OBERBECK CONVECTION IN CHIRAL FLUID THROUGH A VERTICAL CHANNEL IN THE

PRESENCE OF TRANSVERSE MAGNETIC FIELD

ABSTRACT: Nonlinear oberbeck convection in chiral Boussinesq fluid in a vertical permeable channel is investigated in this paper in the presence of transverse magnetic field, convective current and viscous dissipation. The two boundaries are permeable and assumed to be isothermal kept at different temperatures. The nonlinear-coupled momentum and energy equations are solved analytically using regular perturbation method valid for small values of buoyancy parameter N. The velocity and temperature, mass flow rate and rate of heat transfer for various values of electromagnetic thermal number W1. We found that increase W1 increases the velocity and temperature. The effect of perturbation parameter N is shown to decrease the skin friction heat transfer and the mass flow rate. These results are useful in effective control of heat transfer in Biomedical engineering and industrial problems.

1. INTRODUCTION

Mechanical, electrical and electronic devices give off heat as a byproduct of power consumption when a device is on or being used. Fans andregular fluids are used to improve the cooling process, but they increase the device weight, size and bulk. These days the industries are continue to churn out miniature and portable devices for this purpose. Therefore, manufactures are challenged to find new ways to combat the persistence problem of thermal management. In addition, miniaturization has an increasing importance in many fields in the present day modern technological world. As devices continued to decrease in dimension, there is a growing need to miniature devices with thermal management application. When reduced to submillimeter size, the integrity of the material typically used for cooling structure breaks down, materials like silicon become very brittle and easily shatters, while metallic structure used become bendable and weak. Therefore, there is an urgent need of suitable materials to overcome the above deficiencies.

At present nano and smart materials are becoming popular for use in overcoming the above deficiencies. Apart from using this novel material, we can note that chiral material, solid or fluid, can also be used for the above purpose. The literature reveals that solid chiral material have been used to manufacture devices like antennas, but not much attention has been given to chiral fluids like turpentine, sugarcane solution used in sugar industries, body fluids used in designing artificial body organs, organic substances and so on. By definition chiral material is one which cannot be brought into congruence by its mirror image by translation and rotation, i.e. this fluid has the property of handedness, either left-handed or right- handed, which can be efficiently used for cooling purposes involving effective heat transfer. This heat transfer may be due to density changes with temperature, when temperaturegradient is aligned to the direction of gravity, called natural convection. However, if the temperature gradient is applied perpendicular to gravity, fluid movement arises instantaneously, due to the generation of vortices at the boundaries is called oberbeck convection.

Oberbeck convection in ordinary fluid saturated porous media have been investigated by in the literature [1] and [2]. But to our knowledge much attention has not been given to the study ofoberbeck convection in chiral fluids through a vertical channel in spite of its applications in many practical problems sited above. The study of it, in an incompressible and viscous chiral fluid incorporating chirality in the electromagnetic field in the presence of transverse magnetic field, is the objective of this paper. Toachive this objective the plan of it as follows.

2. MATHEMATICAL FORMULATION

The physical configuration considered in this paper is shown in Fig.1. It consists of an infinite vertical permeable channel bounded on both sides by the rigid isothermal plates kept at different temperatures. The temperature of the hotter and cooler plates respectively T1 and T2. A transverse magnetic field is applied in y-direction. Then the required basic equations for an Oberbeck-Boussinesq chiral fluid in the absence of displacement current:

Fig. 1 Physical configuration of flow

. 0q∇ =r

( )0 0[1 ]T T Tρ ρ β= − − ( ).Dq q

q qDt t

∂= + ∇

∂

r r r r

20

Dqp g q J B

Dt=−∇ + + ∇ + ×

r ur ur urrρ ρ µ

2 2p

DTc K T

Dtq= ∇ + ∇ρ µ

These have to be supplemented with the conservation of charges and Maxwell’s equations for chiral fluids

( ). . 0eeq J

t∂

+ ∇ + ∇ =∂

rrρρ

, , , 0BD E H J Be t∂∇ ⋅ = ∇ × = − ∇ × = ∇ =∂

rr r r r riρ

The constitutive equations for chiral fluids are

,D e E i ? B B µ H i µ ? E= + = −r r r r r r

where, is the chirality coefficient, the magnetic permeability, the dielectric constant, re the density of charges and is for convective current. To simplify the problem we assume fully developed, unidirectional flow in the x-direction, so that the velocity and temperature are independent of x and t and function of y only. Then using these approximations in equation (2.2a) and (2.2b) and making dimensionless using the scales b for length for re, for velocity, for temperature, we get

22 2

12 20 , 0

d u d d uW N

d yd y d y

+ + = + =

θθ

where are the electromagnetic thermal number, and the Buoyancy parameter, together with the boundary conditions

(2.6)and the analytical solution of (2.5a) and (2.5b) are obtained.

0 1, 1 1, 1 1u at y at y at y= = ± = = = − = −θ θ

3. ANALYTICAL SOLUTIONS

Analytical solutions of equation (2.5 a, b) are obtained using perturbation technique. 2

0 1 2

20 1 2

..............

..............

u u Nu N u

N Nθ θ θ θ

= + + +

= + + +with N as the perturbation parameter and satisfying boundary conditions (2.6)

3 2 8 7 6 5 231

0 1 1 1 1 1 1 6 7 8,6 2 6 2 6720 840 360 360 2

wY Y Y Y Y Y Y Yu w u W a W Y a A AY A= − + + + = + + + + − + +

6 5 4 3

0 1 1 1 1 5 6,120 20 12 18 36Y Y Y Y Y

Y W a W A Y A= = − − − − − + +θ θ

4. SKIN FRICTION AND RATE OF HEAT TRANSFER

The velocity (u) and temperature (?) can be obtained as above and using (3.1), up to order N

In many practical applications involving separation of flow it is advantages to know the skin friction and the rate of heat transfer at the boundaries. The skin friction t at the walls is in dimensionless form using the scale for t ’ and using the scales for u and y used earlier, and using (3.1a) we get

g b Tρ β ∆

( ) ( )0 11 1/ /

y ydu dy N du dyτ

=± =±= +

Similarly, the rate of heat transfer between the fluid and the plate is given by making this dimensionless using for heat flux is expressed in terms of the Nusselt number and using (3.1a) weget

( )' / y bq k y = ±= − ∂ ∂θ K T− ∆

( ) ( )0 11 1/ /u y y

N d dy N d dy=± =±

= +θ θ

If mf denotes the mass flow rate per unit channel width in the presence of dissipation, then

The t , Nu and mf are computed for different values of N and W1 the results depicted graphically in Fig 4 to 6.

0

b

fb

m u d yρ−

= ∫

5. RESULT AND DISCUSSIONThe analytical solution obtained for velocity and temperature for different values of W1 and the results are depicted graphically in figure 2, for velocity and figure 3, for temperature. From figure 2, we found that the effect of increase in the values of W1 increases the velocity profile. Form figure 3, we found that an increase in W1increases temperature of the system. The skin friction is drawn in figure 4, for different values of W1 and N. we found that the skin friction at hoter plate increases with W1 and decreases with an increase in N. The rate of heat transfer is predicted in figure 5, for different values of W1 and N and we found that increase in W1increases the heat transfer at hot plate while increase in N reduces the heat transfer. The mass flow rate is depicted in figure 6 for different values of W1 and N and we found the increase in W1 and N decrease the mass flow rate. From the results obtained we conclude that the applied magnetic field in chiral fluid controls the heat transfer, which is very useful in manufacture of artificial organs and mechanical, electrical and electronic devices.

-1.0 -0.5 0.0 0.5 1.00

5

10

15

20

25

30

35

W1=50

W 1=40

W1=25

W1=20

W 1=10

Vel

ocity

u

Y

Fig 2: Velocity profiles of various w1and N=0.1

-1.0 -0.5 0.0 0.5 1.0

0

2

4

6

8

10

12

14

16

18

20

22 W1=50

W1=40

W1=25

W1=20

W1=10

Tem

pera

ture

Y

Fig. 3: Temperature profiles for various w1and N=0.1

0 20 40 60 80 100

0

20

40

60

80

100

120

140

160

180

N=0.1

N=0.05

Skin

Fric

tion

W1

Fig. 4: Skin friction of various values of w1

0 20 40 60 80 100

0

50

100

150

200

250

300

350

N=0.1

N=0.05

Rat

e of

hea

ttran

sfer

W1

Fig. 5:Rate of heat transfer for various w1

0 10 20 30 40 500

50

100

150

200

N=0.1

N=0.05

Mas

s flo

w ra

te

W1

Figure 6: Mass flow Rate for various w1

ACKNOWLEDGEMENTThis work is sponsored by ISRO under research project under Respond No. ISRO/RES/2/335/2007-2008. Also the work of Nagaraju is supported bySiddaganga Institute of Technology (SIT), Tumkur and I am grateful to Prof. M. N. Channabasappa, Director, Prof. Basavarajaiah, Dean(planning) ,and Prof. S. M. Shashidhara ,Principal, SIT, Tumkur for their encouragement.

[1] Rudraiah, N. and Nagaraj, S. T., (1977), “Natural convection through vertical porous stratum”, -Int. J. Engg, Sci, Vol.15, pp 589-600.

[2] Rudraiah, N., Masuoka, T.and Malashetty, M. S.,(1983),”Heat transfer by natural convection in a vertical porous layer”, Appeared in int. communications in Heat and Mass transfer, Vol.10, No.1, pp59-76

[3] Brinkman .H.C (1947)., Appl.Sci.Res.A1 27.[4] Chandrasekhar, S.,(1961), “Hydrodynamic and hydromgnetic

stability”, Oxford University Press, London.[5] Rudraiah, N., Venkatachalappa, M. and Malashetty, M. S., (1982),

“Oberbeck convection through vertical porous stratum”, Proc. Indian Acad. Sci (Math. sci) Vol.91, number 1, pp 17-37.

[6] Rudraiah. N, The Text Book on Modeling of Nano and Smart Materials, published by Paradise, India, 2003.

[7] Rudraiah. N, Field equations in Chiral Material and Eddy current losses in transformers, Proc. INSA conference on Math and its applications to Industry, 2000, 1-21.

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