International Journal of Theoretical and Applied Mechanics.

ISSN 0973-6085 Volume 12, Number 1 (2017) pp. 71-81

© Research India Publications

http://www.ripublication.com

Magneto Hydro Dynamics Convective Flow Past a

Vertical Porous Surface in Slip-Flow Regime

G.Dharmaiah1, Uday Kumar.Y.2 and N.Vedavathi3

1Department Of Mathematics,NarasaraoPeta Engineering College, Guntur, A.P, India.

2Department Of Mathematics,Hindu College,Guntur, A.P, India.

3Department Of Mathematics,K.L.University, Guntur, A.P, India.

Abstract

This paper deals with the influences of heat and mass transfer on Two

dimentional MHD free convection flow, laminar and boundary layer of

viscous fluid along a semi vertical permeable moving plate, a uniform

transverse magnetic field, thermal and concentration bouyancy effects.The

governing nonlinear partial difference equations have been decreased to the

coupled nonlinear ordinary differential equations by small perturbation

technique. Numerical evalution of the analytical results is performed and some

graphical results for the velocity, temperature and concentration profiles with

in the boundary layer.

Keywords: MHD, Heat Transfer, Mass Transfer, Slip-flow, vertical plate.

1. INTRODUCTION

Combined heat and mass transfer from different processes with porous media has a

wide range Engineering and Industrial applications such as enhanced oil recovery,

underground energy transport, geothermal reservoirs, cooling of nuclear reactors,

drying of porous solids, packed-bed catalytic reactors and thermal insulation. The

process of heat and mass transfer is encountered in aeronautics, fluid fuel nuclear

reactor, chemical process industries and many engineering applications in which the

72 G.Dharmaiah, Uday Kumar.Y. and N.Vedavathi

fluid is the working medium. The applications are often found in situation the such as

fiber and granules insulation, geothermal systems in the heating and cooling chamber,

fossil fuel combustion, energy processes and Astro-physical flows. Further, the

magneto convection place an important role in the control of mountain iron flow in

the steady industrial liquid metal cooling in nuclear reactors and magnetic separation

of molecular semi conducting materials. In certain porous media applications such as

those involing heat removal from nuclear fuel debris, underground disposal of

radioactive waste material, storage of food stuffs, and exothermic and /or endothermic

chemical reactions and dissociationg fluid in packed-bed reactors, the working fluid

heat generation (source) or absorption (sink) effects are important.

Convection in porous media can be applied to underground ground water hydrology,

iron blast furnaces, cooling of nuclear reactors, solar power collectors, energy

efficient drying processes, cooling of nuclear fuel in shipping flasks, cooling of

electronic equipment’s, coal gasification, and wall cooled catalytic reactors, and

natural convection in earth’s crust. More examinations of the applications related to

convective flows in porous media in Nield and Bejan [1] can be found. The

fundamental problem of flow through porous media has been investigated

extensively [2, 3]. A survey of Magneto Hydro Dynamics revises in the technological

spheres in Moreau [6] can be found. When heat and mass transfer occur

instantaneously between the fluxes, the driving potentials are of more intricate nature.

The uses of magnetic field to control the flow and heat transfer processes in fluids

near different types of boundaries are familiar now. It has led to significant interest in

the study of boundary layer. Hossain [5] analysed on MHD free convective heat

transfer for a Newtonian fluid. The natural convection flows adjacent to both vertical

and horizontal surface, which result from the combined buoyancy effects of thermal

and mass diffusion, was first investigated by Gebhart et al. [7] and Pera et al. [8].

While Soundalgekar [9] investigated the situation of unsteady free convective flows

wherein the effects of viscous dissipation on the flow past an infinite vertical porous

plate was highlighted. In the course of analysis, it was assumed that the plate

temperature oscillates in such a way that its amplitude is small. Later, Chen et al [10]

studied the combined effect of buoyancy forces from thermal and mass diffusion on

forced convection.

The present analysis discussed here in, is based on the study as referred and suggested

by Soundalgekar [11]. Under the assumptions made by Sharma et al. [12] have also

discussed the free convection flow past a vertical plate in slip-flow regime. They had

quoted several applications that occur in several engineering applications wherein

heat and mass transfer occurs at high degree of temperature differences.

In all above presentations, the plate was assumed to be maintained at a constant

temperature, which is also the temperature of the surrounding stationary fluid.

However, in many applications that occur in industrial situations are not those simple

and at high temperatures, quite often the plate temperature starts oscillating about a

non-zero mean temperature. In many practical applications, the particle adjacent to a

solid surface no longer takes the velocity of the surface. The particles at the surface

always possess a finite tangential velocity and it "slips" almost along the surface.

Magneto Hydro Dynamics Convective Flow Past a Vertical Porous Surface … 73

Therefore, the flow regime is called the slip - flow regime and such an effect cannot

be neglected. Hence, in the fitness of the industrial and scientific applications and to

be more realistic the effect of periodic heat and mass transfer on unsteady free

convection flow past a vertical flat porous plate under the influence of applied

transverse magnetic field has been examined. The slip flow regime it is assumed that

the suction velocity oscillates in time about a non-zero constant mean because in

actual practice temperature, species concentration and suction velocity may not

always be uniform.

2. FORMULATION OF THE PROBLEM

An unsteady free MHD convective flow of a viscous incompressible fluid past an

infinite vertical porous flat plate in slip-flow regime, with periodic temperature and

concentration when variable suction velocity distribution * *

0 1 i tV V Ae

is

fluctuating with respect to time is considered. A co-ordinate system is employed

with wall lying vertically in x y plane. The x axis is taken in vertically upward

direction along the vertical porous plate and y axis is taken normal to the plate.

Since the plate is considered infinite in the x direction, hence all physical quantities

will be independent of x . Under these assumption, the physical variables are purely

the functions of y and t only. In the fairness of the realistic situation by neglecting

viscous dissipation and then assuming variation of density in the body force term

(Boussinesq's approximation) the problem can be governed by the following set of

equations:

22

* 0 0

0 21 i tu u uV Ae g T T g C C u u

t y y K

(1)

2

*

0 21 i t

PT T TC V Ae kt y y

(2)

2

*

0 21 i tC C CV Ae D

t y y

(3)

The boundary conditions of the problem are:

*

*

, , at 0

0, , as

i t i t i tw w w wu B e T T T T e C C C C e y

u T T C C y

(4)

74 G.Dharmaiah, Uday Kumar.Y. and N.Vedavathi

We now introduce the following non-dimensional quantities into Eqs. (1) to (4)

2

0 0

2

0 0

3 3

0 0

2 2

0 0

2 2

0 0

4, , , , ,

4

. , ,

, , , , .

w

w w

w

p p

y V t V T Tuy t uV V T T

g T T g C CC CC Gr GcC C V V

C C K V BPr Sc M K Bk k D V V

The subscript denotes the free stream condition. Then equations (1) to (3) reduce

to the following non-dimensional form:

2

2

11

4

i tu u u uAe Gr GcC Mut y Ky

(5)

2

2

1 11

4 Pr

i tAet y y

(6)

2

2

1 11

4

i tC C CAet y Sc y

(7)

The boundary conditions to the problem in the dimensionless form are:

, 1 , 1 at 0

0, 0, 0 as

i t i t i tu Be e C e yu C y

(8)

3. SOLUTION OF THE PROBLEM

Assuming the small amplitude oscillations (ε << 1), we can represent the velocity u,

temperature θ and concentration C near the plate as follows:

0 1, i tu y t u y u y e (9)

0 1, i ty t y y e (10)

0 1, i tC y t C y C y e (11)

Substituting (9) to (11) in (5) to (7), equating the coefficients of harmonic and non

harmonic terms, neglecting the coefficients of 2 , the solutions are given by

Pr

0

yy e (12)

0

ScyC y e (13)

Magneto Hydro Dynamics Convective Flow Past a Vertical Porous Surface … 75

2 Pr

0 1 3 4

m y y Scyu y m e m e m e (14)

6 Pr

1 5 7

m y yy m e m e (15)

9

1 8 10

m y ScyC y m e m e (16)

6 912 2 Pr

1 11 13 14 15 16 17

m y m ym y m y y Scyu y m e m e m e m e m e m e (17)

Using equations (12)-(17), the velocity, temperature and concentration can be

obtained as follows:

6 912

2

2

11 13 14Pr

1 3 4 Pr

15 16 17

,

m y m ym ym y y Scy i t

m y y Scy

m e m e m eu y t m e m e m e e

m e m e m e

6Pr Pr

5 7,m yy y i ty t e m e m e e

9

8 10,m yScy Scy i tC y t e m e m e e

4. RESULTS AND CONCLUSION The effect of Grashof number on velocity profiles is illustrated in Fig. 1. It is noticed

that increase in Grashof number contributes to the increase in velocity of the fluid.

Further, it is noticed in the boundary layer region the velocity increases and thereafter,

it decreases. Also, far away from the plate, not much of significant effect of Grashof

number is noticed.

Fig.1. Effect of Gr on velocity profiles

The variation of velocity with respect to Gc is noticed in Fig. 2. It is observed that as

Gc increases, in general the velocity also increases. As was seen in earlier case, the

76 G.Dharmaiah, Uday Kumar.Y. and N.Vedavathi

rise in velocity is noticed in the boundary layer region and thereafter, it decreases. The

contribution of Schmidt number on the velocity profiles is noticed in Fig. 3. It is seen

that as Schmidt number increases, the velocity decreases. It is seen that the

contribution by Sc is not that significant at the boundary but as we move far away

from the plate, the dispersion due to Sc is found to be more distinct and the effect of

Sc could be noticed. The contribution of the magnetic intensity over the velocity field

is observed in Fig. 4. It is observed that, as the magnetic intensity is increased, the

fluid velocity decreases. Such an observation is in tune with the realistic situation that,

as the magnetic intensity suppresses the fluid velocity. The influence of the porosity

of the fluid bed on the velocity profiles is shown in Fig. 5. It is noted that, as the pore

size of the fluid bed increases, the velocity decreases. This is in agreement with the

real life situation. As the pore size of the fluid bed increases, the fluid over the bed

gets trapped into the pores resulting in the decrease of the fluid velocity. Variation in

the velocity of the fluid medium with respect to suction parameter is illustrated in Fig.

6. It is noticed that as the suction parameter is increased, the velocity of the fluid

medium is found to be decreasing. At the boundary, the suction parameter does not

show any influence. However, as we move far away from the plate the influence

appears to be more predominant. The influence of the amplitude on the velocity

profiles is shown in Fig. 7. It is noticed that, as the amplitude increases, the fluid

velocity decreases and also at times a backward flow is noticed. Due to the

percolation of the fluid into the boundary such a backward flow is noticed. However,

as we move far away from the bounding surface, the effect of such backward flow is

found to be negligible and the influence of amplitude is not seen.

Fig. 8 illustrates the influence of Prandtl number on the temperature field. It is

observed that the prandtl number has significant contribution over the temperature

field. From the illustrations it is seen that as the Prandtl No increases, the temperature

decreases. Further, not much of significant contribution by the Prandtl number is

noticed at the boundary and also as we move far away from the plate. The influence

of suction parameter on the temperature field is illustrated in Fig. 9. It is seen that as

the suction parameter increases, the temperature field decreases. It is observed that the

increase in the suction parameter contributes to the parabolic nature of the

temperature profiles. Further, far away from the plate, it is noticed that the suction

parameter tends to loose its significance. Fig. 10 illustrates the effect of Sc on

concentration profiles. It is seen that as Sc increases, the concentration is found to be

decreasing. Further, increase in Sc contributes to the parabolic nature of the

concentration profiles. Also, the dispersion in concentration is found to be more

distinctive as we move far away from the bounding surface.

Magneto Hydro Dynamics Convective Flow Past a Vertical Porous Surface … 77

Fig. 2. Effect of Gc on velocity profiles

Fig. 3. Effect of Sc on velocity profiles

Fig. 4. Effect of Magnetic field on velocity profiles.

78 G.Dharmaiah, Uday Kumar.Y. and N.Vedavathi

Fig. 5. Effect of Permeability parameter on velocity

Fig. 6. Effect of Suction parameter on velocity

Fig. 7. Effect of B on velocity profiles.

Magneto Hydro Dynamics Convective Flow Past a Vertical Porous Surface … 79

Fig. 8. Effect of Pr on temperature profiles.

Fig. 9. Effect of Suction parameter on temperature.

Fig.10. Effect of Sc on concentration

80 G.Dharmaiah, Uday Kumar.Y. and N.Vedavathi

ACKNOWLEDGMENTS

The authors are very much grateful to the reviewers for their constructive and

valuable suggestions for further improvement in this paper.

REFERENCES:

[1] D .Nield, A. Bejan “Convection in porous media”,Newyork, Springer, 1999.

[2] P.Cheng, “Heat transfer in geothermal system”, Adv Heat Transfer 978,14:1-

105.

[3] N. Rudraiah, “Flow through and past porous media”,Encyclopedia of Fluid

mechanics, Gulf Publ,1986,5:567-647.

[4] Y.J. Kim, “Unsteady MHD convective heat transfer past a semi-infinite

vertical porous moving plate with variable suction”, Int J Eng Sci, 2000,

38:833–45.

[5] M. A. Hossain, “Viscous and Joule heating effects on MHD free convection

flow with variable plate temperature”, Int. J. Heat Mass Transfer, 35, 3485,

1992.

[6] R. Moreau, Magnetohydrodynamics, Kluwar Academic, Dordrecht, The

Netherlands, 1990.

[7] Gebhart, B. and L. Pera, "The nature of vertical natural convection flow from

the combined buoyancy effects on thermal and mass diffusion", Int. J. Heat

Mass Transfer, 14, 2024-2050 (1971).

[8] Pera, L. and B. Gebhart, "Natural convection flows adjacent to horizontal

surface resulting from the combined buoyancy effects of thermal and mass

diffusion", Int. J. Heat Mass Transfer, 15, 269-278 (1972).

[9] Soundalgekar, V.M., "Viscous dissipation effects on unsteady free convective

flow past an infinite vertical porous plate with constant suction", Int. J. Heat

Mass Transfer, 15, 1253-1261 (1972).

[10] Chen, T.S., C.F. Yuh and A. Moutsoglou, "Combined heat and mass transfer

in mixed convection along a vertical and inclined plates", Int. J. Heat Mass

Transfer, 23, 527-537 (1980).

[11] Soundalgekar, V.M. and P.D. Wavre, "Unsteady free convection flow past an

infinite vertical plate with constant suction and mass transfer", Int. J. Heat

Mass Transfer, 20, 1363-1373, (1977a).

[12] Sharma, P.K. and R.C. Chaudhary, "Effect of variable suction on transient free

convection viscous incompressible flow past a vertical plate with periodic

temperature variations in slip-flow regime", Emirates Journal for Engineering

Research, 8, 33-38 (2003).

Magneto Hydro Dynamics Convective Flow Past a Vertical Porous Surface … 81

APPENDIX:

1

1,M M

K

1 3 4 ,m m m

1

2

1 1 4,

2

Mm

3 2

1

,Pr Pr

GrmM

4 2

1

,Gcm

Sc Sc M

5

4 Pr1 ,

iAm

2

6

Pr Pr Pr,

2

im

7

4 Pr,

iAm

8

41 ,

iAScm

2

9 ,2

Sc Sc i Scm

10

4,

iAScm

11 13 14 15 16 17 ,

i tBm e m m m m m

1

12

1 1 4,

2

M im

5

13 2

6 6 1

,/ 4

m Grm

m m M i

8

14 2

9 9 1

,/ 4

m Gcm

m m M i

1 2

15 2

2 2 1

,/ 4

m m Am

m m M i

3 7

16 2

1

Pr,

Pr Pr / 4

m A m Grm

M i

4 10

17 2

1

./ 4

m ASc m Gcm

Sc Sc M i

82 G.Dharmaiah, Uday Kumar.Y. and N.Vedavathi