Effect of Gravity Modulation on the Onset of Rayleigh-Benard Convection in a Weak Electrically Conducting Couple Stress Fluid with Saturated Porous Layer

Effect of Gravity Modulation on the Onset of Rayleigh-Benard Convection in a Weak Electrically Conducting Couple Stress Fluid with Saturated Porous Layer

Sameena Tarannum1

1Department of Professional Studies, Christ University, Bangalore,

India.

S. Pranesp 2Department of Mathematics, Christ University, Bangalore,

India.

Abstract:- The effect of time-periodic body force (TBF, also called gravity modulation) of small amplitude in a weak electrically conducting couple stress fluid with saturated porous layer is investigated by using a linear stability analysis. A regular perturbation method is used 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 TBF. The effect of roles of Couple stress parameter, Hartmann number, Darcy number, Porous parameter and Prandtl number on the onset of convection is studied. It is found that TBF leads to delay in convection. Also the results suggest that instead of taking electrically non-conducting fluid it is better to consider electrically conducting fluid with weak electrical conductivity as this ensures a stable environment in the presence of a magnetic field. The system is most stable with respect to TBF.

Keywords: Gravity modulation, Couple stress fluid, Porous medium, Darcy number, Couple stress parameter, Hartmann number.

1. INTRODUCTION

   The Rayleigh- Bénard problem in its simplest form and one that was the earliest to be investigated is the so called infinite layer case. In such a case, a layer of fluid is constrained between two infinite horizontal surfaces. The surface is heated from below and cooled from above, i.e., the lower surface is at a higher temperature than the upper surface. The heated from below is said to have an adverse temperature gradient because the fluid at the bottom will be lighter than the fluid at the top and this top-heavy arrangement is potentially unstable. When the temperature gradient is below a certain value, the natural tendency of the fluid to move, because of buoyancy, will be inhibited by its own viscosity and thermal diffusivity. Thus the thermal instability will manifest itself only when the adverse temperature gradient exceeds a certain critical value.

   Convection instability in a horizontal porous layer with uniform temperature gradient has been investigated extensively by several authors using Darcy model because of its relevance to variety of situations in science and

   engineering problems. The existence of a fluid layer adjacent to a layer of fluid-saturated porous medium is a common occurrence in both the natural (geophysical) and industrial environments, including such engineering applications as solar collector with a porous absorber, journal bearings, fibrous and granular insulation where the insulation occupies only part of the space separating the heated and cooled walls, etc. The applications also include underground spread of pollutants, convection in earths crust, geothermal energy, extraction of ceramic processing solid, matrix heat exchanger, food storage, porous heat exchangers, etc.

   By a porous medium we mean a material consisting of a solid matrix with interconnected void. We suppose that the solid matrix is either rigid or it undergoes small deformation. The interconnectedness of the void (the pores) allows the flow of one or more fluids through the material. In the simplest situation (single-phase flow) the void is saturated by a single fluid. In two-phase flow a liquid and a gas share the void space.

   Typical studies in this field of flow through and over porous layers consider the porous medium to have constant porosity and permeability. However, two situations arise in this connection:

   1. In the presence of a solid boundary, there is a need to redefine porosity and permeability due to the increase in permeability and porosity near a solid wall, thus giving rise to channelling effects. Channelling effects have an impact on shear stress and heat transfer near the wall.

   2. In flow over porous layers, there exists a permeability discontinuity at the interface between the flow regions. This has an impact on momentum and heat transfer across the interface.

   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. The study of non-Newtonian fluids has attracted much attention, because of their practical applications in industry particularly in extraction of crude oil from petroleum products, applications in biology, physiology and technology. In addition, the effects of heat

   and mass transfer in non-Newtonian fluid also have great importance in engineering applications like the thermal design of industrial equipment dealing with molten plastics, polymeric liquids, foodstuffs, or slurries. Several investigators have extended many of the available convection heat and mass transfer problems to include the non-Newtonian effects. In the category of non-Newtonian fluids, couple stress has distinct features, such as polar effect in addition to possessing large viscosity. The consideration of couple stress in addition to classical Cauchy stress, has led to the recent development of several theories of fluid micro continua. One such couple stress theory of fluids was developed by Stokes (1966) and represents the simplest generalization of the classical theory which allows for polar effects such as the presence of couple stresses and body couples.

   Couple stress in 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. In the classical non-polar theory, moment distribution is not considered and mechanical action is assumed to be equivalent to the force distribution only. One of the applications of couple-stresses in fluid is its use in the study of the mechanism of lubrication of synovial joints, which has become the objective of scientific research. The study of couple stress fluid flows has been the subject of great interest due to its widespread and scientific applications. Few authors P.G. Siddeshwar and S. Pranesh (2004), Veera Krishna et al. (2010) and I.S. Shivakumara (2012) have considered couple stress fluid in investigating Rayleigh- Bénard convection under different situations.

   Couple stress fluid theory developed by Stokes (1966), 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 (1984) in his treatise Theories of fluids with microstructure where in he also presented a list of problems discussed by researchers with reference to this theory.

   Another important class of natural convection problem is concerned with the difficulty in avoiding the

   convection in the earths gravitational field even when the basic temperature gradient is uniform and interfacial instabilities can be ignored. Many theoretical and experimental studies dealing with material processing or physics of fluids under the microgravity condition aboard an orbiting spacecraft have been carried out in recent years. Owing to several unavoidable sources of residual acceleration experienced by a spacecraft, the gravity field in an orbiting laboratory is not constant in a microgravity environment, but is arandomly fluctuating field. This fluctuating gravity is referred to as g-jitter. It is reported in the literature that vibrations can either substantially enhance or retard heat transfer and thus drastically affect the convection.

   The effect of gravity modulation on a convection stable configuration can significantly influence the stability of a system by increasing or decreasing its susceptibility to convection. In general, a distribution of stratifying agency that is convectively stable under constant gravity conditions can be destabilized when a time-dependent component of the gravity field is introduced certain combinations of thermal gradients, physical properties and modulation parameters may lead to parametric resonance and hence, to the stability of the system. P.G. Siddheshwar and S. Pranesh (1999, 2000), Malashetty and Basavaraja (2002), P.G. Siddheshwar and Abraham (2003) and recently by S. Pranesh et al (2014) have studied on the onset of convection in Newtonian and non-Newtonian fluids.

   With the above motivation, the main objective of this paper is to study the effects of fluctuating gravity in a weak electrically conducting couple stress fluid with a saturated porous layer on the onset of Rayleigh- Bénard convection.

2. MATHEMATICAL FORMULATION

   Consider a layer of a weak electrically conducting couple stress fluid with saturated porous layer confined between two infinite horizontal walls distant d apart (fig 1). The uniform magnetic field is directed along the z-axis. A Cartesian co-ordinate system is taken with origin in the lower boundary and z-axis vertically upwards.

   Z = d

   Z HO

   T = TO

   y

   Z = 0

   ( )

   x

   T = TO + T

   Fig 1: Physical configuration of the problem

   The governing equations are:

   Continuity equation:

   . = 0, (1)

   Conservation of Linear Momentum:

   p 1 a + 1 ( . ) = p + p ( )+ µ r2 µe +

   state solution, we get linearized equations governing the infinitesimal perturbations in the form

   . = 0, (14)

   p0 a r = p p [ 1 + cos y ] k + µr2 µe

   O at 2

   k k at

   O k k

   × , (2)

   ( )= O[ 1 + cos y ]k , (3)

   Conservation of energy:

   y aT + ( . )T = x 2T, (4)

   at

   Equation of state:

   p = pO[1 a(T TO)], (5)

   µ2 H2 , (15)

   m O

   m O

   y aTr = T w + x 2T , (16)

   at d

   p = pOaT . (17)

   Operating curl twice to eliminate pressure term on equation

   (15) and using (17), we get

   p0 a (2w )= p a[ 1 + cos y ] 2 T + µ r w

   Constitutive equations:

   at O

   O 1 k

   = × , (6)

   µe 2w µ2 H2 2w . (18)

   k m O

   = µmH, (7)

   where is the velocity, pO is the constant density, p is the pressure, p is the density, is the gravitational force, is the porosity, µ is the couple stress viscosity, µ is the

   The perturbation equations (14) and (16) to (18) are non- dimensionalized by the following definition

   ( , , z)= * , , z , = t , w = sr , T = Tr

   effective viscosity, x is the thermal conductivity, y is the

   d d d

   d2 k

   xd T

   ratio of heat capacity, is the current density, is the magnetic induction vector, k is the permeability of the

   porous media, O is the mean gravity, is the small

   . (19)

   The perturbation equations (14) and (16) to (18) on using (19), we get

   1 a (2w) = R[ 1 + cos ] 2 T + C w 1 2w

   amplitude of gravity modulation, y is the frequency, t is the

   Pr at

   1 Da

   time, T is the temperature, a is the coefficient of thermal expansion, is the magnetic conductivity, µm is the

   magnetic permeability, H is the magnetic field.

   For a weak electrically conducting fluid the Lorentz force × on using the constitutive equations (6) and (7) can be written as

   × = µ2 H2 , (8)

   M22w , (20)

   1

   1

   M a 2 T = w , (21)

   at

   where the asterisks have been dropped for simplicity and the non-dimensional parameters Pr,R, C,Da, M2, M1 and are given as

   2

   where HO

   m

   is the applied

   O

   transverse magnetic field.

   Pr = µeE

   p0x

   R = p0ag0Td3

   (Prandtl number),

   (Rayleigh number),

3. BASIC STATE

   The basic state of the fluid is quiescent and is described by

   C = µ r

   µek

   2

   2

   Da = k d

   µex

   (Couple stress parameter), (Darcy number),

   = (0,0,0), p = p (z), T = T (z), p = p (z). (9)

   µ 2 CJ H2d2

   b b b b

   M2 = m 0

   µe

   (Hartmann number),

   The temperature T , pressure p and density p

   satisfy

   M = yr

   (Porous parameter) and

   b b b

   1

   1

   aT a2T E

   y = x

   at

   a

   , (10)

   2

   2

   az

   = d2Ey

   x

   (Non-dimensional modulation frequency).

   b O

   b O

   = p [ 1 + cos y ]k , (11)

   az

   pb = pO[1 a(Tb TO)]. (12)

4. LINEAR STABILITY ANALYSIS

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

   Equations (20) and (21) are solved subject to the conditions

   w = D2w = T = 0 a z = 0,1. (22)

   Eliminating w from equations (20) and (21), we get an equation for T in the form

   1 a 2 C + 1 2 + M22 M1 a 2 T =

   Pr at Da at

   R[ 1 + cos ] 2 T. (23)

   = b + , p = pb(z)+ p , T = Tb(z)+ T , p = pb(z)+ p . (13)

   The prime indicates that the quantities are infinitesimal perturbations.

   Substituting equation (13) into equations (1) to (5), using equation (8) in equation (2) and using the basic

   1

   In dimensionless form, the velocity boundary conditions for solving equation (23) are obtained from equations (20) to (22) in the form

   T = a2T = a4T = 0 a z = 0,1. (24)

   az2 az4

   4.1 STABILITY ANALYSIS

   We now seek the eigen-function T and eigen-values R of the equation (23) in the form

   (R, T)= (RO, TO)+ (R1, T1)+ 2(R2, T2)+ (25)

   The expansion of (25) is substituted into equation (23) and the coefficients of various powers of are equated on either side of the equation. The resulting system of equation is

   LTO = 0, (26)

   The solution of the homogeneous equation corresponding to equation (32) involves a term proportional to sin rrz. However, addition of such a term to a complete solution of equation (32) merely amounts to a renormalization of T because all the terms proportional to sin rrz can then be grouped to define a new TO with corresponding T1 , T2 , .Hence, we can assume that TO is orthogonal to all other Tns.

   The equation of T2 is

   LT1 = (R1 + ROf )2TO , (27)

   LT = R fa2T R a2T , (35)

   1 2 O 1 2 O

   1 1

   1 1

   LT2 = (R1 + ROf )2T1 + (R2 + R1f )2TO , (28)

   where

   We shall not solve equation (35), but will use this to determine R2. For the existence of a solution of equation (35), it is necessary that the steady part of its right hand side is orthogonal to sin rrz. This gives,

   L = 1 a 2 C + 1 2 + M22 M1 a 2

   Pr at Da at

   1 R fa2T sin rrz dz 1 R a2T

   sin rrz dz = 0 ,

   RO2 . (29)

   O O 1

   O 2 O

   1

   Each of Tn is required to satisfy the boundary condition (24).

   The marginally stable solution of the problem is the general solution of the equation (26), i.e.

   Taking time average, we get,

   O

   O

   R2 = 2RO 1 fT1 sin rrz dz , (36) and finally

   TO = sin rrz ei(l*+m ) . (30) corresponding to the lowest mode of convection with the corresponding eigen value

   k2(17+M2)

   R = R0a2Yl . (37)

   2

   2

   2

   2

   2|L(n)|

5. RESULTS AND DISCUSSION

   In this paper we have made an analytical study of

   where

   RO =

   l , (31)

   2

   2

   a

   the effect of gravity modulation on the onset f convection in a weak electrically conducting couple stress fluid with

   1

   1

   7 = 1 + Ck2 ,

   k2 = rr2 + a2 ,

   saturated porous layer. As per the stated motivation on

   control of convection, the following effects on the classical

   1

   a2 = l2

   + m2 .

   Rayleigh- Bénard problem are considered:

   Equation (27) on using equation (30) becomes

   LT1 = R1a2 sin rrz ROa2f sin rrz . (32)

   If the above equation is to have a solution, the

   1

   1

   right hand side must be orthogonal to the null-space of the operator L. This implies that the time independent part of the RHS of the equation (32) must be orthogonal to sin rrz. Since f varies sinusoidal with time, the only steady term on the RHS of equation (32) is ROa2f sin rrz so that R 0 .

   It follows that all the odd coefficients i.e.

   R1 R 3 …… 0 in equation (25). Using equation (29), we find that

   L[sin rrz ei(l*+m -nt)] = L()sin rrz ei(l*+m -nt) , (33) where

   L() = y1 + i y2 ,

   n2Mlk2

   1. Inhibition of convection by suspended particles, and

   2. Gravity modulation.

These two effects are, respectively, represented by the couple stress parameter C and ( ). Before embarking on a discussion of the results depicted by the Figs. 2-10, we note that oscillatory mode of convection does not come into picture in couple stress fluids. It should be noted that gravity modulation affects the entire bulk of fluid between the boundary plates.

The analysis presented in this paper 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

1

1

1 O

1 O

y1 =

l (k2)2x + R a2 ,

Pr

to the fact that the buoyancy force takes a mean value

k2 2

1

1

2

2

y = l

Pr

+ M1k2x1 ,

leading to equilibrium state of the unmodulated case. In view of this, we choose only moderate value of in our

x1 = 7 + M2,

1

1

7 = 1 + Ck2 ,

1

1

k2 = rr2 + a2 ,

a2 = l2 + m2.

The particular solution of equation (32) is

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.

Figure 2 is the plot of correction Rayleigh number

T1 =

T1 =

1

|L(n)|2

(y1 cos t y2 sin t)( ROa2 sin rrz). (34)

R2C versus frequency of modulation for different values

of couple stress parameter C. In the figure we observe that as C increases, R2C also increases and thus stabilizes the system. C is the indicative of the concentration of the suspended particles. The physical reason for the nature of effect of C on R2C can be given by invoking the Einstein law on viscosity of suspensions. Einstein relation for suspended particles µ = µO(1 + 2.5a¢), where µ and µO are the viscosities of suspension(i.e. clean fluid + suspended particles) and clean fluid respectively, a is the shape factor and ¢ is the volume fraction of the suspended particles. The shape factor a is 1 for spherical particles and greater than 1 for other shapes such as ellipsoids, rods, etc.

In order to understand an increase in the viscosity of a suspension with concentration, it is helpful to realize that the viscosity of any fluid is associated with dissipation of mechanical energy into heat within the fluid. The viscosity of a suspension also depends on various other factors such as viscosity of the carrier liquid (suspending medium), the interaction between the particles, on whether the particles are rigid or deformable and on the temperature and electrical conductivity. We keep the analysis to the bare minimum and do not venture into delineating the above individual contributions to viscosity. We just assume that the viscosity of a suspension is higher than that of the carrier liquid. The assumed viscosity concentration relation is valid for low concentrations. In the case of higher concentrations one will have to add quadratic term in ¢ to the Einstein relation. In the light of the above discussion, we consider values of Prandtl number of fluids with suspended particles to be higher than those of the fluids without suspended particle.

Figure 3 is the plot of R2C versus for different values of Hartmann number M2. We observe from the figure that increase in M2, increases R2C . In making conclusions from the figure, we should also note that C and

M2 influence R2C . We find that R2C increases with increase in C and M2. When the magnetic field strength

increases, it induces viscosity into the fluid, and the magnetic lines are distorted by convection. Then these magnetic lines hinder the growth of disturbances, leading to the delay in the onset of instability.

Figure 4 is the plot of R2C versus for different values of Darcy number Da. We observe that increase in Da, decreases R2C and thus destabilize the system.

Figure 5 is the plot of R2C versus for different values of Porous parameter M1. In the figure we observe that as M1 increases, R2C decreases and thus destabilize the system.

Figure 6 is the plot of R2C versus for different values of Prandtl number Pr. We observe that increase in Pr, increases R2C . We can infer from this that the effect of increase in the concentration of the suspended particles is to stabilize the system. It is appropriate to note that Pr does not affect the RO part of R, it effects only R2 .

Figures 7, 8, 9 and 10 are the plots of R2C verses for different values of C, Da, M1 and Pr respectively for M2 = 0. The effects of these parameters are quantitatively similar to that of the results for M2 0.

Comparing our results for C = 0 with the results of P.G. Siddheshwar and S. Pranesh (2000) for N1 = 0, we found that the critical Rayleigh number in the case of finitely conducting fluid is less than that of weak electrically conducting fluid. This is true for the fluids with suspended particles. These observed results have been analysis bearing in mind that in the case of weak

electrically conducting fluids, the resistance to flow is higher compared to the finitely electrically conducting case.

10 0

M2 = 10 , M1 = 0.1 , Da = 0.05 , Pr = 5

80

60

40

C = 0.5

C = 0.3

C = 0.1

20

0

0 20 40 60 80 10 120 140 160 180 20

R2C

Fig 2: The plot of correction Rayleigh number R2C vs frequency of modulation for different values of Couple stress parameter C.

C = 0.1 , M1 = 0.1 , Da = 0.05 , Pr = 5

10 0

80

M2 = 50

60 M2 = 25

M2 = 10

40

M2 = 0

20

0

0 50 10 150 20 250 30 350

R2C

Fig 3: The plot of correction Rayleigh number R2C vs frequency of modulation for different values of Hartmann number M2.

C = 0.1 , M2 = 10 , Pr = 5 , M1 = 0.1

10 0

80

Da = 0.01

60 Da = 0.05

Da = 0.1

40

20

0

0 50 10 150 20 250 30 350 40 450

R2C

Fig 4: The plot of correction Rayleigh number R2C vs frequency of modulation for different values of Darcy number Da.

C = 0.1 , M2 = 10 , Da = 0.05 , Pr = 5

10 0

80 M1 = 0.1

60 M1 = 0.3

40 M1 = 0.5

20

0

0 20 40 60

R2C

10 120 140 160

80

80

Fig 5: The plot of correction Rayleigh number R2C vs frequency of modulation for different values of Porous parameter M1.

C = 0.1 , M2 = 10 , Da = 0.05 , M1 = 0.1

10 0

80

Pr = 5

60 Pr = 10

Pr = 5

40

60 80

60 80

20

0

0 20 40

R2C

10 120 140 160

Fig 6: The plot of correction Rayleigh number R2C vs frequency of modulation for different values of Prandtl number Pr.

10 0

M2 = 0 , M1 = 0.1 , Da = 0.05 , Pr = 5

80

60 C = 0.5

40 C = 0.3

C = 0.1

20

0

0 20 40 60 80 10 120 140 160

R2C

Fig 7: The plot of correction Rayleigh number R2C vs frequency of modulation for different values of Couple stress parameter C.

10 0

C = 0.1 , M2 = 0 , Pr = 5 , M1 = 0.1

80

60

40

Da = 0.01

Da = 0.05

Da = 0.1

20

0

0 50 10 150 20 250 30 350 40 450

R2C

Fig 8: The plot of correction Rayleigh number R2C vs frequency of modulation for different values of Darcy number Da.

10 0

C = 0.1 , M2 = 0 , Da = 0.05 , Pr = 5

80 M1 = 0.1

60

40

M1 = 0.3

M1 = 0.5

20

0

0 20 40 60 80 10

R2C

Fig 9: The plot of correction Rayleigh number R2C vs frequency of modulation for different values of Porous parameter M1.

C = 0.1 , M2 = 0 , Da = 0.05 , M1 = 0.1

10 0

80 Pr = 15

60 Pr = 10

Pr = 5

40

20

0

0 20 40 60 80 10 120

R2C

Fig 10: The plot of correction Rayleigh number R2C vs frequency of modulation for different values of Prandtl number Pr.

CONCLUSION

From the study we conclude that the gravity modulation or g-jitters leads to delay in convection. Also the results suggest that instead of taking electrically non-conducting fluid it is better to consider electrically conducting fluid with weak electrical conductivity as this ensures a stable environment in the presence of a magnetic field.

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