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CHAPTER - III RADIATION EFFECTS ON MHD CONVECTIVE FLOW PAST A

VERTICAL PLATE WITH VARIABLE SURFACE

TEMPERATURE IN THE PRESENCE OF HEAT SOURCE/SINK

1. INTRODUCTION

Free convection flow and heat transfa problems are of important considerations

in the ~ ~ a l design of a variety of indu&al equipment and atso in nuclear reactors and

gm~h~s ica l fluid dynamics. The transient natural convection flows over vertical bodies

has a wide range of applications in engineering ruid technology. The study of natural

c o n ~ t i o n flow past a semi-imfinite plate was first studied by Pohlhausen [I] using an

integral method, whereas the similarity method was f~ used to study this problem by

Ostmch [2] and the resulting non-linear ordinary differential equations were solved

numerically. When the surface heating is variable, the steady state free convection for a

semi-infinite vertical plate becomes quite complicated. This was studied by Sparrow and

Gregg [3] by assuming the power-law variation of the wal13;: = T' +an, where a is a

constant. Takhar et al. [4] studied transient free convection flow past a semi-infinite

vertical plate with variable surface temperature. The non-dimensional governing partial

differential equations are solved using an implicit finite-difference scheme.

Laminar free convection boundary-layer flow of an electrically conducting fluid

in the presence of magnetic field has been investigated by many researchers because of

its wide applications in industry and technology. The influence of a magnetic field on a

viscous incompressible flow of an electrically conducting fluid is of importance in many

applications such as extrusion of plastics in the manufacture of Rayon and Nylon,

purification of crude oil, magnetic materials processing, glass manufacturing control

processes, the paper industry and different geophysical systems. In many industrial

processes, the cooling of threads or sheets of some polymer materials is of importance in

the production line. Magneto convection plays an important role in various industrial

applications including magnetic control of molten iron flow in the steel industry and

liquid metal cooling in nuclear reactors.

Free convection heat transfer due to simultaneous action of buoyancy and induced

magnetic forces was investigated by Sparrow and Cess [5]. They observed that the free

convection heat transfer to liquid metals may be significantly affected by the presence of

a magnetic field, Kumari and Nath [6] studied the development of asymmetric flow of a

viscous electrically conducting fluid in the forward stagnation point repion of a two-

dimensional body and over a stretching surface with an applied magnetic field, whsn the

external stream or the stretching surface was set into an impulsive motion fmn the rest. MHD natural convection h m a non-isothermal inclined m h c e with multiple

~uc t i~n jec t ion slots embedded in a thumally mtified high-pmity medium has been

studied by' Takhar et al. [7]. The non-linear coupled parabolic partial d i h t i a l

equations are solved numerically using an implicit finite- difference scheme. Laminar

free convection boundary layer flow in the presence of a transverse magnetic field over a

heateddown pointing cone spinning with constant angular velocity about the symmetry

axis was studied by Mehmet [8] using similarity variables.

Radiative-convective heat transfer flows fmd numerous applications in glass

manufacturing, furnace technology, high temperature aerodynamics, fire dynamics and

space craft reentry. Cogley et al. [9] had shown that for an optically thin limit, the fluid

does not absorb its own emitted radiation or there is no self-absorption, but the fluid does

absorb radiation emitted by the boundaries. Grief et al. [lo] have shown that in the

optically thin limit, the physical situation can be simplified and then they derived an

exact solution to fully developed vertical channel flow for a radiative fluid. Soundalgekar

and Takhar [ll] considered the radiative free convective flow of an optically thin gray-

gas past a semi-infinite vertical plate. Radiation effects on mixed convection along an

isothermal vertical plate were studied by Hossain and Takhar [12]. Raptis and Pdik i i

[13] studied the effects of thermal radiation and fit% convection flow past a moving

vertical plate. Radiation effects on free convection flow past a vertical plate with mass

transfer are studied by Chamkha et al. [14]. The governing boundary layer equations for

these problems are reduced to non-similar form and are solved numerically by an implicit

finite difference technique. Muthucumaraswamy and Ganesan [IS] studied radiation

effects on flow past an impulsively started infinite vertical plate with variable

temperatures using the Laplace transform technique.

Many processes in new engineering areas occur at high tempembe8 and

knowledge of radiation heat transfer besides the convective heat transfer becomes very

important for the design of the pertinent equipment. Nuclear power plants, gas turbines

and the various propulsion devices for air crafts, missiles, satellites and space vehich atc

examples of such engineering areas. Accordingly, it i8 of interest to examine the e m of

magnetic field on the flow. Studying swh effect has a great importance in the application

fields, where thermal radiation and magnetic field are correlative. The process of h i o n

of metals and the process of cooling of the fust wall inside the nuclear reactor container

vessel, where the hot plasma is isolated from the wall by applying magnetic field are

examples of such fluids.

Magnetohydrodynamic flow padt a plate by the p m c e of radiation was studied

by Raptis and Massalas [16]. An analytical solution for the mean tempenrture, velocity

and the magnetic field have been anived and the effects of radiation on tern- cue

discussed. The combined effects of thermal radiation flux, thermal conductivity,

Reynolds number and non-Darcian (Forcheimmer drag and Brinkman boundary

resistance) body f m s on a steady laminar boundary layer flow along a vertical suface

in an idealized geological porous medium were investigated by Takhar et al. [17]. The

effects of radiation on flee convection flow and mass transfer past a vertical isothermal

cone surface with chemical reaction in the presence of magnetic field was investigated by

Ahmed [IS] using similarity variables. The effect of radiation on magnetohydrodynamic

unsteady fne-convection flow past a semi-infinite vertical porous plate was studied by

Abd El-Naby et al. [19]. The effects of thermal radiation and porous drag forces on the

natural convection heat and mass transfer of a viscous, incompressible, gray, absorbiig

emitting fluid past an impulsively started moving vertical plate adjacent to a non-Darcian

porous regime was studied by Anwar et al. [20].

The heat soucodsink effects in thermal convection, are significant where there

may exist a high temperature differences between the surface (e.g. space craft body) and

the ambient fluid. Heat generation is also important in the context of exothermic or

endothermic chemical reactions. Sparrow and Cess [21] provided one of the earliest

studies using a similarity approach for stagnation point flow with heat source/sink which

vary in time. Pop and Soundalgekar [22] studied unsteady free convection flow past an

infinite plate with constant suction and heat source. Much later Takhar et al. [23]

presented one of the most robust studies of thermal and concentration boundary layers

with MHD effects for the w e of a point sink. Takhar et al. [24] extended this analysis to

examine combined variable lateral mass flux (wall injection/suction), heat source effects

and hall current effeots on doublediffusive boundary layers under strong magnetic fields.

Sahoo et al. [25] studied magnetohydrodynamic unsteady free convection flow past an

infinite vertical plate with constant suction and heat sink.

However, convective flow under the influence of magnetic field and thermal

radiation past a vertical plate subject to a variable surface temperature in the presence of

heat soundsink has not received the attention of any researcher, Hence, the object of the

present investigation is to study the combined effects of magnetic field and thermal

radiation past a semi-infinite vertical plate subject to a variable surface temperature. The

set of non-dimensional govming equations are solved by an implicit finite difference

method of Crank-Nicolson type. The behavior of the velocity, temperature, skin-friction

and Nusselt number has been discussed for variations in the governing parametem.

2. MATHEMATICAL ANALYSIS

A twodimensional unsteady flow of an electrical conducting, radiating, viscous

incompressible fluid past a semi-infinite vertical plate with temperature~(x) = TA + x",

(where n is a constant) is considered. The surrounding fluid which is at nst has a

temperaturel",. The co-ordinate system is chosen such that x- axis measured along the

plate vertically upward and y- axis is taken n o d to the plate. It is assumed that the

viscous dissipation effect is negligible in the energy equation. A unifonn magnetic field

is applied in the direction perpendicular to the plate. The fluid is assumed to be of small

electrical conductivity so that the magnetic Reynolds number is much less than unity and

hence the induced magnetic field is negligible in comparison with the applied magnetic

field. All the fluid properties are considered constant except the influence of density

variation in the body force term (Boussinesq's approximation). Then, in the absence of an

input electric field, the boundary layer equations which govern the flow field m

Mass conservation

Momentum conservation

Thermal Energy conservation

where u, v are the velocity components in x- and y- directions respectively, t '- the time,

o- the kinematic viscosity, g- the acceleration due to gravity, /I- the coefficient of

volume expansion, T'- the temperature of the fluid in the boundary layer, T, - the

temperature of the fluid far away from the plate, (T- the electrical conductivity of the

fluid, Bo- the magnetic induction, p - the density of the fluid, c p - the specific heat at

constant pressure, k - the thermal conductivity, q, - the radiation heat flux and Qo- the

heat generation constant.

The radiating gas is said tc be non-gray, if the absorption co-efficient K, is

dependent on the wave length. The equation that describes the conservation of radiative

transfer in a unit volume, for all wave lengths is

A; = 1 K A ( T ~ ) [ ~ ~ * A ( T ~ ) - G A M (2.4)

where I,,is the spectral density for a block body, is the radiative heat flux and the

incident radiation G, is defined as

G, = jl*Ja)a R=4x

where R is the solid angle.

Now for an optically thin fluid exchange radiation with plate at the average I

temperature value T, and according to the equation (2.5) and Kirchoffs law, the incident

relation is given by

GA = 41d,,(C) = 4eb,(Ti) . Thus equation (2.4) reduces to

& = 4 ~ ~ L ( f ) [ e b A ( T ' ) - e ~ A ( T i ) ] d i l (2.6)

Expanding eb,(Tf) and KA(T1) in Taylor's series about Ti for small(^ - TL), then we

can rewrite the radiative flux divergence as

A&=4(Tf- ti)[^,($) d = 4 ( T 1 - T L ) T (2.7) W

where KAw = K,(TL) is the mean absorption coefficient, T = % K, ($1 and ebA is

Plank's function.

Hence for an optical thin limit for a non-gray gas near equilibrium, the following

relations hold

A; = 4(T1-T$ (2.8)

and hence

Thus the energy equation (2.3) reduces to

The initial and boundary conditions are

t f < O:u=O, v=O, T' = T: for all x and y

On introducing the following non-dimensional quantities

where Gr, , M , Ra , Pr and pj are the Grehof number, magnetic field parameter, radiation

parameter, Prandtl number and heat generation/absorption coefficient respectively,

equations (2.1), (2.2) and (2.10) reduce to the following dimensionless form

The corresponding initial and boundary conditions in a dimensionless form are

tsO:U=O, V=O, T=O for all x and y

The local nondimensional skin friction and the local Nusselt number are given by

Also, the non-dimensional average skin friction and the average Nusselt number are

given by

3. NUMERICAL TECHNIQUE

The governing equations (2.13)-(2.15) represent coupled system of non-linear

partial differential equations, which are solved numerically under the initial and boundary

conditions (2.16) using Crank-Nicolson implicit finite difference scheme. The fmite

difference equations corresponding to equations (2.13)-(2.15) are

The region of integration is considered as a rectangle with sides X, (= 1) and

Y, (=30), where Y, corresponds to Y =a, which lies very well outside the

momentum and thermal boundary layers:. The maximum of Y was chosen as 30 after

some preliminary investigations so that the last two of the boundary conditions (2.16) are

satisfied. TO obtain an economical and reliable grid system for the computations, a grid

independence is performed. Hence the grid system of 20x120 is selected for all

subsequent analysis with AX = 0.05 and A Y = 0.25. Also the time-step size dependency

is carried out, which yields At = 0.01 for reliable results. Here, the subscript i-designates

the grid point along the Xdirection, j-aiong the Y- direction and the superscript n along

the tdirection. During any one time step, the coefficients U:, and y;appearing in the

difference equations are treated as constants. The values of U, V and T are known at all

grid points at t = 0, from the initial conditions. The computations of U, V and T at time

level (n t l ) using the values at previous time level (n) are carried out as follows. The

finite difference equation (2.21) at every internal nodal point on a particular i-level

constitute a tridiagonal system of equations. Such a system of equations are solved by

Thomas algorithm as described in Camahan et a1 [26]. Thus, the values of T are found at

every nodal point for a particular i at (ntl)' time level. Using the values of T at (ntl)'

time level in the equation (2.20), the values of U at ( n t l ) ~ time level are found in a

similar manner. Thus, the values ofT and U are known on a particular i-level. Finally, the

values of V are calculated explicitly using the equation (2.19) at eveIy nodal point on a

particular i-level at (ntl)' time level. This process is repeated for various i-levels. Thus

the values of T, U and V are known, at all grid in the rectangular region at (ntl)'

time level.

Computations are carried out until the steady-state is reached. The steady-state

solution is assumed to have been reached, when the absolute difference between the

values of U, as well as temperature T at two consecutive time steps are less than 1v5 at

all grid points.

The derivatives involved in equations (2.17) and (2.18) are evaluated using a five

point approximation formula and then the integrals are evaluated using Newton-Cotes

closed integration formula.

4. STABILITY AND CONVERGENCE OF THE FINITE DIFFERENCE

SCHEMI$

The stability criterion of the finite difference scheme for constant mesh sizes are

examined using Von-Neumann technique as explained by Carnahan et a1 [27]. The

general term of the Fourier expansion for U and T at a time arbitrarily called t = 0, are

assumed to be of the form el" ei*(here i =G). At a later time t, these terms will

become

Substituting (2.22) in equations (2.20) and (2.21), under the assumption that the

coefficients U and T are constants over any one time step and denoting the values after

one time step by F' and G' . After simplification, we get

Equations (2.23) and (2.24) can be written as

where

AAer eliminating G'and H' in equation (2.25), using equation (2.26), the resultant

equation is given by

Equations (2.27) and (2.26) can be written in matrix as

Now, for the stability of the finite difference scheme, the modulus of each Eigen value of

the amplification matrix does not exceed unity. Since the matrix (2.28) is triangular, the

Eigen values are diagonal elements. The Eigen values of the amplification matrix are

(1 -A)/(l +A) and (1 -B)/(l t B ) . Assuming that, U is everywhere non-negative and V is

[ I

everywhere non-positive, we get

M A=2a sinZ (aAX/2) + 2csin2 (PAYl2) + i (a sinam-b sin BAY) + - Ar

2

1-A & - - it^ (I+AXI+B)

0 - I-B I t B

2(1+ A)(I + B )

- B )

Since the real part of A is greater than or equat to zero,l(l- A)/(I t A) 1 S1 always.

Similarly [(I - B)l(l t B) 1 51 andl(l- ~ ) l ( l + E) 1 11.

Hence, the finite difference scheme is unconditionally stable. The local truncation error is

0 (At2 t AY' t AX) and it tends to zero as4 ,AY and AX tend to zero. Hence the

scheme is compatible. Stability and compatibility ensures convergence.

5. RESULTS AND DISCUSSION

A representative set of numerical results is shown graphically in Figs.1-9, to

illustrate the influence of physical parameters viz., magnetic parameter M, Prandtl

number Pr, radiation parameter Ra, heat generation/absorption parameter4 and exponent

in the power law variation of the wall temperature n on the velocity, temperature, skin-

friction and Nusselt number. Here the value of Pr is chosen as 0.71, which corresponds to

air. The other parameters are arbitrarily chosen. In order to ascertain the accuracy of our

numerical results, the present result is compared with the work available in the literature.

The present numerical results for steady state velocity profiles at X = 1.0 with 4 = 0,

n = 0, Ra = 0, M = 0 are compared with Takhar et al. [4] which is shown in figure 1. It is

observed that the results are in good agreement with each other.

The transient velocity profiles at X = 1.0 for different values of magnetic field

parameter and exponent n are calculated numerically and are presented in Fig. 2. The

velocity of the fluid increases with time until a temporal maximum is reached and

thereafter a moderate reduction is observed until the ultimate steady state is reached. It is

observed that the time taken to reach the steady state is more for higher values of

magnetic field parameter in comparison with lower values of magnetic field parameter.

The effect of a transverse magnetic field on an electrically conducting fluid gives rise to a

resistive type force called Lorentz force. This force has a tendency to slow down the

motion of the fluid and to increase its temperature. Also we observe that the d y - s t a t e

velocity decreases as n increases at all the positions from the leading edge. T i e taken to

reach the steady state increases as n increases.

Fig. 3 shows that the velocity distribution of air for different values of radiation

parameter Ra and heat source parameter/. It is observed that the velocity of air increases

as the radiation parameter increases. More time is required to reach the steady state for

higher values of radiation parameter Ra.' Due to the presence of heat source energy, the

velocity of air is found to increase. Further, it is also noticed that the more time is

required to reach the steady state for higher values of heat source parameter(.

In Fig. 4, the effect of magnetic field parameter M and n on the transient

temperature is shown for air. It is found that the temperature reduces with the increasing

values of n. The effect of n is more near the leading edge of the plate. The temperature of

the fluid decreases as the magnetic field parameter Mdecreases.

The transient temperature profiles for air at their temporal maximum and steady

state against the coordinate Y at X = 1.0 for different values of the radiation parameter Ra

and heat source parameter / are shown in Fig. 5. It is seen that the t e m p e m increases

as the radiation parameter Ra increases, This result qualitatively agrees with expectations,

since the effect of radiation and surface temperature is to increase the rate of energy

transport to the fluid, thereby increasing the temperature of the fluid. It is noticed that the

time required to reach the steady state flow increases with the increasing value of

radiation parameter Ra, this implies that the existence of radiation helps to achieve the

steady state slowly. The presence of a heat source in the boundary layer generates energy,

which causes the temperature of the fluid to increase.

The study of the effects of the physical parameters on the local as well as average

shearing stress and the rate of heat transfer is important in the heat transfer problems. The

steady state values of local skin friction and Nusselt number for different values of

radiation parameter h, magnetic fikld parameter M, heat source ( and exponent n are

calculated numerically and are depicted graphically in Figs. 6 and 7 respectively.

The local skin-friction decreases with the increasing values of n and the effect of

n over the local skin friction is more and reduces gradually with increasing the distance

along the surface. The local wall $hear stress decreases with the increasing values of M

Further it is noticed that the local skin fiiction increases with the increasing values of

radiation parameter Ra and heat source ( .The local Nusselt number reduces with the

increasing values of exponent n along b e surface. It is observed that the heat transfer

increases with the increasing values of magnetic field parameter M. It is observed that the

local Nusselt number increases with the increasing values of radiation parameter Ra and

heat source(. Average skin friction and Nusselt number are presented in Figs. 8 and 9

for different values of Ra, n, 4 and M. The influence of n on average skin friction is

more when exponent n is reduced. It is also observed that average skin friction for air

increases with the increasing values of Ra and(. Also, it is noticed that average skin

friction decreases as M increases. At an initial time, higher values of average Nusselt

number were observed and the average Nusselt number decreases with time and reaches

steady state after some time. It was observed that for short times, the average Nusselt

number was constant at each level of various parameters. This shows that initially there is

only heat conduction. The average Nusselt number increases as M or Ra or / or n

increases.

0 0 2 4 6 8

Y

Fig. 1 Comparisonn of steady state velocity profiles at X=1.0

Fig.2 Transient Velocity Profiles at X=1.0 for different M and n (*-Steady state)

Fig.3 Transient velocity profiles at X=1.0 for different Ra and 4 (*-Steady state)

0 0 4 8 Y 12 16 Fig4 Transient temperature profiles at X=1.0 for different

M and n (*-Steady state)

49

0 4 R Y 12 16

Fig.5 Transient temperature profiles at X=1.0 for different Ra and 4 (*-Steady state)

0 0.25 0.5 0.75 1

X Fig.6 Local skin friction

50

0 015 0.5 0.75 1 X

Fig.7 Local Nusselt number

0.5 1.0 0.02 0.5 1.0 0.5 0.02 0.5 0.5 0.5 0.04 0.5 0.5 0.5 0.02 0.5

F'r = 0.71

Fig9 Average Nusselt number

REFERENCES

1. Pohlhausen E. [1921], Der Wmeaustausch zwischen festen korpen and

Flussigkeiten mit Kleiner Reibung und Kleiner Wmeleitung, ZAMM, Vol.1,

Pp.115-121.

2. Ostrach S. [1953], An analysis of laminar free convection flow and heat transfer

about a flat plate parallel to the direction of the generating body force, NACA

Report-TR1 I 1 1, Pp. 63-79.

3. Sparrow E.M., and Gregg J.L. [1958], Similar Solutions for free convection from

a non-isothermal vertical plate, Transactions of ASME of Journal of Heat transfer,

Vol.8O,Pp.379-386.

4. Takhar H.S., Ganesan P., Ekambavannan K., and Soundalgekar V.M. [1997],

Transient free convection flow past a semi-infinite vertical plate with variable

surface temperature, International Journal for Numerical methods of Heat and

Fluid flow, Vo1.7, No.4, Pp.280-296.

5. Sparrow E.M. and Cess R.D. [1961], The effect of a magnetic field on free

convection heat transfer, International Journal of Heat and Mass Transfer,

V01.3(4), 4.267-274.

6. Kumari M. and Nath G [1999], Development of two dimensional boundary layer

with an applied magnetic field due to an impulsive motion, Indian J. Pure Appl.

Math. Vo1.30,Pp.695-708.

7. Takhar H.S., Chamkha A.J, and Nath G. [2003], Effects of non-uniform wall

temperature or mass transfer infinite sections of an inclined plate on the MHD

natural convection flow in a temperature stratified high-porosity medium,

Jnt. J. thermal sciences, Vo1.42,Pp.829-836.

8. Mehmet Cem Ece [2006], Free convection flow about a vertical spinning cone

under a magnetic field, Applied Mathematics and Computation, Vol. 179, Pp.231-

242.

9. Cogley A.C., Vincenti W.G., and Gill S.E. [1968], Differential approximation for

radiative transfer in a non-gray gas near equilibrium, AIAA J, Vo1.6, Pp.551-553.

10. Grief R., Habib L.S. and Lin L.C. [1971], Laminar convection of a radiating gas

in a vertical channel, J. Fluid Mech., Vo1.45,Pp.5 13-520.

11. Soundalgekar V.M. and Takhar H.S. [1993], Radiation effects on free convection

flow past a semi-infinite vertical plate, Modelling Measurement and Control, Vol.

B51, Pp.31-40.

12. Hossain M.A .and Talchat H.S. [1996], Radiation effects on mixed convection

along a vertical plate with uniform surface temperature, Heat and Mass Tr.,

VoI.3l,Pp.243-248.

13. Raptis A. and Perdikis C. [1999], Radiation and free convection flow past a

moving plate, App. Mech and Engg., Vo1.4(4), Pp.817-821.

14. Chamkha A.J., Takhar H.S. and Soundalgekar V.M. [2001], Radiation effects on

free convective flow past a semi-infinite vertical plate with mass transfer,

Chemical Engineering Journal, Vo1.84, Pp. 335-342.

15. Muthucumaraswamy R. and Ganesan P.[2003], Radiation effects on flow past an

impulsively started infinite vertical plate with variable temperature,

Int. J. Appl. Mech. Engg., Vo1.8 (I), Pp.125-129.

16. Raptis P. and Massalas C.V. [1998], Magnetohydrodynamic flow past a plate by

the presence of radiation, Heat and Mass Tr., Vo1.34, Pp.107-109.

17. Takhar H.S., Beg O.A., Chamkha A.J., Filip D. and Pop 1. [2003], Mixed

radiation-convection boundary layer flow of an optically dense fluid along a

vertical at plate in a non-Darcy porous medium, Int. J. Applied Mechanics and

Engineering, Vo1.8(3), Pp.483-496.

18. Ahmed A. Afify [2004], The effect of radiation on free convective flow and mass

transfer past a vertical isothermal cone surface with chemical reaction in the

presence of a transverse magnetic field, Can. J. Physics, Pp. 447-458.

19. Abd El-Naby M.A.; Elsayed M. E. Elbarbary and Nayed Y. AbdElazem [2004],

Finite difference solution of radiation on MHD unsteady free-convection flow

over vertical porous plate, Applied Mathematics and Computation, Vo1.151,

4.327-246.

20. Anwar Beg O., Zueco J., Takhar H.S. and Beg T.A. [2008], Network Numerical

Simulation of Impulsively-Started Transient Radiation-Convection Heat and Mass

Transfer in a Saturated Darcy-Forchheimer Porous Medium, Nonlinear Analysis

Modelling and Control, Vo1.13(3), Pp.28 1-303.

21. S p m w E.M. and Cess RD. [1%1], Temperaturedependent heat sources or

sinks in a stagnation point flow, Applied Scientific Research, Vol. A 10, Pp. 185.

22. Pop I, and Soundalgekar V.M. [1962], Viscous dissipation effects on unsteady

free convective flow past an infinite vertical porous plate with variable suction,

Int. J. Heat Mass transfer, ~01.17, Pp. 85-92

23. Takhar H.S., Surma Devi C.D. and Nath G. [1986], Heat and Mass Transfer for a

point sink in a plane with a magnetic field, Mechanics Research Communications,

V01.13, Pp.71-78.

24. Takhar H.S., Ram P.C. and Singh S.S. [1992], Hall effects on heat and mass

transfer flow with variable suction and heat generation, Astrophysics and Space

Science, Vo1.191, Pp.101-106.

25. Sahoo P.K., Datta N. and Biswal S. [2003], MHD unsteady free convection flow

past an infinite vertical plate with constant suction and heat sink, Indian Journal of

Pure and Applied Mathematics, Vo1.34(1), Pp.145-155.

26. Camahan B., Luther H.A. and Wilkes J.O. [1969], "Applied Numerical Methods",

John Wiley and Sons, New York.