Effect of inclined magnetic field on unsteady MHD flow of an incompressible viscous fluid through a porous medium in parallel plate channel

8/10/2019 Effect of inclined magnetic field on unsteady MHD flow of an incompressible viscous fluid through a porous medium in parallel plate channel

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Effect of inclined magnetic field on unsteadyMHD flow of an incompressibleviscous fluid through a porous

medium in parallelplate channel

Dr. M. VEERA KRISHNA* , S.V. SUNEETHA*and S. CHAND BASHA**

* Department of Mathematics, Rayalaseema University,Kurnool (A.P) - 518002 (India)

([email protected]; [email protected])

* * Assistant professor, Department of Mathematics,Kottam College of Engineering,

Chinnatekuru, Kurnool (A.P)-518218 (India)

([email protected])

ABSTRACT

In this paper, we make an initial value investigation of theunsteady flow of incompressible viscous fluid between two rigidnon-conducting rotating parallel plates bounded by a porousmedium taking hall current into account. We discuss a threedimensional flow in a parallel plate channel in a porous mediumunder the influence of inclined magnetic field. The perturbationsare created by a constant pressure gradient along the plates inaddition to the non-torsional oscillations of the upper plate whilethe lower plate is at rest. The flow in the porous medium isgoverned by the Brinkman's equations. The exact solution of thevelocity in the porous medium consists of steady state and transientstate. The time required for the transient state to decay isevaluated in detail and the ultimate quasi-steady state solutionhas been derived analytically, its behaviour is computationallydiscussed with reference to the various governing parameters.

The shear stresses on the boundaries are also obtained analyticallyand their behaviour is computationally discussed.

Key words : Hall effects, unsteady flows, parallel platechannels, incompressible viscous fluids, Brinkman's model.

J . Comp. & Math. Sci. Vol. 1(2), 135-144 (2010).

J ournal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)


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136 M. Veera Krishna et al. , J .Comp.&Math.Sci. Vol.1(2), 135-144 (2010).

1 . INTRODUCTION

The flow between parallel platesis a classical problem that has impor-tant applications in magneto hydrodynamic (MHD) power generators and

pumps, accelerators, aerodynamic heating,electrostatic precipitation polymer tech-nology, petroleum industry, purificationof crude oil and fluid droplets, sprays,designing cooling systems with liquidmetal, centrifugal separation of matterfrom fluid and flow meters. Hartmanand Lazarus 5 studied the influence of a transverse uniform magnetic field onthe flow of a viscous incompressibleelectrically conducting fluid betweentwo infinite parallel stationary andinsulating plates. Then the problem wasextended in numerous ways. Closedform solutions for the velocity fieldswere obtained in Ref 1,3,12&13 under thedifferent physical effects. Some exactand numerical solutions for the heattransfer problem are found in Ref 2&6 .In the above mentioned cases the Hallterm was ignored in applying Ohm'sLaw as it has no marked effect for smalland moderate values of the magneticfield. However, the current trend for theapplication of magneto-hydrodynamicsis to words a strong magnetic field, sothat the influence of electromagneticforce is noticeable by Cramer et a l 3 .Under these conditions, the Hall currentis important and it has a marked effecton the magnitude and direction of thecurrent density and consequently onthe magnetic force. The unsteady hydromagnetic viscous flow through a non-

porous or porous medium has drawnattention in the recent years for possibleapplications in Geophysical andCosmical fluid dynamics. The Hall effectsin the unsteady case were discussedby Vatazhin 14 , Pop 7 and Sakhonovski 11 .

Debnath e t . a l .4

have studied theeffects of Hall current on unsteady hydromagnetic flow past a porous plate in arotating fluid system and the structureof the steady and unsteady flow fieldsis investigated. Rao and Krishna 8 studiedHall effects on the non-torsionallygenerated unsteady hydro magneticflow in semi-infinite expansion of anelectrically conducting viscous rotatingfluid. Krishna and Rao 9 & 10 discussed

the Stokes and Eckmann problems inmagneto hydro dynamics taking Halleffects into account. In this paper, wemake an initial value investigation of the unsteady flow of incompressibleviscous fluid between two rigid non-conducting rotating parallel platesbounded by a porous medium taking hallcurrent into account.

2. FORMULATION AND SOLUTIONOF THE PROBLEM:

We consider the unsteady flowof an incompressible electrically con-ducting viscous fluid bounded by porousmedium with two non-conducting parallelplates. A uniform transverse magneticfield is applied to z-axis. In the presenceof strong magnetic field a current isinclined in a direction normal to theboth electric and magnetic field viz.the hall current of strength H 0 inclined



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M. Veera Krishna et al. , J .Comp.&Math.Sci. Vol.1(2), 135-144 (2010). 137

at angle to the normal to the boun-daries in the transverse xz-plane. Theinclined magnetic field gives rise to asecondary flow transverse to thechannel. The hydro magnetic flow isgenerated in a fluid system by non-

torsional oscillations of the upper plate. The lower plate is at rest. The origin istaken on the lower plate and the x-axisparallel to the direction of the upperplate. Since the plates are infinite inextent, all the physical quantitiesexcept the pressure depend on z and tonly. In the equation of motion alongx-direction, the x-component current

density- sino ze H J and the z-com-

ponent current density sino xe H J .

We choose a Cartesian system0(x, y, z) such that the boundary wallsare at z= 0 and z= l . Z-axis being theaxis of rotation of the plates. The flowthrough porous medium governed bythe Brinkman equations. The unsteadyhydro magnetic equations governingflow through porous medium under theinfluence of a transverse magnetic fieldwith reference to a frame are

sinu

k u

H J

dz

ud

x

P

1

t

u o ze2

2

(2.1)

sinw

k w

H J

dzwd

t w

o xe22

(2.2)

Where, (u, w) is the velocitycomponents along O(x, z) directionsrespectively. is the density of the

fluid, e is the magnetic permeability, , is the coefficient of kinematic viscosity,k is the permeability of the medium,

o H is the applied magnetic field. Whenthe strength of the magnetic field isvery large, the generalized Ohm's lawis modified to include the Hall current,so that

qxH) (E JxH

H

J e0

ee (2.3)

Where, q is the velocity vector,H is the magnetic field intensity vector,E is the electric field, J is the current

density vector, e is the cyclotron freq-

uency, e is the electron collision time, , is the fluid conductivity and, e is the

magnetic permeability.

In equation (2.3) the electronpressure gradient, the ion-slip andthermo-electric effects are neglected.We also assume that the electric fieldE= 0 under assumptions reduces to

sin wsin H J m J 0e z x (2.4)sin usin H J m J 0e x z (2.5)

where eem is the Hall parameter. .

On solving equations (2.4) and(2.5) we obtain

sinsinsin

2 )w(mu

m1

H J 2

0e x (2.6)



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sinsinsin

2 )mw(um1

H J 2

0e z (2.7)

Using the equations (2.6) and(2.7), the equations of the motion withreference to rotating frame are givenby

sinsin

2

2

)m (1 H

dzud

xP

1

t u

2

20

2e

2

2

sin uk

)mw(u

(2.8)

sinsin

2

2

)m (1 H

dzwd

t w

2

20

2e

2

2

sin wk )w(mu

(2.9)Now combining the equations (2.8)and (2.9), we obtain

Let ,iwuq

)sin1(sin1 222

2

2

qk

qim

H

z

q

x

P

t

q oe

(2.10) The boundary and initial conditions are

0,0,0 zt q (2.11)

,0, l zt beaeq t it i (2.12)

We introduce the following non dimen-sional variables are

,,,,

2*

2***

l

lt

t lq

ql

z z

,, 22

** PlPl

Using non-dimensional variables, thegoverning equations are (droppingasterisks)

)sin1(sin 1

22

2

2

q Dqim

M z

q xP

t q

(2.13)

where,

vl H

M e22

02

2is the Hartmann number

k l

D2

1 is inverse Darcy parameter

ee

m is the Hall parameter is the effect of inclined magnetic

field

Also the equation (2.13) reduces to

)sin1(sin 1

22

2

2

q Dim

M z

qP

t q

(2.14)Corresponding initial and boundaryconditions are

00,0 zt q (2.15)

10,, zt beaeq t it i (2.16)

Taking Laplace transform of equation (2.14) using initial condition(2.15) the governing equations in termsof the transformed variable reduces to


,

,



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)sin1(sin 1

22

2

2

sP

qs Dim

M dz

qd

(2.17)

The relevant transformed boundaryconditions are

,1,0 zq (2.18)

,is

bis

aq z = 0 (2.19)

Solving the equation (2.17) andmaking use of the boundary conditions(2.18) and (2.19), we obtain

21

11 s

P zSinh B zCosh Aq

(2.20)

Where A=

s

P2

1 , B= is

aSinh 1

1

s

CoshP

s

Pis

b2

1

12

1

1

22

1 )sin1(sin

Dim M

s

Taking inverse Laplace transform to theequation (2.20), we obtain

aSinha

zaSinhaCoshP

aSinha

zaSinhP

aP

q11

11

11

1

1

t ieaSinh

zaSinha

2

2

t at i ea

zaCoshPe

aSinh

zaSinhb 1

1

1

3

3

n an zb

an za

1 322

222

t nae

annnCos zP )(

12222

221

)()1(

(2.21)

The shear stresses on the upper plateand the lower plate are given by

and

0

z L

1 zU dz

dq

dzdq

3. RESULTS AND DISCUSSION

The flow is governed by thenon-dimensional parameters M theHartman number, D -1 the inverse Darcyparameter and m is the Hall parameter.

The velocity field in the porous regionis evaluated analytically its behaviour withreference to variations in the governingparameters has been computationallyanalyzed. The profiles for u and v havebeen plotted in the entire flow field inthe porous medium. The solution forthe velocity consists of three kinds of terms 1. Steady state 2. The quasi-steadystate terms associated with non-torsionaloscillations in the boundary, 3. thetransient term involving exponentiallyvarying time dependence. From theexpression (2.21), it follows that thetransient component in the velocity inthe fluid region decays in dimen-sionless time.




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t > max //1

,//

122

11 naaWhen the

transient terms decay the steadyoscillatory solution in the fluid regionis given by

(q) steady =11

1

1 aSinha

zaSinhP

aP

11

11

aSinha

zaSinhaCoshP

t ie

aSinh

zaSinha

2

2yoscillator (q)

t ieaSinh

zaSinhb

3

3

We now discuss the quasisteady solution for the velocity fordifferent sets of governing parametersnamely viz. M the Hartman number andD-1 the inverse Darcy parameter m isthe Hall parameter, P the non dimen-

sional pressure gradient, the frequencyoscillations , a and b the constantsrelated to non torsional oscillations of the boundary, for computationalanalysis purpose we are fixing the axialpressure gradient as well as a and b ,

and3

. Figures (1-6) corresponding

to the velocity components u and w along the imposed pressure gradientfor different sets of governing parameters

when the upper boundary plate executesnon-torsional oscillations. The magni-tude of the velocity u and w increasesfor the sets of values 0.1 z 0.3 aswell as which reduces for all values of z with increase in the intensity of the

magnetic field (Fig 1 & 4). The resultantvelocity q decreases with increasingthe Hartmann number M. The magnitudeof the velocity u decreases in the upperpart of the fluid region 0.1 z 0.2while it experiences enhancementlower part 0.3 z 0.9 with increasingthe inverse Darcy parameter D -1(Fig. 2).

The magnitude of the velocity w increases in the upper part of the fluidregion 0.1 z 0.3, while it reducesin lower part 0.4 z 0.9 withincreasing the inverse Darcy parameterD-1 (Fig 5). The resultant velocity q reduces with increasing the inverseDarcy parameter D -1. The magnitude of velocity u decreases in the upper partof the fluid region while it experiencesenhancement lower part 0.3 z 0.9and also the magnitude of velocity v increases through out the fluid region(Fig. 3 & 6). However the resultantvelocity q enhances with increasing

the Hall parameter m.

The shear stresses x and y onthe upper plate have been calculated forthe different variations in the governingparameters and are tabulated in thetables (1-2). On the upper plate we

notice that the magnitudes of xenhances the inverse Darcy parameterD-1 and the hall parameter m decreases




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with increase in the Hartmann number

M (table 1). The magnitude of ydecreases with increase in the Hartmannnumber M, the inverse Darcy parameterD-1 and the Hall parameter m fixing the

other parameters (table 2). The similarbehaviour is observed on the lower plate.We also notice that the magnitude of the shear stresses on the lower plateis very small compare to its values of the upper plate.

-0.3

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

1.3

1.5

0 0.2 0.4 0.6 0.8 1

z

u

M=2

M=5

M=8

M=10

-0.25

-0.05

0.15

0.35

0.55

0.75

0.95

1.15

1.35

1.55

z

u

m=1m=2

m=3

m=4

-0.33

-0.28

-0.23

-0.18

-0.13

-0.08

-0.03

0.02

0 0.2 0.4 0.6 0.8 1

z

v

M=2

M=5

M=8

M=10

Fig. 1. The velocity profile for uwith M.

Fig. 2. The velocity profile for uwith D -1 .

Fig. 3. The velocity profile for uwith m.

Fig. 4. The velocity profile for vwith M.

-0.25

-0.05

0.15

0.35

0.55

0.75

0.95

1.15

1.35

0 0.2 0.4 0.6 0.8 1

z

u

D=1000

D=2000

D=3000

D=4000




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Fig. 5. The velocity profile for v with D -1 .

Fig. 6. The velocity profile for u with m.

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.2 0.4 0.6 0.8 1

z

v

m=1

m=2

m=3

m=4

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0 0.2 0.4 0.6 0.8 1

z

v

D=1000

D=2000

D=3000

D=4000




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4 . CONCLUSIONS

1. The resultant velocity q enhances with increasing hall para-meter m and decreases with increasinginverse Darcy parameter D -1 as well asthe Hartmann number M. 2. On the

upper plate the magnitudes of x enhances with increasing the hallparameter m and the inverse Darcyparameter D -1 decreases with increasein the Hartmann number M. The mag-nitude of y decreases with increase

M I II III IV V

2 0.045256 0.052353 0.668275 0.052876 0.060992

5 0.032828 0.043526 0.050346 0.043387 0.051144

I II III IV V

D-1 1000 2000 3000 1000 1000m 1 1 1 2 3

Table 1. The shear stress ( x ) on the upper plate

M I II III IV V

2 -0.05393 -0.04055 -0.03495 -0.04804 -0.32568

5 -0.04525 -0.03425 -0.02434 -0.03345 -0.02245

I II III IV V

D-1 1000 2000 3000 1000 1000

m 1 1 1 2 3

Table 2: The shear stress ( y ) on the upper plate

in the Hartmann number M, the hallparameter m and the inverse Darcyparameter D -1 fixing the other parameters.3. The similar behaviour is observedon the lower plate. 4. The magnitudeof the shear stresses on the lower plateis very small compare to its values of the upper plate.

REFERENCES

1. Alpher R.A. "Heat transfer in magnetodynamic flow between parallelplates". Int. J. Heat and mass transfer,




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3, 108 (1961).2. Attia H.A. and N.A. Kotb., "MHD

flow between two parallel plateswith heat transfer", Acta . M ech. ,117, 215-220 (1996).

3. Cramer K. and S. Pai., "Magneto

hydro dynamics for engineers andapplied physicists". M c.Graw Hill,New York (1973).

4. Debnath. L., Ray. S.C. and Chatterjee.A.K., ZA M M , 5 9 , 469-471 (1979).

5. Hartmann J . and F. Lazarus., Mat.Fys. Medd., 15, 6-7 (1937).

6. Nigam. S.D. and S.N. Singh., "Heattransfer by laminar flow betweenparallel plates under the action of transverse magnetic field". Q. J .M ech. Appl . Math . , 13, 85 (1960).

7. Pop. I. J. M ath. Phys. Sci., 5 , 375(1971).

8. Prasad Rao. D.R.V. and Krishna.D.V. Ind.J. Pure and Appl.M ath. 12,12, 270 (1981).

9. Rao, D. R. V., Krishna. D.V. andDebnath. L., Acta . M ech. , 43, 49(1982).

10. Rao. D.R.V., SivaPrasad. R and U.Rajeshwara Rao., Bulletin of pure

and applied scienc es, 22E, 2, 501-510 (2003).

11. Sakhnovski. E. G., Zh. Telch on Fiz.,33, 631 (1963).

12. Sutton. G. W. and A. Sherman,"Engineering Magneto hydro dyna-

mics".M c. Graw Hill,

New York(1965).13. Tao. L. N., "Steady motion of

conducting fluids in channels undertransverse magnetic field withconsideration of Hall effects", J .Aerospace Sci. Vol. 29 , p. 287(1960).

14. Vatazhin. Z., Angew. M ath . M ech.43, 127 (1963).

Appendix:

Dim

M a ,

)sin1(sin 1

22

1

i Dim

M a ,

)sin1(sin 1

22

2

PPi Dim

M a ,

)sin1(sin 1

22

3



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Effect of inclined magnetic field on unsteady MHD flow of an incompressible viscous fluid through a porous medium in parallel plate channel