = 1,

3T/Sy= - 1

u~0,

T ~0

y; at y = 0 ,

as y ~ .

(7)

3. Numerical Integration and Differentiation The solution of Equation (6) under conditions (7) has been obtained by Soundalgekar and Patil (1980) and here we have to solve only Equation (5) under conditions (7). This equation has been solved by the Laplace transform method. However, the inverse Laplace transforms of the functions involved are difficult to find and one has to restore to approximations. Hence, we have considered here a finite difference method. Figure 1

j'~l

...... (i-1,

iI

I

!

I

L+_I)J. _ _ .j I

I I

i j-1 (i-1,

I

(i+1, j+l)

I 1

,

L

I I I

I I ]

j-l){

I(i+1, I I

I I

I I I

I

(0,0)

i-1

i+1

Y Fig. 1, Mesh system.

j-l)

306

A. K. SINGH AND N. C. SACHETI

shows the mesh system of the finite difference method and the finite difference equation corresponding to Equation (5) (see Chow, 1979) is as u ( i , j + 1) - u(i,j)

At

= GT(i, j) +

u(i + 1,j) - 2u(i,j) + u(i - 1,j)

(@)2

- Mu(i, j ) ,

(8)

where the indices i and j refer to y and t, respectively. From (7), the initial condition takes the form u(0, 0) = 1 and u(i, 0) = 0 for all i except i = 0, whereas the boundary condition at y = 0 becomes u(0,j) = 1 for all j. The analytical solution of (5) for M = 0 obtained by Soundalgekar and Patil (1980), reduces to zero at y - 4.0 and all other points beyond this. The same trends are also found in most of the flow problems over infinite vertical moving plate (see Soundalgekar, 1977; Georgantopoulos et al., 1979). Hence, corresponding to infinity we can take any values after 4.0. To save computatinal time, it will be convenient to choose the minimum value of y at which u reduces to zero. We, therefore, take y = 4.1 corresponding to the boundary condition y = oo in the numerical computations. In Equation (8), T(i, j) is computed from analytical solution in order to get more accuracy, u(i, j + 1) is computed from (8) in terms of the velocities and temperatures at points on the earlier step-time. This process is repeated till t = 0.6. In the entire computations, we have taken Ay and At as 0.1 and 0.002, respectively, in order to satisfy stability condition, not given here for sake of brevity. To see the accuracy of the result, the numerical values obtained for M = 0 have been compared with the analytical values and it is found that maximum error is less than 1.4~o. Again, to see the justification of taking y = 4.1 corresponding to y = o% we have also run the program for y = 5.1 and obtained almost the same results. Knowing the numerical values of velocity field, we can now calculate the skin-friction at the plate. In dimensionless form, it is given by

z

@ y =o

(9)

where v = z'/pU 2. This has been calculated by numerical differentiation using Newton's interpolation formula. 4. Discussions and Results

For the purpose of discussing the results, the velocity profiles are shown in Figures 2 and 3 corresponding to air (P = 0.71) and water (P = 7.0), respectively. It is seen from these figures that velocity of both air and water increases with Grashof number and decreases with magnetic parameter. The velocity of air is always greater than water for same values of other parameters. The numerical values of skin-friction at the plate, are given in Table I. We observe from this table that it increases with magnetic parameter and decreases with Grashof number. The values of skin-friction for water are always greater than air.

307

U N S T E A D Y H Y D R O M A G N E T I C FREE-CONVEC'TION FLOW

1.2

1.0

0.8 0.6

0.~

0.2

1.0

3.0

2.0 Y

Fig. 2.

Velocity profiles of air (P = 0.71).

1-2

1.0

0.8

t

2 --3

-, 0.6

\

Curve

G

M

t

I

3

2 3

5 6 7

3 1 3 1 3 1

8

1

1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0

0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2

0.4

0.2

- 8--

~ l

.

1.0

2.0

Y Fig. 3.

Velocity profiles of water (P = 7.0).

3.0

308

A. K. SINGH AND N. C. SACHETI TABLE I Numerical values of skin-friction M

1.0

0.0 0.5 1.0

3.0

0.0 0.5 1.0

t

Values of z P = 0.71

P = 7.0

0.2 0.4 0.2 0.4 0.2 0.4

1.0676 0.4781 1.1923 0.6615 1.3128 0.8333

1.1935 0.7178 1.3157 0.8914 1.4338 1.0540

0.2 0.4 0.2 0.4 0.2 0.4

0.6724 - 0.3524 0.8033 - 0.1437 0.9298 0.0515

1.0503 0.3667 1.1735 0.5458 1.2927 0.7137

Acknowledgement This work is being supported by U.G.C. Grant No. F. 8-5(14)/87(SR II). References Chow, Chuen-Yen: 1979, An Introduction to Computational Fluid Mechanics, John Wiley and Sons Inc., New York. Georgantopoulos, G. A., Douskos, C. N., Kafousias, N. G., and Goudas, C. L.: 1979, Letters in Heat Mass Trans. 6, 397. Gupta, A. S.: 1960, J. Phys. Soc. Japan 15, 1894. Kakutani, T.: 1958, J. Phys. Soc. Japan 13, 1504. Ong, R. S. and Nicholls, J. A.: 1959, J. Aerospace Sci. 26, 314. Raptis, A. and Singh, A. K.: 1983, Int. Comm. Heat Mass Trans. 10, 313. Rossow, V. J.: 1957, NACA Rept., No. 3971. Rossow, V. J.: 1958, NACA Rept., No. 1358. Soundalgekar, V. M.: 1965, Appl. Sci. Res. 10A, 141. Soundalgekar, V. M.: 1977, Trans. ASME J. Heat Trans. 99C, 399. Soundalgekar, V. M. and Patil, M. R.: 1980, Astrophys. Space Sci. 70, 179. Yang, H. T.: 1958, Rept of the Inst. for Fluid Dynamics and Appl. Maths., University of Maryland.