7. IJPR - Hall Effect on Unsteady Couette Flow of a Visco

8/20/2019 7. IJPR - Hall Effect on Unsteady Couette Flow of a Visco

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www.tjprc.org [email protected]

HALL EFFECT ON UNSTEADY COUETTE FLOW OF A VISCO-ELASTIC FLUID

WITH HEAT AND MASS TRANSFER INCLUDING HEAT SOURCES,

CHEMICAL REACTION AND SORET EFFECT

P. K. MISHRA1, S. BISWAL

2 & G. S. ROY

3

1 Lecturer in Physics, Bhadrak Auto. College, Bhadrak Odisha, India

2 Retd. Principal, Bharatpur, Odisha, India

3Former Chairman, Council of Higher Secondary Education, Odisha & Retd. Principal,

B. J. B. Autonomous College, Bhubaneswar, India

ABSTRACT

The effect of Hall current on unsteady couette flow of a visco-elastic incompressible and electrically

conducting fluid with heat and mass transfer incorporation with heat source, chemical reaction and Soret effect has

been analysed. The effects of magnetic parameter Pm, Hall parameter m, suction parameter R and elastic parameter Rc

on the flow field and temperature field for visco-elastic fluids of Pr=5.0, 9.0 and 16.0 have been studied with the help of

graphs through computer analysis. The skin friction, rates of heat transfer at the two boundary of flow have been

discussed with the help of tables. It is observed that the flow field and temperature field are greatly influenced by the

Hall currents including other fluid parameters.

KEYWORDS: Unsteady MHD Coutte Flow, Hall Current, Visco-Elastic Fluids, Chemical and Soret Effect

Received: Nov 07, 2015; Accepted: Dec 08, 2015; Published: Dec 12, 2015; Paper Id.: IJPRDEC201507

INTRODUCTION

The study of MHD couette flow of compressible and incompressible fluids has been an important subject

for many researchers for a pretty long time. Pai1 has already studied the problem of unsteady couette flow of an

incompressible viscous fluid between two plates when one of the plates is given a sudden impulse. Nanda2 has

studied the problem on flow formation in couette motion through a porous channel with suction or injection while

Katagiri and Mahuri3 have independently analysed the same flow with magnetic field imposed upon it. Mahuri

4

later extended the problem of couette flow through porous walls with one wall moving with constant acceleration

and with constant suction at the walls.

Lehnert5 has considered MHD couette flow of an incompressible fluid between two walls moving

relative to each other. The heat transfer problem in case of unsteady couette flow between two parallel walls

maintained at different temperature is studied by Rath et al6. Mishra7 extended the problem studied by Dutta8

and Kaloni9 by analyzing plane couette flow of an Oldroyd liquid with equal rates of injection at the lower wall

and suction at the upper wall with the imposition of a uniform transverse magnetic field. Bhatanagar 10 has

discussed the plane couette flow of Rivlin-Ericksen higher order fluid with constant suction at the stationary plate.

The plane couette flow of walter’s B’ liquid with equal rate of injection at one wall and suction at the other

moving wall has be studied by Soundalgekar11. Moreover, Mishra and Mohapatra12 have investigated the problem

Or i gi n al Ar t i c

l e

International Journal of Physics

and Research (IJPR)

ISSN(P): 2250-0030; ISSN(E): 2319-4499

Vol. 5, Issue 6, Dec 2015, 49-66

© TJPRC Pvt. Ltd.


2/18

50 P. K. Mishra, S. Biswal & G. S. Roy

Impact Factor (JCC): 2.3529 Index Copernicus Value (ICV): 3.0

of flow of formation in couette motion between two walls taking a Riener-Rivlin fluid subjected to magnetic field.

Agarwal13

has studied generalized couette flow of an incompressible fluid in hydromagnetics. Chang and Yen14

have

discussed MHD couette flow with wall conductance. Mishra15

has studied the pressure induced flow of an elastic-viscous

electrically conducting incompressible fluid between two relatively moving plane porous walls in the presence of atransverse magnetic field. The commencement of unsteady couette flow of a second order fluid has been analysed by

Padhy16

. The same flow has been studied by Dash and Biswal17

by taking Oldroyd liquid through a porous channel in the

presence of heat sources. Soundalgekar and Haldavnker18

have considered MHD couette flow between conducting walls

with heat transfer. Jana and Datta19

have considered the effect of Hall current on unsteady couette flow, when the velocity

of the moving plate varies as tn. Haller

20 has investigated on the effect of Hall current on unsteady couette flow for a MHD

a.c or d.c generator. Dash and Dash21

have studied the MHD flow through porous medium past a stretched vertical

permeable surface in the presence of heat sources/sinks and chemical reaction. Singh and Garge22

have analysed the

oscillatory heat and mass transfer mixed convective flow in a rotating channel with heat source/sink and Soret effect.

As the study of both Newtonian and non-Newtonian couette flow problems in the presence of a magnetic field is

very important from technological point of view, the literature on it is replete with copious investigation on MHD couette

flows.

In the present problem, Hall effect on unsteady couette flow of a visco elastic fluid with heat and mass transfer

including heat sources, chemical reaction and Soret effect has been analysed. Here, heat and mass transfer effects on the

non-Newtonian flow with Hall current have been investigated under the following physical conditions:

• Internal heat generating source.

•

Varied species concentration.

• Chemical reaction and Sorect effect.

The study is carried out for two positive values of n i.e. (i) n=1, constant acceleration, (ii) n=1/2, variable

acceleration.

FORMULATION OF THE PROBLEM

Consider an unsteady couette flow of a non-Newtonian (visco-elastic) incompressible electrically conducting fluid

confined between two horizontal parallel plates separated by 'y = L apart, where the number L will be defined later.

Let 'x -axis be chosen along the lower wall and 'y -axis be normal to it. The lower and upper walls are specified

by the equations 'Y =0 and 'Y =L respectively. It is also supposed that the two walls have infinite extensions on either

sides of the x-axis. An uniform magnetic field of strength B0 is applied transverse to 'x -axis along 'y -axis. The suction

velocity 'v at the walls is considered to be constant. It is assumed that the magnetic field lines are normal to the free

stream velocity of the fluid and the magnetic permeability is uniform throughout the field. The plates may be assumed non-

conducting. Further, as the plates have infinite extension, all the variables except pressure are functions of y only. 'w

represents the secondary flow velocity along 'z direction because effect of Hall current gives rise to a force in the 'z -

direction that induces a cross-field in that direction. Hence flow becomes three dimensional.


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Hall Effect on Unsteady Couette Flow of a Visco-Elastic Fluid with Heat and 51

Mass Transfer Including Heat Sources, Chemical Reaction and Soret Effect


Now the velocity components 'w,'u and 'v at any point ( 'z,'y,'x ) in the flow field compatible with the

equation of continuity can be given by

)'t,'y('u'u =

)'t,'z('w'w =

and 'v = v = constant 2.1

Following the stress-strain rate relation given by the stress components are given by

2

0

'y'x

'y

'uk 2p

∂

∂= (2.2)

∂∂

∂+

∂

∂−

∂

∂η=

't'y

'u

'y

'uvk

'y

'up

2

2

2

00

'y'x (2.3)

and p x’y’ = 0

where k 0 = η 0 (λ 1-λ 2), the volume co-efficient of elasticity of the fluid. As the motion of fluid is due to shearing

action of the fluid layers, then

0'y

'p=

∂

∂ (2.5)

Thus, the equation of motion and energy including viscous dissipation and heat sources for the viscoelastic fluid

model of Oldroyd’s B’ liquid are given below.

The equations of motion are

+

+

∂

∂

ρ

σ−

∂

∂+

∂∂

∂

ρ−

∂

∂

ρ

η=

∂

∂+

∂

∂2

02

3

3

2

3

0

2

2

0

m1

'mw'u

't

B

'y

'uv

't'y

'uk

'y

'u

'y

'uv

't

'u (2.6)

and

+

−

∂

∂

ρ

σ−

∂

∂+∂∂

∂

ρ−∂

∂

ρ

η=∂

∂+∂

∂2

02

3

3

2

3

0

2

2

0

m1

'w'mu

't

B

'y

'wv't'y

'wk

'y

'w

'y

'wv't

'w (2.7)

The equation of energy is

( )L2

22

p

0

2

p

0

2

2

p

''s'y

'u

'y

'uv

'y

'u

't'y

'u

C

k

'y

'u

C'y

'

C

k

'y

'v

't

'θ−θ+

∂

∂

∂

∂+

∂

∂

∂∂

∂

ρ−

∂

∂

ρ

η+

∂

θ∂

ρ=

∂

θ∂+

∂

θ∂

(2.8)

and the equation of concentration is


4/18



2

2

12

2

'y

'D*

'y

'CD

'y

'C

't

'C

∂

θ∂+λ+

∂

∂=

∂

∂+

∂

∂ (2.9)

where, ρ is the density of fluid, Cp is the specific heat at constant pressure, η0 is the co-efficient of viscosity of the

fluid, k 0 the volume co-efficient of elasticity of the fluid, s is the electrical conductivity of the fluid, D is the diffusion co-

efficient, k is the co-efficient of thermal conductivity of the fluid, B0 is the constant magnetic field applied transverse to the

plate, 's is a source-sink related dimensional constant, θL and CL are the temperature and concentration of the fluid at the

upper plate, 'θ and 'C are the temperature and concentration of the fluid at any point (x’,y’),λ* is the chemical reaction

rate term, and D1 is the thermal diffusitivity.

Initial boundary conditions are

'u =0, 'θ =0, 'c =0, for all y, 'u =Atn

, 'w =0, 0'y

'c

,0'y

'

=∂

∂

=∂

θ∂ for

'y =0 and 'u = 'w =0, 'θ =θL for 'y =L (2.10)

FORMATION OF THE EQUATIONS

Introducing the following non-dimensional parameters:

nn

1AT

'ww,

AT

'uu,

T

'tt,

Tv

'yy ====

v

TVR = , the suction paramete ;

TR 21

c

λ−λ= , the elastic parameter,

k

cvP

p1

r

ρ= , the prandtl number;

Lp

n22

c

TAE

θ= , the Eckert number,

2

1

v

v's4S = , the source paramete ;

ρ

σ=

02

m

BP , the magnetic parameter,

ρ

η= 01v , the kinematic viscosity;

D

vSc 1= , the schmidt number,

L

L'

θ

θ−θ=θ , the non-dimensional temperature parameter,

L

L

C

C'CC

−= , the non-dimensional concentration parameter,


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L1

L1

1Cv

DS

θ= , soret number,

TvVK

1

= , non-dimensional chemical reaction parameter.

with the help of above non-dimensional parameters, the equation 2.6, 2.7. 2.8 and 2.9 are now reduced to their

dimensionless forms as follows:

0t

wm

t

u

m1

Pm

y

uRRc

ty

uRc

y

u

y

uR

t

u23

3

2

3

2

2

=

∂

∂+

∂

∂

++

∂

∂+

∂∂

∂+

∂

∂−

∂

∂+

∂

∂ (3.1)

0t

w

t

u

mm1

Pm

y

w

RRcty

w

Rcy

w

y

w

Rt

w23

3

2

3

2

2

=

∂

∂

−∂

∂

+−∂

∂

+∂∂

∂

+∂

∂

−∂

∂

+∂

∂ (3.2)

0SR4

1

y

uE

ty

u.

y

u

y

u.

y

uRER

yP

1

yR

t

2

22

2

2

c2

2

r

=θ−

∂

∂−

∂∂

∂

∂

∂+

∂

∂

∂

∂+

∂

θ∂−

∂

θ∂+

∂

θ∂ (3.3)

and 0RKCy

Sy

C

S

1

y

CR

t

C2

2

12

2

c

=+∂

θ∂−

∂

∂−

∂

∂+

∂

∂ (3.4)

with modified boundary conditions:

for t = 0, u = w = 0, θ = 0, C = 0 for all y, for t > 0, u = tn w = 0, 0y

C,0

y=

∂

∂=

∂

θ∂ for

y = 0 and u = 0, w = 0, θ = 0, C = 0 for y = 1.

SOLUTION OF THE EQUATION OF MOTION

Now let’s assume F = u+iw. Combining equations 3.1 and 3.2, we have

( ) 0t

F

im1m1

Pm

y

F

RRcty

F

Rcy

F

y

F

Rt

F23

3

2

3

2

2

=∂

∂

−++∂

∂

+∂∂

∂

+∂

∂

−∂

∂

+∂

∂ (4.1)

The above equation is a third order differential equation that requires three boundary conditions for its solution.

But the present problem provides only two boundary conditions. To remove this difficulty, we follow small perturbation

technique given by Beard and walter to obtain the approximate solution of the given equation, Accordingly, F is expanded

in the powers of Rc

for Rc


6/18



etc, we get zeroth order equation as

( )0

t

F

m1

im1Pm

y

F

y

FR

t

F0

22

0

2

00 =∂

∂

+

−+

∂

∂−

∂

∂+

∂

∂ (4.3)

and 1st order equation as

( )0

t

F

m1

im1Pm

ty

F

y

F

y

F

y

FR

t

F1

22

0

3

2

1

2

3

0

3

11 =∂

∂

+

−+

∂∂

∂+

∂

∂−

∂

∂−

∂

∂+

∂

∂ (4.4)

In order to solve equations 4.3, 4.4, 3.3, 3.4 by Galerkin technique subject to the boundary conditions 3.5, we

choose the following approximate infinite expressions for F0, F1, θ and C as

F0 ≅ tn ( 1 - y ) + a 1 t y ( 1 - y ) + a 2 t

2 y 2 ( 1 - y ) 2 + a 3 t3 y 3 ( 1 - y ) 3 + … (4.5)

F1 ≅ b 1 t y ( 1 - y ) + b 2 t2

y2

( 1 - y )2

+ b 3 t3

y3

( 1 - y )3

+ ….. (4.6)

θ ≅ c 1 t ( 1 - y2 ) + c 2 t

2 y ( 1 - y 2 ) 2 + c 3 t3 y 2 ( 1 - y 2 ) 3 + ….. (4.7)

and C ≅ D 1 t ( 1 - y2 ) + D 2 t

2 y ( 1 - y 2 ) 2 + D 3 t3 y 2 ( 1 - y 2 ) 3 + ….. (4.8)

where a j, b j, c j and D j (j=1,2,3…..) are arbitrary constants to be determined later.

Solution of Zeroth Order Equation (4.3)

Substituting eq. 4.5 in eqn. 4.3, the defect function DF0 is determined as

DF0 = -Rtn + nt

n-1 (1-y) ( )

−+

+ im1m1

Pm1

2

( ) ( )

−

++−+−++ )}im1(

m1

Pm1{y1yy21Rtt2a

21

−+

+−+−+−−++ )im1(m1

Pm1)y1(ty)y6y61(t)y3y2y(Rta2

2

2222232

2

)y12y20y10y2(t)y4y5y2y(Rt[a3 2343345233 −+−+−+−+

)}]im1(m1

Pm1{)y1(yt

2

332 −+

+−+ (4.9)

The defect function DF0 is then minimized by Galerkin technique of orthogonalisation leading to the following

three double integrals.

∫∫ ==−1

0

j j j

0

1

0

3,2,1 jwhere0dydt)y1(ytDF

It is note worthy here that t ∈ [0,1] as t is not large.


7/18




After performing the above integration we obtain the following three algebraic equations involving the parametric

constants a j, j = 1,2,3 as

A1a1 + A2a2 + A3a3 = d1

B1a1 + B2a2 + B3a3 = d2

and C1a1 + C2a2 + C3a3 = d3 (4.10)

The three linear equations in 4.10 are solved by Cramer’s rule to give a1, a2, a3

Substituting values of a1, a2 and a3 in eqn. 4.5, we can get

F0 ≅ tn ( 1 - y ) + a 1 t y ( 1 - y ) + a 2 t

2 y 2 ( 1 - y ) 2 + a 3 t3 y 3 ( 1 - y ) 3 + ….. (4.11)

Solution of First Order Equation (4.4)

The first order equation (4.4) is

0t

F)im1(

m1

Pm

ty

F

y

F

y

F

t

FR

t

F 122

0

3

2

1

2

3

0

3

11 =∂

∂−

++

∂∂

∂+

∂

∂−

∂

∂+

∂

∂+

∂

∂ (4.12)

and F1 ≅ b 1 t y ( 1 - y ) + b 2 t2

y2

( 1 - y )2

+ b 3 t3

y3

( 1 - y )3

+ …..

Substituting F0 and F1 in eqn. 4.12, we get the defect function DF1 as

DF1 = -2a1+4a2 t{Rt(6y-3)+(6y2 -6y+1)}

+6a3 t

2

{Rt(1-12y+30y

2

+20y

3

)+(3y+18y

2

+30y

2

-15y

4

)}

+b1[y(1-y)+Rt(1-2y)+2t+ 2m1

Pm

+(1- im)y(1-y)]

−−

+++−−+−+−+ 22

2

2232222

2)y1(ty)im1(

m1

Pm)y6y61(t)y2y3y(Rt)y1(tyb2

[ ])y10y20y12y2(t)y2y5y4y(Rt)y1(ytb3 4323543233323

−+−−−+−+−+

−−

++ 3322 )y1(yt)im1(m1

Pm (4.13)

The defect function DF1 is minimized by Galerkin technique of orthogonilisation resulting the following three

double integrals stated as

∫∫ ==−1

0

jy j

0

1

0

3,2,1 jwhere0dydt)y1(ttDF

Performing the above integrations, we obtain the following three algebraic equations involving the constants bj’s,

j=1,2,3 as

A ′ 1b 1+A ′ 2b 2+A ′ 3b 3=d ′ 1


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B ′ 1b 1+B ′2b 2+B ′3b 3=d ′2

and C ′1b 1+C ′ 2b 2+C ′3b 3=d ′3 (4.15)

The three linear equations (4.15) are solved by Cramer’s rule to get the value of b1, b2 and b3.

Substituting b1,b2 and b3 in eqn. (4.6), we obtain,

F1 ≅ b 1 t(1-y)+b 2t2 y2(1-y)2+b 3 t

3 y3(1-y)3+….. (4.16)

Consequently, F = F0 + RcF1

∴ F=t n(1-y)+(a1+R c b1) ty(1-y)+(a2 +Rc+b 2) t2 y2+(1-y)2+(a 3+Rcb 3) t

3 y3(1-y)3 (4.17)

Calculation of a1,a2,a3,b1,b2 and b3

Separating real and imaginary parts from the constants A1, A2, A3, B1, B2, B3, C1, C2, C3, d1, d2 and d3 we obtain

24 constants from p1 to p24 and substituting these constants, we obtain the values a1, a2 and a3 as

a1=p29+ip30,a2=p33+ip34 and a3=p37+ip38

Similarly, separating real and imaginary parts from the constants'

1A ,

'

2A ,

'

3A ,

'

1B ,

'

2B ,

'

3B ,

'

1C ,

'

2C ,

'

3C ,

'

1d , '

2d , '

3d , we obtain 24 constants from P39 to P62 and substituting these constants in the expressions for b1, b2 and b3,

we obtain their values as b1 = p37 + ip68, b2 = p71 + ip72 and b3 = p75 + ip76

Substituting the values of a1, a2, a3, b1, b2 and b3 in eqn. 4.17 we obtain

F = u + iw

=tn(1-y)+(p2 9+Rcp 6 7) ty(1-y)+(p3 3+Rcp 7 1)t2y2(1-y)2

+(p3 7 +Rcp7 5) t3y

3(1-y)

3+i[(p3 0+Rcp 6 8 ) ty(1-y)+(p3 4 +Rcp 7 2) t

2y

2(1-y)

2

+(p3 8 +Rcp7 6) t3 y3(1-y)3] (4.18)

Separating real and imaginary parts from F, we obtain

u(y)=t n(1-y)+p7 7 ty(1-y)+p7 8 t2y2 (1-y)2 +P 7 9 t

3 y3(1-y)3 (4.19)

[where p 7 7=p 2 9+Rcp 6 7 ,p7 8 =p3 3 +Rcp 7 1 , and p 7 9 =p 3 7+Rcp 7 5 ]

and w(y)=p 8 0 ty(1-y)+p8 1 t2 y2(1-y)2+p 8 2 t

3 y3(1-y)3 (4.20)

where, p8 0=p 3 0 +Rcp 6 8 ,p8 1=p3 4+Rcp 7 2 and p 8 2=p 3 8+R cp 7 3

Equations (4.19) and (4.20) represent the expressions for velocity of the fluid along the horizontal plates and

along the direction transverse to the plates respectively with boundary conditions.

for t = 0, u = w = 0 for all y

for t > 0, u = tn, w = 0 for y = 0

u = 0, w = 0 for y = 1


9/18




Solution of Equation of Energy

The defect function Dθ is obtained from equation (3.3) with help of eqn’s (4.7) and (4.19) as

−−+−−=θ )y1(stR

41t

p2Rty2y1cD 22

r

21

−++−+−+ )y5y3(tP

4)y5y61(Rt)y1(ty2c 22

r

42222

2

)y4y9y6y(Rt2)y1(yt3{c)y1(yStR4

1 753332223

2222 −+−+−+

−−

−−−+−−

322326423

r

)y1(yStR41)y28y45y181(t

P2

)y21)(1n(ptEREtntER77

n

c

n21n2

c −+−−+ −

)}y21(p)y2y3y)(2n(pRpRR{E277

32

78c77c

1n −++−+−+ +

)y2y5y4y)(3n(pR3)y2y3y(p4{Et5432

79c

32

78

2n −+−+−+−+ +

)y2y5y4y(p6{Et)}y6y61(RRcP25432

79

3n2

78 −+−++−− +

22

77

432

79)y21(ERctp)}y5y10y6y(RRcP6 −+−+−−

)}y4y41(p)y21(RRcp2)y4y8y5y(pRcp6{Et 2277

2

77

432

7877

2 +−−−−−+−+

)y16y24y101(ppRR2{Et 327877c

3 −+−+

)y4y12y13y6y(ppR1265422

7977c +−+−+

)}y4y8y5y(pp4)y4y12y13y6y(Rcp8432

7877

654322

78 −+−−+−+−+

)y12y30y26y9y)(RRcp4pRRcp6(Et54322

787977

4 +−+−++

)y4y16y25y19y7(pRcp30 876547978

−+−++

4322

78

65432

7977y13y6y(p4)y4y12y13y6y(pp6 +−−+−+−

5432

7978

565y150y95y28y3(pRRcp6{Et)}y4y12 −+−+−


10/18



9876542

79

76y20y41y44y26y8y(Rcp27)y32y112 −+−+−+−+

)}y4y16y25y19y7y(pp12)y4 8765437978

10 −+−+−−+

)y10y45y82y77y39y10y(pRR18{Et 9876543279c

6 +−+−+−+

)y4y20y41y44y26y8y(p9109876542

79 +−+−+−− (4.21)

The defect function is minimized by Galerkin technique of orthogonalisation resulting the following three double

integrals as

∫∫ ==−−

1

0

j21 j j

1

0

3,2,1 jwhere0dtdy)y1(yDt

Performing the above integration, we obtain the following three parametric equations involving constants cj’s,

j=1,2,3 as

"

13

"

32

"

21

"

1dcAcAcA =++

"

23

"

32

"

21

"

1dcBcBcB =++

"

33

"

32

"

21

"

1dcCcCcC =++ (4.22)

The set of three linear equations in (4.22) are solved by Cramer’s rule to obtain the constants c1, c2 and c3. Now,

with the values of c1, c2 and c3 the expression for temperature θ becomes.

3223

3

222

2

2

1)y1(ytc)y1(ytc)y1(tc)y( −+−+−=θ (4.23)

Solution of Equation of Concentration

The defect function DC is obtained from equation (3.4) with the help of equation (4.8) as

−++−−= )y1(RKtt

S

2Rty2y1DDC

2

c

2

1

−−−++−+−+ 22232

c

32222

2)y1(yRKt)y5y3(t

S

4)y5y61(Rt)y1(ty2D

)y4y9y6y(Rt2)y1(yt3{D75333222

3 −+−+−+

−+−+−− 32236423

c

)y1(yRKt)y28y45y181(tS

2

)y28y45y181(tcS2)y5y3(tcS4tcS26423

31

32

2111 −+−−−++


11/18




The defect function DC is then minimized by Galerkin technique of orthogonalisation leading to the following

three double integrals

∫∫ =−−

1

0

j21 j j

1

00dydt)y1(yDCt ,

where, j = 1,2,3. Performing the above integration, we obtain the following three parametric equations with the

constants D js,j = 1,2,3, as

'''

13

'''

32

'''

21

'''

1dDADADA =++

'''

23

'''

32

'''

21

'''

1dDBDBDB =++ (4.24)

'''

33

'''

32

'''

21

'''

1 dDCDCDC =++

The three linear equations in (4.24) are solved by Cramer’s rule to obtain the constant D1, D2 and D3

With the values of D1, D2 and D3 the expression for concentration (C) as

3232

3

222

2

2

1)y1(tyD)y1(ytD)y1(tD)y(C −+−+−= (4.25)

The non-dimensional skin-friction is given by

∂∂

∂+

∂

∂−

∂

∂=τ

ty

u

y

uRR

y

u2

2

2

cxy (4.26)

The skin-friction at the lower and the upper plates are determined as

0yxy0 =τ=τ

)pnttRp2tRp2(Rtpt77

1n2

7877c77

n −+−++−= − (4.27)

and1yxy1 =τ=τ

)pnttRp2tpR2(Rtpt77

1n2

7877c77

n ++−+−−= − (4.28)

The rate of heat transfer at the lower and the upper plates are respectively

2

20y0tC

yNu −=

∂

θ∂−= = (4.29)

and tC2y

Nu 11y1 −=∂

θ∂−= = (4.30)

The concentration gradient at the two plates are


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2

20y0GtD

y

CC −=

∂

∂= = (4.31)

and tD2yCC 11y1G =∂∂−= = (4.32)

The constants involved in the above equations are omitted here in order to save space.

RESULTS AND DISCUSSIONS

The present investigation deals with the effect of Hall current on unsteady couette flow of a visco-elastic fluid

with heat and mass transfer incorporation with heat generating sources, chemical reaction and Soret effect. The flow

behavior of the fluid has been analysed with the help of graphs and tables designed under numerical and mathematical

computation on varying various related fluid parameters like suction parameter (R), non-Newtonian elastic parameter

(Rc>0), Hall parameter (m), magnetic Prandtl number (Pm), Prandtl number (Pr), source parameter (S), Schmidt number

(Sc), Eckert number (E), Chemical reaction parameter (K) and time parameter (t).

Velocity

Figure 1 illustrates the velocity profile (u) with the variations of R, R c, Pm, m, n and t. It is observed that the flow

velocity rises with the rise of suction parameter R (Curves I, II & III). Interestingly, no appreciable change is noticed in the

flow on the variation of elastic parameter (Rc) (curves III, IV). Further, it is marked that with imposition of external

magnetic field, velocity of the fluid is decelerated with Rc (curves IV, V). However, it is accelerated with Hall parameter m

(curves V, VI). It is also revealed from the velocity pattern that as time increases, velocity of flow also increases near the

lower plate while it decreases with the increase of the numerical value n. With the height of the fluid in the channel

velocity gradually decreases and becomes zero at the upper plate for the rise of both t and n (curves VI, VII, VIII).

Figure 1: Velocity (u) Profile for Pr = 9.0, Sc = 2, S = 0.1, E = 0.01

The effects of R, Rc, Pm, m, t and n on the secondary velocity flow (w) are depicted in figure 2. The graph

indicates that the rise of suction parameter doesn’t bring any appreciable change in the secondary flow (curves I, II, III).

However, rise of elastic parameter reduces the flow which attains even more negative for higher value of elasticity (curves

III, IV). Further, it is noticed that with the increase of magnetic filed strength the flow velocity increases form negative to

positive value (curves IV & V). On the other hand, rise of Hall parameter rises the flow throughout the region between the

plates (curves V, VI). It is also observed that as n increases the flow is decelerated (curves VI, VII) while it is accelerated

significantly as t increases (curves VII, VIII).


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Figure 2: Velocity (w) Profile for Pr = 9.0, Sc = 2, S = 0.1, E = 0.01

Temperature

The effects of prandtl number Pr, source parameter S and Eckert number E on the temperature field is illustrated

in Figure 3. It is revealed that the fluid temperature falls with the rise of prandtl number within the boundary conditions

imposed (curves I, II). As the source strength increases, temperature is reduced indicating cooling effect in couette flow

(curves II, III). Again, curves III, IV, V unveil the fact that the rise in Eckert number E produces a significant fall in

temperature.

Figure 4. explains the influence of R, Rc, n and t on the temperature field within the boundary conditions. From

the temperature profiles it is revealed that the rise of suction parameter reduces the temperature slowly for Newtonian

fluids, Rc=0 (Curves I, II, III), but as Rc attains value greater than zero temperature of the flow is gradually reduced (curves

III, IV, V). Further it is noticed that as the time t passes temperature falls sharply (curves V, VI) while reversal effect is

noticed as n rises.

Figure 3: Temperature (

) Profile for R = 5.0, Rc = 0.05, Sc = 25, t = 0.05, n = 0.5


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Figure 4: Temperature (

) Profile for Pr = 9.0, S = 0.1, Sc = 25, E = 0.001

Concentration

Finally, concentration profiles are presented under the variation of fluid parameters R, Sc, S1 and K in figure 5,

keeping all other parameters fixed. It is observed that the rise of suction parameter reduces the concentration with greater

reduction for greater value of the parameter (curves I, II, III). Further, the rise in schmidt number Sc reduces the rate ofmass transfer, However, for large Sc, the rate of reduction is insignificant (curves III, IV, V) throughout the flow. It is also

observed that as Soret number S1 and chemical reaction parameter K are increased, concentration of flow field increases

(curves V, VI, VII).

Figure 5: Concentration (C) Profile for Pr = 9.0, Rc = 0.05, S = 0.1, Pm = 0.3, n = 0.5, t = 0.05, E = 0.01

Shear Stresses

The values of shear stresses (skin frictions) for different values of R, Rc and m are entered into table 1, keeping all

other related parameters fixed. It is observed that with the rise of R, skin friction increases at both the plates for both

viscous (Rc=0) and viscoelastic fluids (Rc>0) with exception that it decreases at the upper plate for viscous (or Newtonian)

fluids (Rc=0) only. Further, it is marked that skin friction increases with R at both the plates for both types of fluids with

the rise of Hall current except at the upper plate for Newtonian fluid (Rc=0) where it reduces with the rise of Hall current.

Table 1: Effects of R, Rc and m on Skin Friction for n=0.5, t=0.05, S=0.5, Pr=5.0, Pm=0.3, E=0.02

MRc 0.00 0.05 0.10

R 0 1 0 1 0 1

0.5

5.0 -0.1065 -0.3408 -0.0556 -0.0556 -0.0076 0.2232

10.0 0.0269 -0.4741 0.1287 0.1287 0.2141 0.7076

15.0 0.1603 -0.6075 0.4409 0.4409 0.6810 1.4372

1.0

5.0 -0.1039 -0.3433 -0.0541 -0.0541 -0.0068 0.2291

10.0 0.0309 -0.4781 0.1336 0.1336 0.2216 0.7233

15.0 0.1658 -0.6130 0.4512 0.4512 0.7001 1.4677

1.55.0 -0.1023 -0.3449 -0.0534 -0.0534 -0.0074 0.231710.0 0.0336 -0.4808 0.1355 0.1355 0.2216 0.7284

15.0 0.1694 -0.6166 0.4551 0.4551 0.7014 1.4758


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Table 2: Effects of t, n, R and Rc on Skin Friction for m=0.5, S=0.5, P r=5.0, Pm=0.3, E=0.02

R tn

Rc

0.5 1.0

0 1 0 1

0.5 0.05

0.00 -0.2265 -0.2207 -0.0619 -0.0381

0.05 -0.1119 -0.1119 -0.0006 -0.00060.10 0.0026 -0.0031 0.0606 0.0373

5.0 0.10

0.00 -0.0819 -0.5505 0.0640 -0.2640

0.05 -0.0122 -0.2447 0.1079 -0.0550

0.10 0.0424 0.0424 0.1419 0.1419

10.0 0.05

0.00 0.0269 -0.4741 0.1363 -0.2363

0.05 0.1287 0.1287 0.1793 0.1793

0.10 0.2141 0.7076 0.2108 0.5782

Similarly, from Table 2 which records the effects of R, Rc, n and t on skin friction it is concluded that as t

increases, skin friction rises with R and Rc at both the boundary of flow. On increasing n from 0.5 to 1.0 it is noticed that

skin friction also increases with R and Rc at both the plates.

Rate of Heat Transfer

Table 3 exhibits the dependence of rates of heat transfer (or Nusselt Numbers) on R, R c, n and t, keeping other

related parameters fixed. It is revealed that as t increases, rate of heat transfer decreases at both the plate for both viscous

(Rc=0) and visco-elastic (Rc>0) fluids for both t=0.05 and n=1.0. Further, it is noticed that the rise in n results in increase of

rate of heat transfer at the lower plate for both Rc=0 and Rc>0 and reduction of rates at the upper plate for both R c=0 and

Rc>0 with exception for higher value of Rc. It is also observed that the rates of heat transfer first decreases then increases at

the two plates for both types of fluids. All the above conclusions drawn are in the presence of a uniform external magnetic

field and Hall current.

Table 3: Effects of t, n, R and Rc on the Rates of Heat Transfer

(Nusselt Number) m = 0.5, S=0.2, Pr=5.0, Pm=0.3, E=0.001

R t

n

Rc

0.5 1.0

Nu0 Nu1 Nu0 Nu1

0.5 0.05

0.00 -1.127E-06 -4.538E-05 2.190E-07 -5.619E-05

0.05 -1.645E-06 -3.407E-05 -4.679E-07 -4.162E-05

0.10 -2.435E-06 -1.814E-05 -1.742E-06 -1.697E-05

5.0 0.10

0.00 -4.508E-06 -9.075E-05 8.762E-07 -1.124E-04

0.05 -6.579E-06 -6.814E-05 -1.872E-06 -8.323E-05

0.10 -9.739E-06 -3.627E-05 -6.969E-06 -3.393E-05

10.0 0.05

0.00 -6.539E-07 3.637E-05 -2.298E-07 3.053E-05

0.05 -2.736E-06 5.163E-05 -7.004E-07 1.700E-05

0.10 -7.246E-06 1.244E-04 -2.581E-06 3.570E-05

5.0 0.10

0.00 -2.616E-06 7.274E-05 -9.193E-07 6.106E-05

0.05 -1.094E-05 1.033E-04 -2.801E-06 3.400E-05

0.10 -2.898E-05 2.488E-04 -1.032E-05 7.141E-05

10.0 0.05

0.00 -2.637E-06 3.454E-05 -1.256E-06 2.409E-05

0.05 -1.113E-05 3.827E-05 -4.615E-06 1.060E-05

0.10 -2.448E-05 6.741E-05 -1.106E-05 1.250E-05

Table 4: Effects of Sc on the Concentration Gradient CG0 & CG1 for R = 5.0, K = 0.5,

S1 = 1.0, Rc=0.05, t=0.05, n=0.5, m=0.5, S=0.1, Pr=5.0, Pm=0.3, E=0.001

Sc CG0 CG1

2.0 3.864E-06 -8.404E-05


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Table 4: Contd.,

25.0 9.288E-06 -1.044E-04

425.0 9.979E-06 -1.043E-04

Concentration Gradient

And finally it is observed that the concentration gradient reduces with height of the fluid in the channel for a given

rate of mass transfer. But as Schmidt number (rate of mass transfer) is increased the concentration gradient sharply

increases with marginal increase for high value of Sc. This revelation has been recorded in the Table 4.

CONCLUSIONS

This paper investigates the effect of heat and mass transfer on MHD couette flow of compressible and

incompressible fluids with heat and mass transfer including heat sources, chemical reaction and Soret effect. The effect of

Hall current on unsteady couette flow of a visco-elastic incompressible and electrically conducting fluid with heat and

mass transfer incorporation with heat source, chemical reaction and Soret effect has been analysed. As the study, Hall

effect on unsteady couette flow of a visco elastic fluid with heat and mass transfer including heat sources, chemical

reaction and Soret effect has been analyzed.

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Documents

7. IJPR - Hall Effect on Unsteady Couette Flow of a Visco