Rakesh IJAMM 1

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Int. J. of Appl. Math and Mech. 8 (6): 49-68, 2012.

MATHEMATICAL MODELLING OF SORET AND HALL EFFECTS

ON OSCILLATORY MHD FREE CONVECTIVE FLOW OF

RADIATING FLUID IN A ROTATING VERTICAL POROUS

CHANNEL FILLED WITH POROUS MEDIUM

R. Kumar1

and K. D. Singh2

1 Department of Mathematics, Govt. College for Girls (RKMV), Shimla-17 1001, India.

2 Department of Mathematics (ICDEOL), H.P. University, Shimla-171 005, India.

Email: [email protected]

Received 29 March 2011; accepted 28 November 2011

ABSTRACT

Heat and mass transfer in MHD (Magnetohydrodynamic) oscillatory free convective flow of a

viscous incompressible fluid through a highly porous medium bounded between two infinite

vertical porous plates has been studied. The two porous plates are subjected to a constant

injection and suction. A uniform magnetic field is applied in the direction normal to the

planes of the plates. The entire system rotates about the axis normal to the planes of the plates

with uniform angular velocity . It is also assumed that the conducting fluid is gray,

absorbing-emitting radiation and non-scattering. Using perturbation technique, the

dependence of steady and unsteady resultant velocities and their phase differences on variousparameters are obtained. The effects of thermal radiation, Soret number and Hall current have

been discussed.

Keywords: Oscillatory, Porous medium, Vertical porous channel, MHD, Hall current,

Thermal radiation, Soret number.

1 INTRODUCTION

Radiative convective flows have gained attention of many researchers in recent years. This is

justified by the fact that the radiative flows of an electrically conducting fluid with hightemperature in the presence of magnetic field plays a vital role in many engineering, industrial

and environment processes e.g. heating and cooling chambers, fossil fuel combustion energy

processes, evaporation from large open water reservoirs, astrophysical flows, solar power

technology and space vehicle re-entry. More applications and a good insight into the subject

are given by Rashad (2009), Sanyal and Adhikari (2006), Muthucumaraswamy and

Kulandaivel (2008), Prasad and Reddy (2008b), Singh and Kumar (2010) and Raptis and

Perdikis (1999). Chamkha (2000) considered the problem of steady, hydromagnetic boundary

layer flow over an accelerating semi-infinite porous surface in the presence of natural

radiation, buoyancy and heat generation or absorption. Analytical model of MHD mixed

convective radiating fluid with viscous dissipative heat have been presented by Ahmed and

Batin (2010). Soundalgekar (1984) investigated oscillatory MHD flow and heat transfer

effects on the channel. Ali et al. (1984) studied the radiation effect on free convection



R. Kumar and K. D. Singh


50

boundary layer flow over horizontal surfaces, using the Rosseland diffusion approximation.

Soundalgekar and Takhar (1993) have studied radiation effects on free convection flow of a

gas past a semi-infinite flat plate. Theoretical analysis of radiative effects on transient free

convection heat transfer past a hot vertical surface in porous media was presented by Ghosh

and Beg (2008). Kim and Fedorov (2003) studied transient mixed radiative convection flowof a micropolar fluid past a semi-infinite vertical porous plate. Hossain and Rees (1998)

investigated free convection from isothermal inclined plates to horizontal plates. The

interaction of free convection and radiation on boundary layer flows with fluid suction

through the porous wall was investigated by Hossain et al. (1999). Yih (1999) studied the

radiation effect on natural convection about a truncated cone. EL-Hakim and Rashad (2007)

used Rosseland diffusion approximation in studying the effect of radiation on free convection

from a vertical cylinder embedded in a fluid-saturated porous medium. Raptis and Massalas

(1998) analyzed the effects of radiation on the oscillatory flow of a gray gas, absorbing-

emitting in the presence of induced magnetic field. Beg and Ghosh (2010) investigated an

analytical study for MHD flow of radiating fluid with oscillatory surface temperature and

secondary flow effects. Prasad and Reddy (2008b) studied radiation and mass transfer effectson an unsteady MHD free convection flow past a semi-infinite plate through porous medium.

On the other hand, simultaneous heat and mass transfer from different geometries embedded

in porous media has many engineering and geophysical applications such as drying of porous

solids, thermal insulations, cooling of nuclear reactors and underground energy transport.

Attia and Kotb (1996) investigated the two dimensional MHD flow between two porous,

parallel and infinite plates. Recently Singh and Mathew (2008) studied the injection/suction

effect on a hydromagnetic oscillatory flow in a horizontal channel in a rotating system. Singh

(2004) studied the effects of transversely applied uniform magnetic field on oscillatory flow.

When the strength of the magnetic field is strong, one cannot neglect the effects of Hall

current. The Hall current gives rise to a cross flow making the flow three-dimensional. Very

recently,Rajesh and Varma (2010) investigated heat source effects on MHD flow past an

exponentially accelerated vertical plate with variable temperature through a porous medium.

Radiation effect on convective heat transfer through a porous medium in a vertical channel

with quadratic temperature variation has been studied by Reddaiah and Prasada Rao (2010).

Alam et al. (2009) investigated the effects of variable reaction, thermophoresis and radiation

on MHD free convection flows.

If two regions in a mixture are maintained at different temperatures so that there is a flux of

heat, it has been found that a concentration gradient is set up. In a binary mixture, one kind of

a molecule tends to travel toward the hot region and the other kind toward the cold region.This is called the “Soret effect”. Eckert and Drake (1972) have pointed out that in a

convective fluid when the flow of mass is caused by a temperature difference one cannot

neglect the thermal diffusion effect (commonly known as Soret effect) due to its practical

application in engineering and science. Usually this effect has a negligible influence on mass

transfer, but it is useful in the separation of certain mixtures. Thermal diffusion effect or Soret

effect has been utilized for isotope separation and in mixtures between gases with very light

molecular weight ) He , H ( 2 and medium molecular weight ( 2 N , air) and it was found to be

of a magnitude that it cannot be neglected. More physical insight into the problem is given by

Sparrow and Cess (1962) and Renuka et. al. (2009). Reddy and Reddy (2010) investigated

Soret and Dufour effects on steady MHD free convective flow past an infinite plate. Soret

effects due to natural convection between heated inclined plates have been investigated byRaju et al. (2008).



Mathematical Modelling of Soret and Hall Effects


51

As the importance of radiation in the fields of aerodynamics as well as in space science

technology, the present study is motivated towards this direction. The objective of the present

paper is to analyze the effects of radiation, Soret, permeability variation and injection/suction

on heat and mass transfer of free convective flow through porous medium in the presence of

Hall current in a vertical porous channel when the entire system rotates about an axisperpendicular to the planes of the plates. In this work we assume that the temperature

differences with in the flows are sufficiently small so that4

*T may be expressed as a linear

function of temperature.

2 MATHEMATICAL MODEL

Consider an unsteady flow of an electrically conducting, viscous, incompressible fluid

through a highly porous medium bounded between two insulated infinite vertical porous

plates at distance d apart in the presence of thermal radiation. A constant injection velocity,

0w , is applied at the stationary plate 0* z and the same constant suction velocity,

0w , is

applied at the plate d z* , which is oscillating in its own plane with a velocity )t (U **

about a non-zero constant mean velocity0

U . The origin is assumed to be at the plate 0* z

and the channel is oriented vertically upward along the * x axis. The channel rotates as a

rigid body with uniform angular velocity * about the * z axis. A strong magnetic field of

uniform strength 0 H is applied along * z axis taken perpendicular to the planes of the plates.

The magnetic Reynolds number is considered to be small so that the induced magnetic field is

neglected. It is also assumed that the radiation heat flux in the * x direction is negligible as

compared to that in the

*

z direction. The temperature and concentration oscillate about aconstant mean

0T and

0C respectively at the plate 0* z and the species concentration is

assumed at low level. The fluid considered here is a gray, absorbing-emitting radiation but a

non-scattering medium. All physical quantities depend on*

z and *t for this problem of fully

developed laminar flow. The physical configuration of the problem is shown in Figure1.

The equation of continuity 0

V . , gives on integration0ww

* where )w ,v ,u(V ***

and

solenoidal relation for the magnetic field 0

H . , gives0 H H

*

z (constant) everywhere in

the flow field. The equation of conservation of electric charge 0

J . gives

*

z J constant.This constant is zero i.e. 0 J * z at the plates which are electrically non-conducting. Taking

Hall current into account the generalized Ohm’s law (Cowling [3]) is of the form is

) H V E ( H J H

J e

ee

0

, (1)

where

V is the velocity vector,

H is the magnetic field,

J is the current density,

E is the

electric field, is the electrical conductivity, e is the magnetic permeability, e is the

cyclotron frequency, and e is the electron collision time.





52

Figure 1: Physical configuration of the problem.

For very large magnetic field, the*

x and*

y components of Ohm’s law (2.1), which includeHall current, are

)v H E ( J J *

e

*

x

*

yee

*

x 0

and

)u H E ( J J *

e

*

y

*

xee

*

y 0 .

Since the external electric field arising due to polarization of charges is negligible. Hence

0* y

* x E E . Therefore, solving for *

x J and * y J , we get

)m(

)vmu( H J

**

e*

x 2

0

1

and

)m(

)umv( H J

**

e*

y 2

0

1

. (2)

Thus within the frame work of these assumptions and making use of (2), the heat and mass

transfer flow of radiative fluid in the presence of Hall current through porous medium is

governed by the following equations:

*

***

e**

*

*

*

*

*

*

*

*

K

u

)m(

)umv( H v

z

u

x

p

z

uw

t

u

2

2

0

22

01

21

2

)C C (g)T T (g d

*

d

* , (3)

z

y

0 H

Porous Medium

x





53

*

***

e**

*

*

*

*

*

*

*

*

K

v

)m(

)vmu( H u

z

v

y

p

z

vw

t

v

2

2

0

22

01

21

2, (4)

*

*

p*

*

p

*

*

*

*

zq

C zT

C k

zT w

t T

12

2

0 , (5)

22

2

1

2

0 *

*

*

*

*

*

*

*

z

T D

z

C D

z

C w

t

C

, (6)

where is the fluid density, *t is the time, *K is the permeability of the porous medium,

eem is the Hall parameter, is volumetric coefficient of thermal expansion,

is

volumetric coefficient of thermal expansion with concentration, k is thermal conductivity, pC

is specific heat at constant pressure, *q is the radiative heat flux, D is molecular diffusivity

and D1 is thermal diffusivity.

The boundary conditions for the problem are

0 0

0 0

0

0

0

1 0

* * * * *

d

*

* * *

d

* * * * * *

*

* *

d d

u v ,T T ( T T ) cos t ,

at z

C C ( C C ) cos t

u U ( t ) U ( cos t ), v ,

at z d

T T , C C

(7)

where * is the frequency of oscillations,0

U is the mean velocity,0

T is the mean

temperature,0

C is the mean concentration and is a very small positive quantity.

The quantity*

q on the right-hand side of equation (5) represents the radiative heat in the * z

direction. The local radiant for the case of an optically thin gray gas is expressed by

)T T (a z

q *

d

*44

4

, (8)

where

a is the absorption coefficient and is the Stefan-Boltzmann constant.

We assume that the temperature differences with in the flow are sufficiently small such that4

*T may be expressed as a linear function of the temperature. This is accomplished by

expanding4*

T in a Taylor series about d T and neglecting higher-order terms, thus





54

4334

4

d

*

d

* T T T T . (9)

By using equations (8) and (9), equation (5) reduces to

p

*

d d

*

*

*

p

*

*

*

*

C

)T T (T a

z

T

C

k

z

T w

t

T

3

2

2

0

16. (10)

Eliminating the modified pressure gradient under the usual boundary layer approximations,

equations (3) and (4) become

)m(

)U umv( H v

z

u

t

U

z

uw

t

u***

e**

*

*

*

*

*

*

*

*

2

2

0

22

01

22

)C C (g)T T (gK

)U u(d

*

d

*

*

**

, (11)

*

****

e***

*

*

*

*

*

*

K

v

)m(

v)U u(m H )U u(

z

v

z

vw

t

v

2

2

0

22

01

22

. (12)

Introducing the following non-dimensional quantities ,d

z*

,t t ** ,

U

uu

*

0

0

U

vv

*

,

2d *

is the rotation parameter,

2d

*

is the frequency parameter,2d

K K

*

is the

permeability parameter,

d w

0 is the injection/suction parameter,

d H M

0 is the

Hartmann number, D

Sc

is the Schmidt number,

)C C (

)T T ( DS

d

d

0

01

0

is the Soret number,

k

C Pr

p

is the Prandtl number,2

00

0

wU

)T T (gGr d

is the thermal Grashoff number,

2

00

0

wU

)C C (gGm d

is the mass Grashoff number,

d

d

*

T T

T T

0

,

d

d

*

C C

C C C

0

and

p

d

*

C

d T a R

2316

is the radiation parameter, into equations (6), (10), (11) and (12) and taking

ivuq , we get

C GmGr )U q(Sdt

dU qq

t

q 22

2

2

, (13)

R

Pr t

2

21

, (14)





55

2

2

2

21

C

S

C

t

C

c

, (15)

where

K m

im M iS

1

1

12

2

2 .

The boundary conditions (7) can be written in complex notation as

0 1 1 02 2

1 0 0 12

it it it it

it it

q , ( e e ), C ( e e ) at

q U( t ) ( e e ), , C at

. (16)

3 METHOD OF SOLUTION

Now we look for a solution of equations (13) (14) and (15) under the boundary conditions

(16) of the form

it it e)(qe)(q)(q)t ,(q

210

2

, (17)

it it e)(e)()()t ,(

2102

, (18)

it it e)(C e)(C )(C )t ,(C

2102

. (19)

Substituting equations (17), (18) and (19) into equations (13), (14) and (15) and comparing

the harmonic and non-harmonic terms, we get

0

2

0

2

000 C GmGr SSqqq , (20)

1

2

1

2

111C GmGr )iS(q)iS(qq , (21)

2

2

2

2

222C GmGr )iS(q)iS(qq , (22)

0000

Pr RPr , (23)

0111 )i R(Pr Pr , (24)





56

0222 )i R(Pr Pr , (25)

0000

cc SSC SC , (26)

10111 ccc SSC SiC SC , (27)

20222

ccc SSC SiC SC . (28)

The corresponding transformed boundary conditions are

0 1 2 0 1 2 0 1 2

0 1 2 0 1 2 0 1 2

0 1 1 0

1 0 0 1

q q q , , C C C at

q q q , , C C C at

(29)

where dashes denote differentiation with respect to ' ' .

The solutions of equations (20) to (28) under the boundary conditions (29) are obtained as:

1221

21

10

mmmm

mmee

ee)(

, (30)

3443

43

11

mmmm

mmee

ee)(

, (31)

5665

65

12

mmmm

mmee

ee)(

, (32)

21

2121210

1 mm

mm

Se Ae A

eee B B)(C c

, (33)

43

43

87

43431

1 mm

mm

mme Ae A

eee Be B)(C

, (34)

65

65

109

65652

1 mm

mm

mme Ae A

eee Be B)(C

, (35)

1 2

1 2

1 2

1 2

0 7 8

2

7 8 9 10

11

1c

n n

n n

S m m

m m

q ( ) B e B ee e

A A e A e A ee e

, (36)





57

3 4

3 4

3 7 84

3 4

1 9 10

211 12 13 14

11

1

n n

n n

m m mmm m

q ( ) B e B ee e

A e A e A e A ee e

, (37)

5 6

5 6

5 6 9 10

5 6

2 11 12

2

15 16 17 18

11

1

n n

n n

m m m m

m m

q ( ) B e B ee e

A e A e A e A ee e

. (38)

4 RESULTS AND DISCUSSIONS

Now for the resultant velocities and shear stresses of the steady and unsteady flow, we write

)(q)(iv)(u 000

(39)

and

it it e)(qe)(q)(iv)(u

2111 . (40)

The solution (36) corresponds to the steady part which gives0

u as the primary and0

v as the

secondary velocity components. The amplitude and the phase difference due to these primary

and secondary velocities for the steady flow are given by

2

0

2

00vu R , )u / v(tan 00

1

0

. (41)

Similarly the solutions (37) and (38) together give the unsteady part of the flow. The unsteady

primary and secondary velocity components )(u 1 and )(v 1 , respectively, for the

fluctuating flow can be obtained as

t sin)(q Im)(q Imt cos)(qal Re)(qal Re)t ,(u 21211 , (42)

t cos)(q Im)(q Imt sin)(qal Re)(qal Re)t ,(v 21211 . (43)

The resultant velocity or amplitude and the phase difference of the unsteady flow are given by

2

1

2

11vu R , )u / v(tan

11

1

1

. (44)

The resultant velocities0

R ,1

R and the phase angles10

, for the steady and unsteady part of

the flow are respectively shown graphically in Figures (2) to (5). To be realistic the two

values of Prandtl number Pr as 0.71 and 7.00 are chosen to represent air and water





58

respectively. The values of Schmidt number cS are taken for water-vapor ( cS =0.60) and

Ammonia ( cS =0.78). Figures (2) and (3) for the steady and unsteady resultant velocities,0

R

and1

R respectively show that both these resultants increase with the increase of Grashoff

number Gr or mass thermal Grashoff number Gm or Hartmann number M or Hall parameterm or injections/suction parameter or permeability of the porous medium K or Soret

number0

S or Schmidt number cS , while these resultants decrease with the increase of

radiation parameter R or Prandtl number Pr . It is interesting to note that with the increase of

the rotation of the channel , both these resultants increase near the stationary plate and then

decrease near the oscillating plate. Figure (3) shows clearly that1

R decreases with the

increase of frequency of oscillation .

It is also found from Figures (4) and (5) that the phases differences0

and1

for the steady

and unsteady parts of the flow respectively decrease with increasing Gr or Gm or M or or

or 0S or cS . However, 0 and 1 both increase with increasing m or R or Pr or K . Figure

(5) shows clearly that1

increases with the increase of frequency of oscillations .

For the steady flow the amplitude and the phase difference of shear stresses at the stationary

plate 0 can be obtained as

) / (tan , x yr y xr 00

1

0

2

0

2

00 , (45)

Where

0

0

00

q

i y x.

Here x0 and y0

are, respectively, the shear stresses at the stationary plate due to the primary

and secondary velocity components. The numerical values for the resultant shear stress r 0

and the phase angle r 0 are listed in Table 1. This Table shows that r 0

and r 0 goes on

increasing with increasing rotation of the channel. The amplitude of shear stress r 0 also

increases with the increase of Gr or Gm or M or or 0S or cS and decreases with the

increase of m or K . The increase of these flow parameters has opposite effect on r 0 . It is

interesting to note that both x0 and r 0 increase with the increase of radiation parameter R

and Prandtl number Pr .





59

Figure 2: Resultant velocity0

R due to0

u and0

v .

Figure 3: Resultant velocity1

R due to1

u and1

v at4

t .

0 R

1 R

Gr Gm M m R Pr K S0 Sc

5 4 2 1 2 0.2 10 0.71 1 1 0.60 5 I

15 4 2 1 2 0.2 10 0.71 1 1 0.60 5 II5 12 2 1 2 0.2 10 0.71 1 1 0.60 5 III

5 4 4 1 2 0.2 10 0.71 1 1 0.60 5 IV

5 4 2 3 2 0.2 10 0.71 1 1 0.60 5 V5 4 2 1 10 0.2 10 0.71 1 1 0.60 5 VI

5 4 2 1 2 1.0 10 0.71 1 1 0.60 5 VII

5 4 2 1 2 0.2 20 0.71 1 1 0.60 5 VIII5 4 2 1 2 0.2 10 7.00 1 1 0.60 5 IX

5 4 2 1 2 0.2 10 0.71 5 1 0.60 5 X

5 4 2 1 2 0.2 10 0.71 1 5 0.60 5 XI5 4 2 1 2 0.2 10 0.71 1 1 0.78 5 XII

5 4 2 1 2 0.2 10 0.71 1 1 0.60 15 XIII

5 4 2 1 2 0.2 40 0.71 1 1 0.60 5 XIV5 4 2 1 2 0.2 80 0.71 1 1 0.60 5 XV


5 4 2 1 2 0.2 10 0.71 1 1 0.60 I

15 4 2 1 2 0.2 10 0.71 1 1 0.60 II

5 12 2 1 2 0.2 10 0.71 1 1 0.60 III

5 4 4 1 2 0.2 10 0.71 1 1 0.60 IV

5 4 2 3 2 0.2 10 0.71 1 1 0.60 V

5 4 2 1 10 0.2 10 0.71 1 1 0.60 VI5 4 2 1 2 1.0 10 0.71 1 1 0.60 VII

5 4 2 1 2 0.2 20 0.71 1 1 0.60 VIII

5 4 2 1 2 0.2 10 7.00 1 1 0.60 IX5 4 2 1 2 0.2 10 0.71 5 1 0.60 X

5 4 2 1 2 0.2 10 0.71 1 5 0.60 XI

5 4 2 1 2 0.2 10 0.71 1 1 0.78 XII5 4 2 1 2 0.2 40 0.71 1 1 0.60 XII

5 4 2 1 2 0.2 80 0.71 1 1 0.60 XIV





60

Figure 4: Phase angle0

due to0

u and0

v .

Figure 5: Phase angle1

due to1

u and1

v at4

t .

0

1


5 4 2 1 2 0.2 10 0.71 1 1 0.60 5 I

15 4 2 1 2 0.2 10 0.71 1 1 0.60 5 II

5 12 2 1 2 0.2 10 0.71 1 1 0.60 5 III5 4 4 1 2 0.2 10 0.71 1 1 0.60 5 IV

5 4 2 3 2 0.2 10 0.71 1 1 0.60 5 V

5 4 2 1 10 0.2 10 0.71 1 1 0.60 5 VI5 4 2 1 2 1.0 10 0.71 1 1 0.60 5 VII

5 4 2 1 2 0.2 20 0.71 1 1 0.60 5 VIII

5 4 2 1 2 0.2 10 7.00 1 1 0.60 5 IX5 4 2 1 2 0.2 10 0.71 5 1 0.60 5 X

5 4 2 1 2 0.2 10 0.71 1 5 0.60 5 XI

5 4 2 1 2 0.2 10 0.71 1 1 0.78 5 XII5 4 2 1 2 0.2 10 0.71 1 1 0.60 15 XIII

5 4 2 1 2 0.2 40 0.71 1 1 0.60 5 XIV

5 4 2 1 2 0.2 80 0.71 1 1 0.60 5 XV


5 4 2 1 2 0.2 10 0.71 1 1 0.60 I

15 4 2 1 2 0.2 10 0.71 1 1 0.60 II

5 12 2 1 2 0.2 10 0.71 1 1 0.60 III5 4 4 1 2 0.2 10 0.71 1 1 0.60 IV

5 4 2 3 2 0.2 10 0.71 1 1 0.60 V5 4 2 1 10 0.2 10 0.71 1 1 0.60 VI5 4 2 1 2 1.0 10 0.71 1 1 0.60 VII

5 4 2 1 2 0.2 20 0.71 1 1 0.60 VIII

5 4 2 1 2 0.2 10 7.00 1 1 0.60 IX5 4 2 1 2 0.2 10 0.71 5 1 0.60 X

5 4 2 1 2 0.2 10 0.71 1 5 0.60 XI

5 4 2 1 2 0.2 10 0.71 1 1 0.78 XII5 4 2 1 2 0.2 40 0.71 1 1 0.60 XII

5 4 2 1 2 0.2 80 0.71 1 1 0.60 XIV





61

Table 1: Amplitude r 0 and the phase angle r 0

due to0

u and0

v .

Gr Gm M m R Pr K 0S cS r 0

r 0

5

155

5

5

5

5

5

5

5

5

5

55

4

412

4

4

4

4

4

4

4

4

4

44

2

22

4

2

2

2

2

2

2

2

2

22

1

11

1

3

1

1

1

1

1

1

1

11

2

22

2

2

10

2

2

2

2

2

2

22

0.2

0.20.2

0.2

0.2

0.2

1.0

0.2

0.2

0.2

0.2

0.2

0.20.2

10

1010

10

10

10

10

20

10

10

10

10

4080

0.71

0.710.71

0.71

0.71

0.71

0.71

0.71

7.00

0.71

0.71

0.71

0.710.71

1

11

1

1

1

1

1

1

5

1

1

11

1

11

1

1

1

1

1

1

1

5

1

11

0.60

0.600.60

0.60

0.60

0.60

0.60

0.60

0.60

0.60

0.60

0.78

0.600.60

4.6641

4.68634.6814

5.3684

4.5637

4.6742

4.9791

6.4247

4.6693

4.6544

4.6638

4.6671

8.991212.660

0.7186

0.70490.7065

0.6313

0.7533

0.7193

0.4863

0.7527

0.7195

0.7365

0.7179

0.7186

0.77070.7795

For the unsteady part of the flow, the amplitude and the phase difference of shear stresses at

the stationary plate 0 can be obtained, for4

t , as

x

y

r y xr tan ,1

11

1

2

1

2

11

, (46)

Where

0

2

0

1

11

v

iu

i y x.

The amplitude r 1 and phase difference r 1

of the unsteady shear stresses are shown in

Figures (6) and (7). It is clear from Figures (6) and (7) that r 1 increases with the increasing

Gr or Gm or M or or or0

S . However, these flow parameters have opposite effect on

r 1 . The r 1 decreases and r 1 increases with the increase of Hall parameter m. It is interesting

to note that an increase in K or cS leads to an increase in r 1 and r 1

. It is also observed from

Figures (6) and (7) that with the increase of radiation parameter R and Prandtl number Pr , r 1

increase for smaller values of oscillations and decreases for larger values of oscillations ,

while r 1 increases with the increase of radiation parameter R or Prandtl number Pr . There

always remains a phase lead.





62

Figure 6: The amplitude r 1 of unsteady shear stress at

4

t .

5 PARTICULAR CASES

(I) Our results reduce to the results of Singh and Kumar (2010) in the absence of free

convection, heat and mass transfer, radiation and Soret effect

(II) In the absence of porous medium, mass transfer and radiation and Soret effect, our

solutions are similar to those of Singh and Kumar (2009).

(III) Our results are found in good agreement in the absence of Hall currents ( m = 0) and

heat and mass transfer for an ordinary medium )( K as our solution reduces to the

one obtained by Singh and Mathew (2008).

r 1


5 4 2 1 2 0.2 10 0.71 1 1 0.60 I

15 4 2 1 2 0.2 10 0.71 1 1 0.60 II

5 12 2 1 2 0.2 10 0.71 1 1 0.60 III5 4 4 1 2 0.2 10 0.71 1 1 0.60 IV

5 4 2 3 2 0.2 10 0.71 1 1 0.60 V5 4 2 1 10 0.2 10 0.71 1 1 0.60 VI5 4 2 1 2 1.0 10 0.71 1 1 0.60 VII

5 4 2 1 2 0.2 20 0.71 1 1 0.60 VIII

5 4 2 1 2 0.2 10 7.00 1 1 0.60 IX

5 4 2 1 2 0.2 10 0.71 5 1 0.60 X

5 4 2 1 2 0.2 10 0.71 1 5 0.60 XI

5 4 2 1 2 0.2 10 0.71 1 1 0.78 XII5 4 2 1 2 0.2 40 0.71 1 1 0.60 XII

5 4 2 1 2 0.2 80 0.71 1 1 0.60 XIV





63

Figure 7: The phase r 1 of unsteady shear stress at

4

t .

6 CONCLUDING REMARKS

Both0

R and1

R increase rapidly from zero near the stationary plate and then approach unity

in the form of damped oscillations. For large values of rotations and injection/suction at the

plates, a phase lag is also observed for both steady and unsteady phase angles0

and1

. The

resultant velocities0

R and1

R both increase with the increase of thermal Grashoff number,

mass Grashoff number, Hartmann number, Hall current, injection/suction parameter, Soret

number and Schmidt number, while these resultants decrease with the increase of radiation

and Prandtl number. The resultant r 0 of the steady part of the shear stress and the phase

angle r 0 both go on increasing with increasing rotation of the channel. The resultant r 1

of

the unsteady shear stress goes on increasing with the increase of thermal Grashoff number,

mass Grashoff number, Hartmann number, injection/suction parameter, Soret number and

rotation of the channel; however these flow parameters have opposite effect on unsteady

phase angle r 1 .

r 1


5 4 2 1 2 0.2 10 0.71 1 1 0.60 I

15 4 2 1 2 0.2 10 0.71 1 1 0.60 II

5 12 2 1 2 0.2 10 0.71 1 1 0.60 III5 4 4 1 2 0.2 10 0.71 1 1 0.60 IV

5 4 2 3 2 0.2 10 0.71 1 1 0.60 V

5 4 2 1 10 0.2 10 0.71 1 1 0.60 VI5 4 2 1 2 1.0 10 0.71 1 1 0.60 VII

5 4 2 1 2 0.2 20 0.71 1 1 0.60 VIII

5 4 2 1 2 0.2 10 7.00 1 1 0.60 IX5 4 2 1 2 0.2 10 0.71 5 1 0.60 X

5 4 2 1 2 0.2 10 0.71 1 5 0.60 XI

5 4 2 1 2 0.2 10 0.71 1 1 0.78 XII

5 4 2 1 2 0.2 40 0.71 1 1 0.60 XII

5 4 2 1 2 0.2 80 0.71 1 1 0.60 XIV





64

APPENDIX

2

422

1

Pr RPr Pr m

,

2

422

2

Pr RPr Pr m

,

2

422

3

)i RPr(Pr Pr m

,

2

422

4

)i RPr(Pr Pr m

,

2

422

5

)i RPr(Pr Pr m

,

2

422

6

)i RPr(Pr Pr m

,

2

422

7

cccSiSS

m

,

2

422

8

ccc SiSSm

,

2

422

9

ccc SiSSm

,

2

422

10

ccc SiSSm

,

2

42

1

Sn

,

2

42

2

Sn

,

2

42

3

)iS(n

,

2

42

4

)iS(n

,

2

42

5

)iS(n

,

2

42

6

)iS(n

,

c

m

c

Sm

emSS A

1

10

1

2

,c

m

c

Sm

emSS A

2

20

2

1

,cc

m

c

SimSm

emSS A

3

2

3

2

30

3

4

,

cc

m

c

SimSm

emSS A

4

2

4

2

40

4

3

,cc

m

c

SimSm

emSS A

5

2

5

2

50

5

6

,

cc

m

c

SimSm

emSS A

6

2

6

2

60

6

5

,

S

BGm A 1

7 ,

S)S(S

BGm A

cc

12

2

8

,

1

1

2

1

92

1 AGmeGr

Smm A

m

, 2

1

2

1

101

1 AGmeGr

Smm A

m

,

3

3

2

3

114

1 AGmeGr

)iS(mm A

m

, 4

4

2

4

123

1 AGmeGr

)iS(mm A

m

,

)iS(mm

BGm A

7

2

7

3

13 ,)iS(mm

BGm A

8

2

8

4

14 ,





65

5

5

2

5

156

1 AGmeGr

)iS(mm A

m

,

6

6

2

6

165

1

AGmeGr )iS(mm Am

,

)iS(mm

BGm A

9

2

9

5

17 ,

)iS(mm

BGm A

10

2

10

6

18 ,

21

1211 11

1 mms

S ee Ae A Ae Bc

c

,

)e( A)e( Ae

Bmm

Sc

21 1111

1212

,

)ee( A)ee( Aeee

Bmmmmm

mm

84388

87433

1

,

)ee( A)ee( Aeee

Bmmmmm

mm

47737

87434

1

,

)ee( A)ee( Aeee

Bmmmmm

mm

10651010

109655

1

,

)ee( A)ee( Aeee

Bmmmmm

mm

69959

109656

1

,

)ee( A)ee( A)ee( A)e( Ae BnmmnnSnn c 2212222

109877 1 ,

)ee( A)ee( A)ee( A)e( Ae B

mnnmSnnn c 2111111

109878 1

,

)ee( A)ee( A)ee( A)ee( Ae Bnmnmnmmnn 484744344

141312119 ,

)ee( A)ee( A)ee( A)ee( Ae Bmnmnmnnmn 837343333

1413121110 ,

)ee( A)ee( A)ee( A)ee( Ae Bnmnmnmmnn 6106966566

1817161511 ,

)ee( A)ee( A)ee( A)ee( Ae Bmnmnmnnmn 1059565555

1817161512 .





66

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