hazem ali attia- effect of hall current on the velocity and temperature distributions of couette...

8/3/2019 Hazem Ali Attia- Effect of Hall current on the velocity and temperature distributions of Couette flow with variable pr

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Volume 28, N. 2, pp. 195212, 2009

Copyright 2009 SBMAC

ISSN 0101-8205

www.scielo.br/cam

Eect o Hall current on the velocity and

temperature distributions o Couette fow with

variable properties and uniorm suction and injection

HAZEM ALI ATTIA

Department o Engineering Mathematics and Physics

Faculty o Engineering, El-Fayoum University

El-Fayoum-63111, Egypt

E-mail: [email protected]

Abstract. The unsteady hydromagnetic Couette fow and heat transer between two parallel

porous plates is studied with Hall eect and temperature dependent properties. The fuid is acted

upon by an exponential decaying pressure gradient and an external uniorm magnetic eld. Uni-

orm suction and injection are applied perpendicularly to the parallel plates. Numerical solutions

or the governing non-linear equations o motion and the energy equation are obtained. The eect

o the Hall term and the temperature dependent viscosity and thermal conductivity on both the

velocity and temperature distributions is examined.

Mathematical subject classication: 76D05, 76M25.

Key words: Fluid Mechanics, nonlinear equations, Hall eect, numerical solution.

1 Introduction

The fow between parallel plates is a classical problem that has important ap-

plications in magnetohydrodynamic (MHD) power generators and pumps, ac-

celerators, aerodynamic heating, electrostatic precipitation, polymer technol-

ogy, petroleum industry, purication o crude oil and fuid droplets and sprays.

Hartmann and Lazarus [1] studied the infuence o a transverse uniorm magnetic

eld on the fow o a viscous incompressibleelectrically conducting fuid between

#756/08. Received: 21/III/08. Accepted: 01/II/09.


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196 VELOCITY AND TEMPERATURE DISTRIBUTIONS OF COUETTE FLOW

two innite parallel stationary and insulating plates. The problem was extended

in numerous ways. Closed orm solutions or the velocity elds were obtained

[2-5] under dierent physical eects. Some exact and numerical solutions or

the heat transer problem are ound in [6, 7]. In the above mentioned cases the

Hall term was ignored in applying Ohms law as it has no marked eect or small

and moderate values o the magnetic eld. However, the current trend or the

application o magnetohydrodynamics is towards a strong magnetic eld, so that

the infuence o electromagnetic orce is noticeable [5]. Under these conditions,

the Hall current is important and it has a marked eect on the magnitude and

direction o the current density and consequently on the magnetic orce. Tani

[8] studied the Hall eect on the steady motion o electrically conducting and

viscous fuids in channels. Soundalgekar et al. [9, 10] studied the eect o Hall

currents on the steady MHD Couette fow with heat transer. The temperatures

o the two plates were assumed either to be constant [9] or varying linearly along

the plates in the direction o the fow [10]. Abo-El-Dahab [11] studied the eect

o Hall currents on the steady Hartmann fow subject to a uniorm suction and

injection at the bounding plates. Attia [12] extended the problem to the unsteady

state with heat transer.

Most o these studies are based on constant physical properties. It is known

that some physical properties are unctions o temperature [13] and assuming

constant properties is a good approximation as long as small dierences in tem-

perature are involved. More accurate prediction or the fow and heat transer

can be achieved by considering the variation o the physical properties with tem-

perature. Klemp et al. [14] studied the eect o temperature dependent viscosity

on the entrance fow in a channel in the hydrodynamic case. Attia and Kotb[7] studied the steady MHD ully developed fow and heat transer between two

parallel plates with temperature dependent viscosity in the presence o a uniorm

magnetic eld. Later Attia [15] extended the problem to the transient state. The

infuence o Hall current and variable properties on unsteady Hartmann fow

with heat transer was given in [16]. The infuence o variations in the physical

properties on steady Hartmann fow was studied by Attia without taking the Hall

eect into considerations [17]. The eect o variable properties on Couette fowin a porous medium was done by Attia [18].

Comp. Appl. Math., Vol. 28, N. 2, 2009


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HAZEM ALI ATTIA 197

In the present work, the unsteady Couette fow o a viscous incompressible

electrically conducting fuid is studied with heat transer. The viscosity and

thermal conductivity o the fuid are assumed to vary with temperature. The

fuid is fowing between two electrically insulating plates and is acted upon by

an exponential decaying pressure gradient while a uniorm suction and injection

is applied through the surace o the plates. The upper plate is moving with a

constant velocity while the lower plate is kept stationary. An external uniorm

magnetic eld is applied perpendicular to the plates and the Hall eect is taken

into consideration. The magnetic Reynolds number is assumed small so that the

induced magnetic eld is neglected [4]. The two plates are kept at two constant

but dierent temperatures. This conguration is a good approximation o some

practical situations such as heat exchangers, fow meters, and pipes that connect

system components. Thus, the coupled set o the equations o motion and the

energy equation including the viscous and Joule dissipation terms becomes non-

linear and is solved numerically using the nite dierence approximations to

obtain the velocity and temperature distributions.

2 Formulation o the problem

The fuid is assumed to be fowing between two innite horizontal plates located

at the y = h planes. The two plates are assumed to be electrically insulating

and kept at two constant temperatures T1 or the lower plate and T2 or the upper

plate with T2 > T1. The upper plate is moving with a constant velocity Uo while

the lower plate is kept stationary. The motion is produced by an exponential

decaying pressure gradient d P/d x = Get in the x-direction, where G and

are constants. A uniorm suction rom above and injection rom below, with

velocity vo, are applied at t = 0. A uniorm magnetic eld Bo is applied in the

positive y-direction. This is the only magnetic eld in the problem as the induced

magnetic eld is neglected by assuming a very small magnetic Reynolds number.

The Hall eect is taken into consideration and consequently a z-component or

the velocity is expected to arise. The viscosity o the fuid is assumed to vary

exponentially with temperature while its thermal conductivity is assumed to

depend linearly on temperature. The viscous and Joule dissipations are taken

into consideration. The fuid motion starts rom rest at t = 0, and the no-slip



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condition at the plates implies that the fuid velocity has neither a z nor an x-

component at y = h. The initial temperature o the fuid is assumed to be equal

to T1. Since the plates are innite in the x andz-directions, the physical quantities

do not change in these directions and the problem is essentially one-dimensional.

The fow o the fuid is governed by the Navier-Stokes equation

D v

Dt= p + ( v) + J Bo (1)

where is the density o the fuid, is the viscosity o the fuid, J is the current

density, and v is the velocity vector o the fuid, which is given by

v = u(y, t)i + vo j + w(y, t)k

I the Hall term is retained, the current density J is given by the generalized

Ohms law [4]

J =

v Bo

J Bo

(2)

where is the electric conductivity o the fuid and is the Hall actor [4]. Using

Eqs. (1) and (2), the two components o the Navier-Stokes equation are

u

t+ vo

u

y= Get +

2u

y2+

y

u

y

B2o

1 + m2(u + mw) (3)

w

t+ vo

w

y=

2w

y2+

y

w

y

B2o

1 + m2(w mu) (4)

where m is the Hall parameter given by m = Bo. It is assumed that the

pressure gradient is applied at t = 0 and the fuid starts its motion rom rest.

Thus

t = 0 : u = w = 0 (5a)

Fort >0, the no-slip condition at the plates implies that

y = h : u = w = 0 (5b)

y = h : u = Uo, w = 0 (5c)



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HAZEM ALI ATTIA 199

The energy equation describing the temperature distribution or the fuid is

given by [19]

cp Tt

+ cpvo Ty

=

y

k T

y

+

uy

2

+

w

y

2

+B2o

1 + m2(u2 + w2)

(6)

where T is the temperature o the fuid, cp is the specic heat at constant pres-

sure o the fuid, and k is thermal conductivity o the fuid. The last two terms

in the right-hand-side o Eq. (6) represent the viscous and Joule dissipations

respectively.

The temperature o the fuid must satisy the initial and boundary conditions,

t = 0 : T = T1 (7a)

t > 0 : T = T1, y = h (7b)

t > 0 : T = T2, y = h (7c)

The viscosity o the fuid is assumed to vary with temperature and is dened as, = o f1(T). By assuming the viscosity to vary exponentially with temper-

ature, the unction f1(T) takes the orm [7], f1(T) = exp(a1(T T1)). In

some cases a1 may be negative, i.e. the coecient o viscosity increases with

temperature [7, 15].

Also the thermal conductivity o the fuid is varying with temperature as k =

ko f2(T). We assume linear dependence or the thermal conductivity upon the

temperature in the orm k = ko(1 + b1(T T1)) [19], where the parameter b1may be positive or negative [19].

The problem is simplied by writing the equations in the non-dimensional

orm. To achieve this dene the ollowing non-dimensional quantities,

y =y

h, t =

tUo

h, G =

hG

U2o, (u, w) =

(u, w)

Uo, T =

T T1

T2 T1,

f1(T) = ea1(T2T1)T = eaT, a is the viscosity parameter,

f2(T) = 1 + b1(T2 T1)T = 1 + bT, b is the thermal conductivity parameter,

Re = Uoh/o is the Reynolds number,



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S = voh/o is the suction parameter,

H a2 = B2o h2/o, H a is the Hartmann number,

Pr = ocp/ ko is the Prandtl number,

Ec = U2o /cp(T2 T1) is the Eckert number.

In terms o the above non-dimensional quantities the velocity and energy equa-

tions (3) to (7) read (the hats are dropped or convenience)

u

t+

S

Re

u

y

= Get +1

Re

f1(T) 2u

y2+

1

Re

f1(T)

y

u

y

H a2

Re(1 + m2)

(u + mw)(8)

w

t+

S

Re

w

y

=1

Ref1(T)

2w

y2+

1

Re

f1(T)

y

w

y

H a2

Re(1 + m2)(w mu)

(9)

t = 0 : u = w = 0 (10a)

t > 0 : y = 1, u = w = 0 (10b)

t > 0 : y = 1, u = w = 0 (10c)

T

t+

S

Re

T

y=

1

Prf2(T)

2T

y2+

1

Pr

f2(T)

y

T

y

+Ecf 1(T)

uy

2+

wy

2+ EcH a

2

Re(1 + m2)(u2 + w2)

(11)

t = 0 : T = 0 (12a)

t > 0 : T = 0, y = 1 (12b)

t > 0 : T = 1, y = 1 (12c)

Equations (8), (9), and (11) represent a system o coupled non-linear partialdierential equations which can be solved numerically under the initial and



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HAZEM ALI ATTIA 201

boundary conditions (10) and (12) using nite dierence approximations. The

Crank-Nicolson implicit method is used [20]. Finite dierence equations relat-

ing the variables are obtained by writing the equations at the mid point o the

computational cell and then replacing the dierent terms by their second order

central dierence approximations in the y-direction. The diusion terms are

replaced by the average o the central dierences at two successive time levels.

The non-linear terms are rst linearized and then an iterative scheme is used at

every time step to solve the linearized system o dierence equations. All calcu-

lations have been carried out or G = 5, = 1, Pr = 1, Re = 1, and Ec = 0.2.

Step sides t = 0.001 and y = 0.05 or time and space respectively, are cho-

sen and the scheme converges in at most 7 iterations at every time step. Smaller

step sizes do not show any signicant change in the results. Convergence o the

scheme is assumed when any one ou, w, T, u/y, w/y and T/y or the

last two approximations dier rom unity by less than 106 or all values o y

in 1 < y < 1 at every time step. Less than 7 approximations are required

to satisy this convergence criteria or all ranges o the parameters studied here.

The transient results obtained here converges to the steady state solutions given

in [17] in the case m = 0.

3 Results and Discussion

Figure 1 presents the velocity and temperature distributions as unctions o y

or various values o time t starting rom t = 0 up to steady state. The gure

is evaluated or H a = 1, m = 3, S = 0, a = 0.5 and b = 0.5. It is clear

that the velocity and temperature distributions do not reaches the steady state

monotonically. They increase with time up till a maximum value and decrease

up to the steady state due to the infuence o the decaying pressure gradient.

The velocity component u reaches the steady state aster than w which, in turn,

reaches the steady state aster than T. This is expected as u is the source o w,

while both u and w are sources o T.

Figure 2 depicts the variation o the velocity component u at the centre o the

channel (y = 0) with time or various values o the Hall parameter m and the

viscosity parameter a and or b = 0 and H a = 3. The gure shows that u

increases with m or all values o a. This is due to the act that an increase in



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a)0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

b)0

0.5

1

1.5

-1 -0.5 0 0.5 1

c)0

0.5

1

1.5

-1 -0.5 0 0.5 1

t=0.5 t=1 t=2

Figure 1 The evolution o the prole o: (a) u; (b) w and (c) T. (H a = 3, m = 3,

a = 0.5, b = 0.5).

m decreases the eective conductivity (/(1 + m2)) and hence the magnetic

damping. The gure shows also how the eect oa on u depends on the param-

eterm. For small and moderate values oa, increasing a decreases u, however,

or higher values o m, increasing a increases u. It is observed also rom the

gure that the time at which u reaches its steady state value increases with in-

creasing m while it is not greatly aected by changing a.

Figure 3 presents the variation o the velocity component w at the centre o

the channel (y = 0) with time or various values o m and a and or b = 0,

S = 0 and H a = 3. The gure shows that w increases with increasing m or

all values oa. This is because w, which is the z-component o the velocity is a

result o the Hall eect.



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HAZEM ALI ATTIA 203

a)0

0.2

0.4

0.6

0 1 2 3 4

b)0

0.2

0.4

0.6

0.8

0 1 2 3 4

c)

0

0.5

1

1.5

2

0 1 2 3 4

a=-0.5 a=0 a=0.5

Figure 2 The evolution o u at y = 0 or various values o a and m: (a) m = 0;

(b) m = 1 and (c) m = 5. (H a = 3, b = 0).

Although the Hall eect is the source or w, a careul study o Figure 3 shows

that, at small times, an increase in m produces a decrease in w. This can be

understood by studying the term ((w mu )/(1 + m2)) in Eq. (10), which

is the source term o w. At small times w is very small and this term may be

approximated to (mu/(1 + m2)), which decreases with increasing m i m > 1.

Figure 3 shows also that or small values om the eect o a on w depends on

t. For small t, increasing a increases w, but with time progress, increasing a

decreases w.

Figure 4 presents the evolution o the temperature T at the centre o the channel

or various values o m and a when b = 0 and H a = 3. The eect o m on T



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a)

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4

b)

0

0.5

1

1.5

0 1 2 3 4

a=-0.5 a=0 a=0.5

Figure 3 The evolution o w at y = 0 or various values o a and m: (a) m = 1 and

(b) m = 5. (H a = 3, b = 0).

depends on t. When m >1, increasing m decreases T slightly at small times but

increases T at large times. This is because when t is small, u and w are small

and an increase in m results in an increase in u but a decrease in w, so the Joule

dissipation which is proportional also to (1/(1 + m2)) decreases. When t is

large, u and w increase with increasing m and so do the Joule and viscous

dissipations. Increasing a increases T due to its eect on decreasing thevelocities

and the velocity gradients and the unction f1.Figure 5 presents the evolution o T at the centre o the channel or various

values o m and b when a = 0, S = 0 and H a = 3. The gure indicates that

increasing b increases T and the time at which it reaches its steady state or all

m. This occurs because the centre o the channel acquires heat by conduction

rom the hot plate. The parameter b has no signicant eect on u or w in spite

o the coupling between the momentum and energy equations.

Table 1 shows the dependence o the steady state temperature at the centre o

the channel on a and m or b = 0 and S = 0. It is observed that T decreases



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HAZEM ALI ATTIA 205

a)

0

0.2

0.4

0.6

0.8

0 1 2 3 4

b)0

0.2

0.4

0.6

0.8

0 1 2 3 4

c)

0

0.2

0.4

0.6

0.8

0 1 2 3 4

a=-0.5 a=0 a=0.5

Figure 4 The evolution o T at y = 0 or various values o a and m: (a) m = 0;

(b) m = 1 and (c) m = 5. (H a = 3, b = 0).

with increasing m as a result o decreasing the dissipations. On the other hand,

increasing a decreases T as a consequence o decreasing the dissipations. Table 2

shows the dependence o T at the centre o the channel on m and b or a = 0,

S = 0 and H a = 1. The dependence o T on m is explained by the same

argument used or Table 1. Table 2 shows that increasing b increases T since

the centre gains temperature by conduction rom hot plate. Table 3 shows the

dependence o T on a and b or H a = 1 and m = 3. Increasing a decreases T

and increasing b decreases T.

Figures 6, 7, and 8 present the time development o the velocity components u

and w and the temperature T, respectively, at the centre o the channel (y = 0) or



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a)

0

0.2

0.4

0.6

0.8

0 1 2 3 4

b)0

0.2

0.4

0.6

0.8

0 1 2 3 4

c) 0

0.2

0.4

0.6

0.8

0 1 2 3 4

b=-0.5 b=0 b=0.5

Figure 5 The evolution o T at y = 0 or various values o b and m: (a) m = 0;

(b) m = 1 and (c) m = 5. (H a = 3, a = 0).

T m = 0.0 m = 0.5 m = 1.0 m = 3.0 m = 5.0

a = 0.5 0.5541 0.5511 0.5452 0.5338 0.5314

a = 0.1 0.5459 0.5439 0.5398 0.5310 0.5291

a = 0.0 0.5435 0.5417 0.5382 0.5300 0.5281

a = 0.1 0.5412 0.5396 0.5365 0.5289 0.5272

a = 0.5 0.5325 0.5318 0.5301 0.5253 0.5240

Table 1 Variation o the steady state temperature T at y=

0 or various values om and a (H a = 1, b = 0).



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HAZEM ALI ATTIA 207

T m = 0.0 m = 0.5 m = 1.0 m = 3.0 m = 5.0

a = 0.5 0.5541 0.5511 0.5452 0.5338 0.5314

a = 0.1 0.5459 0.5439 0.5398 0.5310 0.5291

a = 0.0 0.5435 0.5417 0.5382 0.5300 0.5281

a = 0.1 0.5412 0.5396 0.5365 0.5289 0.5272

a = 0.5 0.5325 0.5318 0.5301 0.5253 0.5240

Table 2 Variation o the steady state temperature T at y = 0 or various values o

m and b (H a = 1, a = 0).

T m = 0.0 m = 0.5 m = 1.0 m = 3.0 m = 5.0

b = 0.5 0.4805 0.4781 0.4732 0.4621 0.4596

b = 0.1 0.5329 0.5311 0.5273 0.5187 0.5167

b = 0.0 0.5435 0.5417 0.5382 0.5300 0.5281

b = 0.1 0.5528 0.5512 0.5478 0.5401 0.5383

b = 0.5 0.5795 0.5781 0.5382 0.5690 0.5677

Table 3 Variation o the steady state temperature T at y = 0 or various values o

a and b (H a = 1, m = 3).

various values o the suction parameter S and the viscosity variation parameter

a when H a = 3, m = 3, and b = 0. Figures 6 and 7 indicate that increasing S

decreases both u and w or all a due to the convection o the fuid rom regions

in the lower hal to the centre which has higher fuid speed. It is also clear that

the infuence o the parametera on u and w becomes more pronounced or lower

values o the parameter S. Figure 8 shows that increasing the suction parameter

decreases the temperature T or all a as a result o the infuence o convection

in pumping the fuid rom the cold lower hal towards the centre o the channel.Figure 9 presents the time development o the temperature T at the centre

o the channel (y = 0) or various values o the suction parameter S and the

thermal conductivity variation parameter b when H a = 3, m = 3, and a = 0.

The gure shows that increasing S decreases T or all b. Figure 9a indicates

that, orS = 0, the variation oT with the parameterb depends on time as shown

beore in Figure 5c or higher values o the Hall parameterm. Figures 9b and 9c

present an interesting eect or the suction parameter in the suppression o thecrossover points occurred in the T t graph due to changing o the parameterb.



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a)

0

0.5

1

1.5

0 1 2 3 4

b)

0

0.5

1

1.5

0 1 2 3 4

c)0

0.5

1

1.5

0 1 2 3 4

a=-0.5 a=0 a=0.5

Figure 6 The evolution o u at y = 0 or various values o a and S: (a) S = 0;

(b) S = 1; (c) S = 2. (H a = 3, m = 3, b = 0).

Note that the infuence o increasing the parameter b on T is more apparent orhigher values o suction velocity.

4 Conclusions

The transient MHD Couette fow between two parallel plates is studied with the

inclusion o the Hall eect. The viscosity and thermal conductivity o the fuid

are assumed to vary with temperature. The eects o the Hartmann number H a,

the Hall parameter m, the viscosity parameter a and the thermal conductivity

parameter b on the velocity and temperature elds at the centre o the channel



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HAZEM ALI ATTIA 209

a)0

0.5

1

1.5

0 1 2 3 4

b)0

0.2

0.4

0.6

0.8

0 1 2 3 4

c)

0

0.2

0.4

0.6

0 1 2 3 4

a=-0.5 a=0 a=0.5

Figure 7 The evolution o w at y = 0 or various values o a and S: (a) S = 0;

(b) S = 1; (c) S = 2. (H a = 3, m = 3, b = 0).

are discussed. Introducing the Hall term gives rise to a velocity component

w in the z-direction and aects the main velocity u in the x-direction. It was

ound that the parameter a has a marked eect on the velocity components u

and w or all values om. On the other hand, the parameterb has no signicant

eect on u orw.

REFERENCES

[1] J. Hartman and F. Lazarus, Kgl. Danske Videnskab. Selskab. Mat.-Fys. Medd., 15(6-

7) (1937).[2] L.N. Tao, Magnetohydrodynamic eects on the ormation o Couette fow. Journal o

Aerospace Sci., 27 (1960), 334.



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a)0

0.2

0.4

0.6

0.8

0 1 2 3 4

b)0

0.1

0.2

0.3

0.4

0 1 2 3 4

c)

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4

a=-0.5 a=0 a=0.5

Figure 8 The evolution o at y = 0 or various values o a and S: (a) S = 0;

(b) S = 1; (c) S = 2. (H a = 3, m = 3, b = 0).

[3] R.A. Alpher, Heat transer in magnetohydrodynamic fow between parallel plates. Interna-

tional Journal o Heat and Mass Transer, 3 (1961), 108.

[4] G.W. Sutton and A. Sherman, Engineering Magnetohydrodynamics. McGraw-Hill (1965).

[5] K. Cramer and S. Pai, Magnetofuid dynamics or engineers and applied physicists.

McGraw-Hill (1973).

[6] S.D. Nigam and S.N. Singh, Heat transer by laminar fow between parallel plates under

the action o transverse magnetic eld. Quart. J. Mech. Appl. Math., 13 (1960), 85.

[7] H.A. Attia and N.A. Kotb, MHD fow between two parallel plates with heat transer. Acta

Mechanica, 117 (1996), 215220.

[8] I. Tani, Steady motion o conducting fuids in channels under transverse magnetic elds with

consideration o Hall eect. Journal o Aerospace Sci., 29 (1962), 287.



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HAZEM ALI ATTIA 211

a)

0

0.2

0.4

0.6

0.8

0 1 2 3 4

b)

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4

c)

0

0.1

0.2

0.3

0 1 2 3 4

b=-0.5 b=0 b=0.5

Figure 9 The evolution o at y = 0 or various values o b and S: (a) S = 0;

(b) S = 1; (c) S = 2. (H a = 3, m = 3, a = 0).

[9] V.M. Soundalgekar, N.V. Vighnesam and H.S. Takhar, Hall and ion-slip eects in MHD

Couette fow with heat transer. IEEE Transactions on Plasma Sciences, PS-7(3) (1979).

[10] V.M. Soundalgekar and A.G. Uplekar, Hall eects in MHD Couette fow with heat transer.

IEEE Transactions on Plasma Science, PS-14(5) (1986).

[11] E.M.H. Abo-El-Dahab, Eect o Hall currents on some magnetohydrodynamic fow prob-

lems. Master Thesis, Dept. o Math., Fac. o Science, Helwan Univ., Egypt (1993).

[12] H.A. Attia, Hall current eects on the velocity and temperature elds o an unsteady Hart-

mann fow. Can. J. Phys., 76 (1998), 739746.

[13] H. Herwig and G. Wicken, The eect o variable properties on laminar boundary layer fow.

Warme-und Stoubertragung, 20 (1986), 4757.



18/18


[14] K. Klemp, H. Herwig and M. Selmann, Entrance fow in channel with temperature depen-

dent viscosity including viscous dissipation eects. Proceedings o the Third International

Congress o Fluid Mechanics, Cairo, Egypt, 3 (1990), 12571266.

[15] H.A. Attia, Transient MHD fow and heat transer between two parallel plates with temper-

ature dependent viscosity. Mechanics Research Communications, 26(1) (1999), 115121.

[16] H.A. Attia and A.L. Aboul-Hassan, The eect o variable properties on the unsteady Hart-

mann fow with heat transer considering the Hall eect. Applied Mathematical Modelling,

27(7) (2003), 551563.

[17] H.A. Attia, On the eectiveness o variation in the physical variables on the mhd steady

fow between parallel plates with heat transer. International Journal or Numerical Methods

in Engineering, John Wiley & Sons, Ltd., 65(2) (2006), 224235.

[18] H.A. Attia, On the eectiveness o variation in the physical variables on generaized Couette

fow with heat transer in a porous medium. Research Journal o Physics, Asian Network or

Scientic Inormation (ANSI), 1(1) (2007), 19.

[19] M.F. White, Viscous fuid fow, McGraw-Hill (1991).

[20] W.F. Ames, Numerical solutions o partial dierential equations, Second Ed., Academic

Press, New York (1977).


hazem ali attia- effect of hall current on the velocity and temperature distributions of couette...

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