An Analytical Study of Linear Gravity Modulated Rayleigh-Benard Convection in a Rotating Boussinesq Stokes Suspension with Weak Electric Field

An Analytical Study of Linear Gravity Modulated Rayleigh-Benard Convection in a Rotating Boussinesq Stokes Suspension with Weak Electric Field

Anjana K

Dept. of Mathematics,

Mount Carmel College (Autonomous), Bangalore-560052

Abstract – The effect of Coriolis force and gravity modulation of small amplitude on a weak electrically conducting Boussinesq- Stokes suspension is investigated by using regular perturbation method to arrive at an expression for the correction Rayleigh number. The Venezian approach is adopted in arriving at the critical Rayleigh and wave number for small amplitudes of gravity modulation. The effect of role of Couple stress parameter, Taylor number and Hartmann number on the onset of convection is studied. The system is most stable with respect to gravity modulation.

Keywords: Rayleigh-Benard Convection, Boussineq-Stokes suspension, Gravity modulation, Coriolis Force.

INTRODUCTION

The last century saw a thorough understanding of the principles of fluid mechanics and knowledge of how to apply them to many practical problems. Aeronautical, biomedical, civil, marine and mechanical engineers as well as astrophysicists, geophysicists, space researchers, meteorologists, physical oceanographers, physicists and mathematicians have used this knowledge to tackle a multitude of complex flow phenomena. An important class of fluid differs from that of Newtonian fluids, in that the relationship between the shear stress and flow field is more complicated. Such fluids are non-Newtonian.

Couple stress is the consequence of assuming that mechanical action of one part of a body on another across a surface is equivalent to a force and moment distribution. Couple stress fluid theory developed by Stokes[1], is one among the polar fluid theories which considers couple stresses in addition to the classical Cauchy stress. It is the simplest generalization of the classical theory of fluids which allows for polar effects such as the presence of couple stresses and body couples. This fluid theory is discussed in detail by Stokes[2] in his treatise Theories of fluids with microstructure wherein he also presented a list of problems discussed by researchers with reference to this theory. The first paper on Rayleigh- Benard situation in Boussinesq-Stokes Suspension is by Siddheshwar and Pranesh[3]. They investigated the effect of RaleighBenard situation in BoussinesqStokes suspensions using both linear and non-linear stability

analyses. Some of the problems of recent interest can also be seen in Neduvinamani et al[4].

Convection in porous media has recently been studied since it has many practical, especially biological significance. Bhadauria B. S, Sherani and Md. Aalam[5] investigated the effect of gravity modulation on the onset of Darcy convection in a rotating porous medium. It was concluded that gravity modulation delays the onset of convection as does rotation. T. Sivakumar and S. Saravanan[6] investigated the effect of Gravity Modulation on the Onset of Convection in a Horizontal Anisotropic Porous Layer. A linear stability theory was used to investigate the effect of gravity modulation on the onset of convection in a homogeneous anisotropic porous layer heated from above. The Brinkman model with anisotropic permeability was considered. Free-convection flow past an infinite vertical porous plate with periodic suction and gravity modulation was investigated by S. Baljinder[7]. The double-diffusive convection in a horizontal layer of nanofluid under rotation in a porous medium was studied by Rana et al.[8] using the Darcy model. For the case of stationary convection, it was observed that rotation and solute gradient have a stabilizing effect on the system.

Buoyancy-driven convection in microgravity resulting from gravity uctuations has gained considerable attention owing to the possibility of conducting research in the low gravity environment of space, and because of interest in the fundamental eects of gravity modulation on uid systems. The effect of such forces on fluid motion is known as Gravity Modulation or g-jitter induced flow and it comes from crew motions, mechanical vibrations (pump, motors, and excitations of natural frequencies of space craft structure), atmospheric drag, solar drag, earth's gravity gradient and other sources.

An extensive overview of the above and other sources of unsteady gravitational accelerations, derived from estimates based on measurements in the space lab engineering model, have been documented by Gresho and Sani[9]. The uctuating accelerations act on density gradients in the uid caused by heat and/or mass transfer between the uid and boundaries, producing convective

motions. These motions may increase heat transfer signicantly. Previous studies of buoyancy-induced uid motion and heat transfer resulting from gravity modulation under microgravity have focused on a few basic uid systems, as well as some specic applications. If an imposed modulation can destabilize another stable state, then there can be a major enhancement of heat, mass and momentum transport. Consequently, this has led to research into the possibility of processing materials in

space where the low-level background gravitational acceleration can eliminate buoyancy driven convection. However, research has shown that time-dependent accelerations or g-jitter of substantial amplitude resulting from orbital maneuvers and inherent mechanical vibrations may alone induce buoyant convection. It is also of interest to understand how vibration might be used to control convective instabilities.

Equation of continuity:

.q 0

MATHEMATICAL FORMULATION

(1)

Equation of conservation of linear momentum:

1 q 1 (q.)q 2 ( q) p g ' 2 q eff

q 2 H 2q

t

t

0 2

1 1

1

m 0

(2)

Equation of conservation of energy:

T (q.)T 2T

t

Equation of state:

0 1 (T T0 )

(3)

(4)

where, q is the velocity, p pressure, H magnetic field, B magnetic induction, T temperature, density , g acceleration due to gravity, thermal conductivity, coefficient of thermal expansion, 0 reference density, T0 reference temperature, m magnetic permeability, dynamic viscosity, couple stress viscosity, µeff effective viscosity, electrical conductivity, porosity and t time. The above equations are solved for free-free isothermal boundary conditions. The gravity modulation is considered as () = 0(1 + ), where is the small amplitude and is the frequency.

Initially we assume that the fluid is at rest and is described by

b

b

q q (0,0,0), (z),T T (z), p p (z),

b b b b

0 k

(5)

When these quantities are substituted in the governing equations we get the following set of equations:

dpb

dz b

2T

g(1 cos t) 0

(6)

b 0

z 2

b[1 (Tb T0 )]

Solving eqn (3) using the boundary condition Tb T0 at z=0 and Tb T1

at z=d we get,

(7)

(8)

T T z T

where,

T (T0 T1 )

(9)

b d 0

Linear Stability Analysis

Let the basic state be disturbed by an infinitesimal thermal perturbation. We now have,

b

b

b

b

q q q', ',T T T ', p p p', '

b b b

(10)

<>The prime indicates that the quantities are infinitesimal perturbations.

Substituting these into governing equations and using the basic state equations, we get linearized equations governing the infinitesimal perturbations in the form:

1 (2 w') 2

v

u 2

0

t 0 z x y) 0g(1 cos t)1 T

' 4 w' eff 2 w' 2 H 22 w'

m 0

(11)

1 V 2

w ' 2V eff V 2 H 2V

0 t

0 z

m 0

(12)

T ' (q.)T

2T '

t b

(13)

The perturbation equations are non-dimensionalized using the following definitions:

d

d

d

d

d

d

(x* , y* , z* ) x , y , z ,t * t ,

w* w , T * T ,

* d,

V * V

d 2 /

/ d T

/ d 2

(14)

Eqn (14) is used to get a set of dimensionless equations given below:

1

1

M / t 2 T w ,

(15)

1 (2 w) 1/ 2 V

Ta

Ta

R(1

2

C 4

1 2

22

pr t

z cos

t) 1 T

a

w w' M w' Da

(16)

D

D

1 V Ta1/ 2 w

C 2V

1 V M 2V

pr t

z Da Da

2

d 2 2

(17)

gd 3T

Where the non-dimensional parameters

Pr ,

0 0

0 0

Ta ,

R 0 C

0

, d

2

, M

2 H 2 d 2

m 0

m 0

, M ,

1

Da d 2

are Prandtl number, Taylor number, Darcy-Rayleigh number, Coupling stress parameter, Hartmann number, Darcy number respectively.

On simplification,

1 2 C 4 1 2

2 2 1 C 2 1 2

2 2

a

a

pr t D

M

Da

pr t D

• M Da

  TaD

  M1 t

  T

  a

  a

  1 C 2 1 2 2

  a

  a

  pr t D

• M Da

R(1 cos t)1 T

(18)

Each of Tn is required to satisfy the boundary conditions. The marginally stable solution of the problem is the general solution

of the above equation i.e. T0 sin(z)e ,

i(lxmy)

corresponding to the lowest mode of convection with the corresponding eigen value.

(k 2 X 2 2Ta)k 2

R 1

1

1

0 a 2 X

(19)

Where

X C k 2

1 D

1 D

1 M 2

Da

(20)

The consequent equations when solved gives the correction Rayleigh number, R2c

a 2 R 2

2Y

2Y

Y X

n 1

n 1

R 0 1 1 2 n X

Y Pr

(21)

2c 2 Pr L(, n2

X1 Pr X1 X1

L(, n) Y1 iY2

(22)

22 M k 2 X

2 4 2k 4 2 2

Y1 1 n n Xn kn n

Takn aR0 Xn

(23)

pr pr 2

3 M k 2 2k 4 X

a 2 R

Y M k 2 X 2 1 n n n 2TaM

0

(24)

2 1 n n

pr 2 pr

1 pr

RESULT AND DISCUSSIONS

The value of R obtained by this procedure is the eigenvalue corresponding to the eigen function W that, though oscillating, remains bounded in time. Since R is a function of the horizontal wavenumber a and the amplitude of perturbation, we have (, ) = 0() + 22() . It was shown by Venezian[10]

that the critical value is determined by (2), by evaluating R0 and R 2 at a a0 . It is only when one

wishes to evaluate R 4

that

a2 must be taken into account where

a a2 minimizes

R 2 .To evaluate the

critical value of R 2

is minimum.

one has to substitute a a0 in R2, where a0 is the value at which R0 given by equation

500 2 = 10, = 100, = 10, = 0.5

400

300

200

C = 0.1, 0.3, 0.5

100

0

0.0 -0.5 -1.0 -1.5 -2.0

R2c

Fig 1: Plot of correction Rayleigh number R2c versus for different values of C

500

500

= 0.1, = 100, = 10, = 0.5

400

300

200

2 = 50,25,10

100

0

0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2 -1.4

2

Fig 2: Plot of correction Rayleigh number R2c versus for different values of M2

500

500

= 0.1, 2 = 10, = 10,

= 0.5

400

300

200

= 500, 300, 100

100

0

0.0 -0.2 -0.4 -0.6 -0.8 -1.0

2

Fig 3: Plot of correction Rayleigh number R2c versus for different values of Ta

500

500

C = 0.1, Ta = 0, Pr = 10 , 2 = 10

400

300

a

a

200 D 0.5,1,2

100

0

0 -5 -10 -15 -20 -25

2

Fig 4: Plot of correction Rayleigh number R2c versus for different values of Da

We make an analytical study of the effect of Coriolis force and gravity modulation on the onset of convection in a weak electrically conducting couple stress fluid in a porous medium. Double diffusive convection in porous medium is studied under Coriolis force, gravity modulation and the inhibition of convection by suspended particles. It should be noted that gravity modulation affects the entire bulk of fluid between the boundary plates.

The analysis presented in this dissertation is based on the assumption that the amplitude of the gravity modulating is small. The validity of the results obtained

here depends on the value of the modulating frequency . When 1, the period of modulation is large. The gravity modulation affects the entire volume of the fluid, resulting in the growth of the disturbance. On the other hand, the effect of modulation disappears for the large frequency. This is due to the fact that the buoyancy force

takes a mean value leading to equilibrium state of the unmodulated case. In view of this, we choose only moderate value of in our study. It must be noted here

that because of the presence of suspended particles in the fluid and according to Einsteins relation for viscosity, the value of Prandtl number is taken higher than those of clean fluid.

CONCLUSION:

Figure 1 is the plot of correction Rayleigh number R2c versus for different values of C. We see that as C increases the value of R2c becomes more and more negative. C is the indicative of the concentration of the

suspended particles. Therefore, the couple stress parameter

stabilizes the system.

Figure 2 is the plot of correction Rayleigh number R2c versus for different values of M2. From the figure

we observe that with increase in M2, R2c becomes more negative. Magnetic field induces viscosity into the fluid

and the magnetic lines are distorted. These magnetic lines hinder the growth of disturbances.

Figure 3 is the plot of correction Rayleigh number

R2c versus

for different values of Ta. We note that as

R2c versus

for different values of Ta. We note that as

ACKNOWLEDGEMENT

The author expresses gratitude to the management of the institution for the encouragement received for this work and would like to express heartfelt thanks to the reviewers whose comments were critical. Special gratitude is expressed towards Dr. S. Pranesh who helped from the inception to the completion of this paper and also to Dr.

K. Hemalatha whose constant support led to this work.

REFERENCES

1. Stokes.V. K., Couple stress in fluids., vol. 9. Phys. Fluid., 1966, pp. 1079-1715./p>

2. Stokes. V. K., Theories of fluids with microstructure., Publ: Springer, New York. 1984.

3. Siddheshwar P.G. and Pranesh S., An analytical study of linear and non-linear convection in Boussinesq-Stokes suspensions. vol. 39. Int. J. Non-Linear Mec. 2004. , pp. 165-172.

4. Naduvinamani N. B. , Hiremath P. S. and Gurubasavaraj G., Effects of surface roughness on the Couple stress squeeze film between a sphere and a flat plate. vol. 38. Tribology International. 2005, pp. 451- 458.

5. Bhadauria B. S., Onset of magneto-convection in an electrically conducting-fluid-saturated porous medium under gravity modulation, vol.3. Bull. Pure Appl. Math. 2009, pp. 142-57.

6. Sivakumar T. and Saravana S., Effect of Gravity Modulation on the

   Onset of Convection in a Horizontal Anisotropic Porous Layer. AIP

   Onset of Convection in a Horizontal Anisotropic Porous Layer. AIP

   R2c versus

   for different values of Ta. We note that as

   R2c versus

   for different values of Ta. We note that as

   Ta increases, R2c increases, that is it becomes less negative. This implies that rotation causes convection to delay.

   Figures 4 is a plot of correction Rayleigh number R2c versus for different values of Da. From the graph it is clear that the Darcy number causes a delay in

   convection thus stabilizing the system.

   Onset of Convection in a Horizontal Anisotropic Porous Layer. AIP

   Onset of Convection in a Horizontal Anisotropic Porous Layer. AIP

   Conf. Proc. 1146. 2009, pp. 472-478.

7. Baljinder S. and Rajvanshi S. C., Three-dimensional MHD free- convection flow past an infinite vertical porous plate with periodic suction and gravity modulation., vol. 55. Acta Tech., 2010, pp. 237 258.

8. Rana G. C, Thakur R.C and Kango S. K., On The Onset Of Double- Diffusive Convection In A Layer Of Nanofluid Under Rotation Saturating A Porous Medium., vol.17. J. Por. Media. 2014, pp. 657- 667

9. Gresho P. M. and Sani R. L., The effects of gravity modulation on the stability of a heated fluid layer, vol. 40. J. Fluid Mech., 1970, pp. 783-806.

10. Venezian G., Effect of modulation on the onset of thermal convection, vo. 35. J. Fluid Mech., 1969, pp. 243-254.