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7.1 INTRODUCTION

Heat and mass transfer phenomenon occurs in a variety of problems. In nature this

occurs in buoyancy induced motions in the atmosphere. This finds variety of applications

in science and technology. Singh and Rana (1992) investigated the three–dimensional

flows and heat transfer through a porous medium. They observed that heat transfer

decreases with increase in Prandtl number and permeability parameter, while it increases

with increase in Eckert number. Sacheti et al (1994) investigated the unsteady

hydromagnetic flow past a vertical plate subject to constant heat flux. They obtained an

exact solution of this problem using Laplace transform. Velocity and skin friction of the

flow have been presented for water. They noted that the magnetic field has a retarding

effect on the velocity while skin friction at the plate increases with it. Das et al (2001)

described the three–dimensional fluctuating free convective flow through a porous

medium bounded by an infinite vertical porous plate. Sharma et al (2002) and Chaudhary

and Chand (2002) have solved the problems of three–dimensional viscous flow and heat

transfer along a porous plate in the presence of sinusoidal suction.

The study of porous media together with the magneto - hydrodynamics has been

studied by many researchers. The flow past a vertical plate embedded in a saturated

porous media has been studied by Na and Pop (1983). They prescribed non-uniform wall

temperature and a non-uniform wall heat flux. The governing equations were solved

numerically by using two-point finite-difference method. Singh et al (2003) have

investigated Hydromagnetic heat and mass transfer in MHD flow of an incompressible,

electrically conducting, viscous fluid past an infinite vertical porous plate embedded with

porous medium of time dependent permeability under oscillatory suction velocity normal

to the plate. Chaudhary and Jain (2007) studied the MHD flow past an infinite vertical

oscillating plate embedded in porous medium. The governing equations were solved by

using Laplace transform technique. They observed that the skin-friction increased with an

increase in Schmidt number, Prandtl number, magnetic parameter, while it decreases with

an increase in the value of Grashof number, modified Grashof number, permeability

parameter and time. Nusselt number increases with an increase in Prandtl number while

temperature decreased with an increase in the value Prandtl number.

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Zueco and Ahmed (2010) analyzed MHD flow of an incompressible viscous

electrically conducting fluid past an infinite vertical porous plate with combined heat and

mass transfer. The porous plate is subjected to a constant suction velocity as well as a

uniform stream velocity. The governing equations are solved by the perturbation

technique and a numerical method. An increase in the heat source/sink or the Eckert

number is found to strongly enhance the fluid velocity. It is found that the velocity, the

fluid temperature, and the induced magnetic field decrease with the increase in the

destructive chemical reaction. Das and Jana (2010) investigated heat and mass transfer on

an infinite vertical plate embedded in porous medium, which moves with time dependent

velocity in a viscous, electrically conducting incompressible fluid. A uniform magnetic

field is applied normal to the plate. They solved the problem analytically by Laplace

transform technique. They obtained the expressions for velocity, temperature,

concentration, skin friction, rate of heat and mass transfer. They observed that skin

friction decreases with increase in permeability of the porous medium and increases with

increase in magnetic parameter.

In this study, the effects of permeability variation and oscillatory suction velocity

on free convection and mass transfer in MHD flow of a viscous incompressible

electrically conducting fluid past on infinite vertical porous non–conducting plate

embedded in a porous medium are presented. The plate is subjected to oscillatory suction

velocity normal to the plate in the presence of a uniform transverse magnetic field. The

rate of change of temperature and concentration species is prescribed on the boundary. In

certain situations these may be more convenient than prescribing the temperature and

concentration of species. The expressions for the velocity, temperature, concentration of

species and skin-friction are obtained. The effect of different parameters entering into the

expressions is shown graphically.

7.2 CO-ORDINATE SYSTEM AND EQUATIONS

Unsteady hydromagnetic flow past an infinite vertical porous plate bounded by a

porous medium of time dependent permeability and suction velocity is considered. The

fluid is assumed to be incompressible, viscous and electrically conducting. The x'-axis is

168

taken along the direction in which fluid is flowing and y'-axis is normal to it. A uniform

magnetic field is applied normal to the direction of flow. The magnetic Reynolds number

is taken to be very small so that the induced magnetic field is small in comparison to the

applied magnetic field and hence can be neglected. The fluid properties are assumed to be

constant. The temperature difference between the wall and the medium develops

buoyancy force which induces the basic flow. Initially, the fluid as well as the plate are

taken to be at the same temperature. It is further assumed that the concentration of the

species is very low. Following Gebhart and Pera (1971), the Soret and Dofour effects are

neglected. For 0*t , the temperature of the plate and the concentration of species are

changed to *

wT and *

wC respectively. Taking usual Boussinesq approximation into

account, the governing equations for conservation of mass, momentum, energy and

concentration are

0

y

v (7.2.1)

t

u v

y

u= g )(

**

TT )(*

CCg +

2

2

y

u

uBK

u 2

0

(7.2.2)

v

t

T

y

T =

pC

2

2

y

T (7.2.3)

2

2

*

**

*

y

CD

y

Cv

t

C (7.2.4)

where *

v is the constant suction velocity, *u the fluid velocity, *T the fluid

temperature, *

T the fluid temperature in free stream, *C the species concentration in the

fluid, *

C the species concentration in free stream, the coefficient of thermal

expansion, the coefficient of thermal expansion with concentration, g the

acceleration due to gravity, D the chemical molecular diffusivity, pC the specific heat

at constant pressure, *K the permeability of porous medium, the thermal

conductivity, the kinematic viscosity, *

the fluid density, the electric

permeability and 0B the magnetic field intensity.

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The suction velocity on the vertical plate is imposed in the form

)1(***

0

tievv

(7.2.5)

The permeability of the porous medium is taken in the form

)1( **

0

tieKK (7.2.6)

where *

0v is the constant suction velocity of the fluid through the porous surface, 0K is

the constant permeability of the porous medium and is a constant )10( .

The following non-dimensional quantities are introduced

y =

0vy, t =

4

2*

0vt

, 2*

0

4

v

, *

0v

uu

, **

**

CC

CCC

w

, **

**

TT

TT

w

,

Grashof Number Gr = g3*

0

**)(

v

TTw ,

Modified Grashof Number Gc =

g3*

0

**)(

v

CCw ,

Prandtl Number Pr = )/( pC

, Magnetic parameter

0

0

v

BM ,

Schmidt number Sc =D

, Permeability parameter K =

2

*2*

0

Kv ,

(7.2.7)

Using (7.2.5) - (7.2.7) into the equations (7.2.1) – (7.2.4), we have

)1(ti

ev

(7.2.8)

uMeK

u

y

uCGcGr

y

ue

t

uti

ti 2

0

2

2

)1()1(

4

1

(7.2.9)

2

2

Pr

1)1(

4

1

yye

t

ti

(7.2.10)

2

21

)1(4

1

y

C

Scy

Ce

t

C ti

(7.2.11)

such that u is the velocity along the x-axis, the fluid temperature, C the species

concentration, the frequency of oscillation, t the time.

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The boundary conditions in the non-dimensional form are

u = 0, titi

ey

Ce

y

1,1 at y = 0 , (7.2.12)

u 0, 0, C 0 as y . (7.2.13)

7.3 SOLUTION OF EQUATIONS

The solution of governing equations is obtained by separating the steady and

unsteady parts in the following manner

ti

eyCuyCutyCu

))(,,())(,,(),)(,,( 111000 (7.3.1)

Substituting (7.3.1) into the equations (7.2.9) to (7.2.11) and separating the steady and

unsteady components, we obtain

000100 CGcGruauu , (7.3.2)

0

0

0111211K

uuCGcGruauu , (7.3.3)

00 Pr = 0, (7.3.4)

0111 PrPr4

Pr

i

, (7.3.5)

000 CScC , (7.3.6)

01114

CScCSci

CScC

, (7.3.7)

where prime denotes differentiation with respect to y.

The corresponding boundary conditions are

1,,1,,0, 101010 CCuu at 0y ,

0,,,,, 101010 CCuu as y . (7.3.8)

The solutions of equations (7.3.2) to (7.3.7) under the boundary conditions (7.3.8) are

recorded as under

tiyymyee

ie

i

mety

Pr

1

Pr 41

Pr41

Pr

1),( 1 (7.3.9)

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tiScyymScyee

ie

iSc

me

SctyC

41

411),( 2

2

(7.3.10)

ymymyScyymeaeaeaeaeaatyu 143

10154

Pr

343 )(),(

tiyScyymymeeaeaeaea

14

Pr

13121132 (7.3.11)

where 1m to 4m and 1a to 15a are constants depending on physical parameters and are

recorded in the APPENDIX - IX.

It is the real part alone of the complex quantities which have physical significance

in the flow problems. The velocity, temperature and concentration fields are expressed in

the following form

u (y, t) = u0(y) + (Mr tcos – Mi tsin ) (7.3.12)

(y, t) = )(0 y + (Kr tcos – Ki tsin ) (7.3.13)

C (y, t) = C0(y) + (Lr tcos – Li tsin ) (7.3.14)

where Mr , Mi , Kr , Ki , Lr and Li are constants recorded in the APPENDIX - IX.

7.4. RESULTS AND DISCUSSION

In this section, velocity field, temperature field, concentration field and skin-

friction co-efficient at the plate are discussed by assigning numerical values to various

parameters appearing in the solution.

The values of Prandtl number Pr are taken for Air ( Pr = 0.71), Water ( Pr =7.0)

and water at freezing point ( Pr =11.4). The values of Schmidt number Sc are taken for

Hydrogen ( Sc =0.22), Helium ( Sc = 0.30), water-vapour ( Sc = 0.60), Oxygen

( Sc =0.66) and Ammonia ( Sc = 0.78). The values of Grashoff number are chosen to be

Gr = 10.0, 20.0 and values of modified Grashoff number are taken as Gc = 10.0, 20.0.

Magnetic parameter are selected as M = 0.5, 1.0, permeability parameter as K0 = 10.0,

20.0 and frequency parameter = 5.0. The perturbation parameter is taken as 0.005.

The various profiles have been drawn for 2/ t .

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Velocity Field

The transient velocity for 2

t is given by

iMyuyu

)(

2, 0 (7.4.1)

Fig.7.1 shows transient velocity field for Gr = 10.0 and 20.0 which correspond to

the cooling of the plate. It is observed that an increase in Gc or Gr or K0 increases

numerically the transient velocity, while an increase in Sc and M decreases numerically

the transient velocity field. It is further noted that the transient velocity field first

increases numerically and then the curves fall gradually after attaining a maximum value.

The maxima occur in the near vicinity of the plate.

Temperature Field

The transient temperature field for 2

t is given by

iKyTyT

)(

2, 0 (7.4.2)

Fig. 7.2 shows transient temperature field due to variation in Prandtl number Pr

for air, water and water at freezing point at = 0.005 and 2/ t . It is observed that

an increase in Pr , increases numerically the transient temperature. The temperature

approaches zero for small values of y in case of air and water. In the present case the

slope of the curves does not indicate much variation.

173

174

Concentration Field

The transient concentration field at 2

t is given by

iLyCyC

)(

2, 0 (7.4.3)

Fig. 7.3 shows transient concentration field due to variation in Schmidt number

Sc for Hydrogen, Helium, Water-vapour, Oxygen and Ammonia at = 0.005 and

2/ t . It is observed that the magnitude of transient concentration field is greater for

Hydrogen and Helium. With increase in Schimdt number the magnitude decreases. It is

seen that for Ammonia, it is least. With increase in y the magnitude goes on decreasing.

Out of the cases considered here Hydrogen maintains larger concentration level.

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Skin-Friction

Skin-friction coefficient () at the plate in terms of amplitude and phase is

)(cos||0

0

tNy

u

y

, (7.4.4)

where

0 = m3 (a3 + a4) – a3 Pr – a4 Sc , N = Nr + i Ni = 0

1

yy

u, tan =

r

i

N

N,

Ni = – B1N1 – A1N2 – B2N3 – A2N4 – m3N6 – PrN8 – ScN10 – B3N11 – A3N12,

Nr = –A1N1 + B1N2 – A2N3 + B2N4 – m3N5 – PrN7 – ScN9 – A3N11 + B3N12.

Table-1 represents the numerical values of skin-friction co-efficient () in terms of

amplitude N and phase angle () for variation in Gr , Gc , Sc , M, K0, and Pr

respectively for cooling of the plate. It is observed that increase in Gr , Gc , K0 leads to

an increase in the value of N while an increase in Sc , M, , Pr leads to decrease in

the value of N . However, the increase in Gr brings greater changes in N . It is

observed that there is a phase lag in skin-friction due to increase in Gr , Gc , Sc , K0, ,

Pr . The magnitude of phase lag varies for variation of different parameters as shown in

Table–1. Out of the cases which are considered in this study Prandtl number affects the

phase lag most.

Table – 2 shows the effect of Gr , Gc , Sc , K0, , M and Pr on skin –friction

coefficient (0) due to steady part of velocity and skin–friction coefficient () due to

cooling of the plate at = 0.005 and 2/ t . It is observed that the magnitude of skin-

friction coefficients 0 and increase due to increase in Gr , Gc , K0 while decreases due

to increase in Sc , M, Pr . However, with variation in , there is slight increase in

magnitude of .

176

Table – 1 Values of Amplitude N and phase angle ( ) for Skin–friction

coefficient due to cooling of the plate.

Gr Gm Sc M 0K ω Pr N (Degree)

10.0 4.0 0.22 0.5 10.0 5.0 0.71 50.3271 –81.81

20.0 4.0 0.22 0.5 10.0 5.0 0.71 73.9358 –79.86

10.0 8.0 0.22 0.5 10.0 5.0 0.71 77.1288 –83.67

10.0 4.0 0.66 0.5 10.0 5.0 0.71 33.3457 –76.26

10.0 4.0 0.22 1.0 10.0 5.0 0.71 38.8981 –81.87

10.0 4.0 0.22 0.5 10.0 10.0 0.71 27.4392 –80.81

10.0 4.0 0.22 0.5 10.0 5.0 7.00 26.9494 –86.83

10.0 4.0 0.22 0.5 10.0 5.0 7.00 26.9494 –86.82

Table – 2 Values of 0 and due to cooling of the plate at 2/ t .

Gr Gm Sc M 0K ω Pr

0

10.0 4.0 0.22 0.5 10.0 5.0 0.71 51.0658 50.8167

20.0 4.0 0.22 0.5 10.0 5.0 0.71 65.3706 65.0067

10.0 8.0 0.22 0.5 10.0 5.0 0.71 87.8267 87.4434

10.0 4.0 0.66 0.5 10.0 5.0 0.71 20.7896 20.6276

10.0 4.0 0.22 1.0 10.0 5.0 0.71 30.8832 30.6907

10.0 4.0 0.22 0.5 10.0 10.0 0.71 54.1876 53.9197

10.0 4.0 0.22 0.5 10.0 5.0 7.00 51.0658 50.9303

10.0 4.0 0.22 0.5 10.0 5.0 7.00 36.9573 36.8227

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7.5 CONCLUSION

From this study, it is concluded that Grashoff number increases the transient

velocity, Schmidt number decreases the same. Grashoff number also shows greater

influence on skin-friction in comparison to other parameters. Schmidt number is also

responsible for decreasing the magnitude of concentration field .The effect of Prandtl

number is to increase the transient temperature.