 Open Access
 Total Downloads : 153
 Authors : Anjana K. , R. K. Vanishree
 Paper ID : IJERTV5IS060487
 Volume & Issue : Volume 05, Issue 06 (June 2016)
 DOI : http://dx.doi.org/10.17577/IJERTV5IS060487
 Published (First Online): 17062016
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Study of Effects of Temperature Modulation on Double Diffusive Convection in OldroydB Liquids
K. Anjana1 1Dept. of Mathematics, Mount Carmel College,
Bengaluru
Abstract – The effect of modulation of temperature has been studied on double diffusive convection in viscoelastic Oldroyd B liquid. A linear analysis has been done. The thermal Rayleigh number for the modulation problem is obtained using regular perturbation technique. It is observed that the effect of the stress relaxation parameter is to destabilize the system whereas strain retardation parameter and the Lewis number stabilize the system. Modulation is shown to give rise to supercritical motion.
Key words: RayleighBÃ©nard convection, Double diffusion, OldroydB liquids, Temperature modulation.

INTRODUCTION
Convection in NonNewtonian fluids varies significantly from that in Newtonian fluids, one of which is viscoelastic fluids which are made up of two components viscous component and elastic component. Viscoelastic fluids are the type of fluids in which the stressstrain relationship depends on time; polymers, human tissues and metals at high temperatures being classic examples. It has come to light that viscoelastic liquids are a working media in many problems in chemical and nuclear industries, geophysics, and engineering in biological systems etc. There have been quite a few works on convection and its onset in viscoelastic liquids (see Sekhar and Jayalatha [1] and references therein). Stationary and oscillatory instabilities for the OldroydB viscoelastic model was studied by Li and Khayat [2,3]which gave a much needed information about the formation of pattern in viscoelastic fluid convection. Oscillatory convection in these fluids was also studied by Green. It was found that in a thin layer of the fluid when heated from below a large restoring force sets up an oscillating convective motion. Siddheshwar and Krishna
[4] investigated the linear stability analysis of the RayleighBenard convection problem in a Boussinesquian, viscoelastic fluid. They found that the strain retardation time should be less than the stress relaxation time for convection to set in as oscillatory motions in highporosity media. Recently, nonlinear stability of thermal convection under gjitter in a layer of viscoelastic liquid was studied by Siddheshwar [5]. Sharma [6] found that in OldroydBR. K. Vanishree2
2Dept. of Mathematics, Maharani Science College for Women,
Bengaluru
liquids rotation has a destabilizing as well as a stabilizing effect in contrast to a Maxwell fluid. However, there are fewer studies on nonlinear convection as compared to that of linear studies in the case of these liquids.
Buoyancy driven convection occurs due to two components with different diffusivitiestemperature and solute. This is popularly known as double diffusive convection (Mojtabi and CharrierMojtabi [7]). The study of double diffusive convection has emerged due to Stern [8]. Prior to this, Stommelet al. [9] noted that there was a significant potential energy available in the decrease of salinity with depth found in much of the tropical ocean. Modulation of thermal convective instability was initially studied by Rosenblat and Tanaka [10]. If a system has two diffusing components, instabilities occur depending on whether solute components is stabilizing or destabilizing. Further advancement took place when Siddheshwar and Pranesh[11] investigated the effect of temperature/gravity modulation on the onset of magnetoconvection in weak electrically conducting fluids with internal angular momentum. Siddheshwar and Sri Krishna [12] investigated Rayleigh BÃ©nard convection in a viscoelastic fluid filled high porosity medium with non uniform basic temperature gradient. Bhadauria[13] studied temperature modulation of double diffusive convection in a horizontal fluid layer. A study on double diffusive magneto convection in viscoelastic fluids was done by Narayana et. al. [14]. Recently, Milo et. al.[15] made an analysis of Rayleigh BÃ©nard convective instability in the presence of spatial temperature modulation. It was found that modulation delays the onset of convection. Pranesh and Sangeetha [16] investigated the effect of imposed timeperiodic boundary temperature of small amplitude on electro convection under AC electric field in dielectric couple stress liquids.
In the present paper the effects of temperature modulation on double diffusive convection in a viscoelastic fluid satisfying the OldroydB constitutive equation is studied.
Nomenclature
d thickness of the liquid
k dimensionless wave number Pr Prandtl number
q velocity
Ra thermal Rayleigh number
Rs solutal Rayleigh number t time
T temperature
To constant temperature of the upper boundary
TR reference temperature Le Lewis number
Nomenclature
d thickness of the liquid
k dimensionless wave number Pr Prandtl number
q velocity
Ra thermal Rayleigh number
Rs solutal Rayleigh number t time
T temperature
To constant temperature of the upper boundary
TR reference temperature Le Lewis number
Greek symbols
Greek symbols
s
1
s
1
thermal expansion coefficient amplitude of modulation thermal diffusivity
solutal diffusivity
stress relaxation coefficient strain retardation coefficient
thermal expansion coefficient amplitude of modulation thermal diffusivity
solutal diffusivity
stress relaxation coefficient strain retardation coefficient
2
2
0
0
elastic ratio( / 1 )
elastic ratio( / 1 )
2
2
viscosity
frequency of modulation
viscosity
frequency of modulation
density
density
reference density
reference density

MATHEMATICAL FORMULATION
Consider a layer of a viscoelastic fluid, namely OldroydB liquid, confined between two finite horizontal walls distance d apart. The parallel plates are maintained at different temperatures and concentrations so that there is density gradient. A Cartesian coordinate system is taken with origin in the lower boundary (fig 1).
The governing equations are: Continuity Equation:
.q 0
Conservation of momentum:
Fig 1: Physical configuration of the problem
(1)
q p g . '
0 t
(2)
Rheological Equation:
'
q q tr
1 1 t
1 2 t
(3)
Conservation of Energy:
T (q.)T 2T
t
Conservation of Species:
(4)
s
s
S (q.)S
t
Equation of State:
2 S
(5)
0 1 (T T0 ) '(S S0 )
The surface temperatures are:
(6)
T TR (1/ 2)T (1 cos t) at z 0,
T TR (1/ 2)T (1 cos(t ))
at z d
(7)

BASIC STATE
In the basic state the fluid is at rest. Therefore, the parameters take the following form:
q qb 0, p pb (z), b (z), S Sb (z),T Tb (z,t)
The temperature Tb, pressure pb and density b satisfy
(8)
dpb
dz
T
g 0
b
b
2T
(9)
b b
t z 2
(10)
0 (1 (T T0 )),
The rheological equation takes the form
(11)
q
2
1 1 t 0 t p g 1 2 t q
(12)
The solution of eqn.(10) that satisfies the thermal boundary conditions is
z z
T (z, t) T (z) Rea()e d
s
s
a()e d
eit
b
where
1
(13)
d 2 2
(1 i)
2
(14)
T ei e
a()
2 e e

STABILITY ANALYSIS
(15)
We now superimpose the infinitesimal perturbations on the quiescent basic state to study the stability of the system. Let the basic state be perturbed by an infinitesimal perturbation as follows, where the primes denote perturbations.
b
b
q q', p p
p', b
',T Tb
T ', S Sb

S'
(16)
Substituting these equations in the governing equations and using the basic state solutions, we obtain the following equations for the perturbations
T ' w' T ' w' Tb
2T '
t z z
(17)
S' w' S' w' Sb
2 S'
t z
z s
(18)
(2 w') 2
2
4
1 1 t 0 t 0 g1T ' ' 0 g1 S' 1 2 t w'
(19)
We introduce the stream function such that
u , w and all the terms are independent of y. Using the
following dimensionless new variables
z x
w*
w' ,t *
t ,T *
T ' , S * S , * d,(x* , y* , z* ) x , y , z
/ d
d 2 /
T S
d
d
d
d
d
d
, (20)
the resulting nondimensional equations for the problem are:
1 (1/ Pr)2
1
4
1 t
2
t
2
2 t x
(21)
1 1 t Ra1 T Rs1 S ,
2
(q.)T f (z, t) 1 ,
t
T
x
1 2 S (q.)S ,
(22)
t Le x
(23)
the dimensionless parameters that appear in these equations are the stress relation parameter, the strain retardation parameter, Prandtl number, Thermal Rayleigh number and Solutal Rayleigh number which are respectively given in eqn. (27).
1
2
0 gTd 3
' 0 gSd 3
1 d 2 , 2
, Le ,
d 2 s
Pr , Ra
0
, Rs
(24)


LINEAR STABILITY ANALYSIS
In this section, we discuss the linear stability analysis considering marginal and overstable states. To this end we neglect the Jacobians in eqns. (21) to (23). The linear version of these equations are:
1 (1/ Pr)2
1
4
1 t
t
2 t x
2 2
(25)
1 1 t Ra1 T Rs1 S ,
t
2
T
T
f (z, t) 1 ,
x
(26)
(1/ Le)2 S ,
t
x
(27)
Eliminating T and S between eqns. (25) – (27), an equation for is obtained in the form
2
1 2 1
1
1
2 2
t
t
t
Le
Le
1 t pr t
2 t
1 2
2
2
2 2
1 1
Ra
2

f1Ra
1 1
2
2
Rs 2
t t Le
x
x
t t
x
(28)


PERTURBATION PROCEDURE
A regular perturbation technique is followed and the following expansions are used
2
2
2
2
0 1 2 ….
R R0
R1
2 R
….
(29)
Equation (29) is substituted into eqn. (28) and the coefficients of various powers of are equated to obtain the following system of equations
L 0 0
(30)
1 2
2
0
0
L 1 1
1 t t Le
1
Ra1
Ra 0 f1
2
x 2
2
2
2
(31)
L 1
2 Ra
1 Ra
f 1 Ra
0 Ra f
0
2 1 t t Le
1
x 2
0 1 x 2
2 x 2
1 1 x 2
(32)
where,
L 2 1 2 1
1 1
2 2
t Le
t
t
1
1 t pr t
1
2
2 t
2
2 2
1 t t Le
Ra0
x 2 1
1 t t
Rs
x 2
(33)
Each n is required to satisfy the boundary condition
2
4
0 at z = 0, 1.

Solution to the zeroth order problem
The zerothorder problem is equivalent to the double diffusive problem of OldroydB liquid in theabsence of temperature modulation.The marginally stable solution of the problem is the general solution of the equation. Eq. (30), obtained at o(0) is the one used in the study of convection in a layer of OldroydB liquid subjected to uniform temperature modulation.
The marginal stable solutions are
0 = sin(x)sin(z) (34)
2
2
with the corresponding eigenvalue Ra0 given by
6
Ra0 a 2

LeRs R2
(35)


Solution to the first order problem
Substituting eqn. (34) in eqn. (31), we get
L 1

a 2 k 2
Le
Ra1 0
a 2 k 2 f Le
Ra0 0
(36)
where,
L(, n) Y1 iY2
(37)

k 8

k 6 2
4 k 2
2 k 4 pr
2 k 4
Y n n 1 k 4 2 1 n n 2 k 6 n
1 Le pr
2 k 6
n pr
0
0
k 2
2 n Lepr
(38)
2 n a 2 Ra
n 2 a 2 Rsk 2 2
Le
k 6
k 8
Le 1
3 k 2
n 1
3 k 4
k 6
(39)
Y n 2 n n 3 k 2 k 6 1 n n
2 Lepr Le pr 2 n n pr Le
3 k 4 k 2
0
0
1 n a 2 Ra 1 n a 2 Rs k 2
Lepr
Le 1 n
The solvability condition requires that the timeindependent part of the right hand side of eqn. (36) should be orthogonal
to the null operato L and this implies that
a 2 k 2 f
Ra1 Ra3 Ra5…. 0. Therefore, eqn. (36) becomes
L 1
Le Ra0 0
(40)
We use eqn.(32) to determine Ra2, the first nonzero correction to R0. The steady part of eqn.(32) is orthogonal to sinz.
Taking time average of eqn.(40) and using eqn.(32) we get the following expression for the correction Rayleigh number.
R 2 4 4 2 R 2 2 2  B () 2
0 0
0 0
n
n
R2  L(, n)   L (, n) 
*
2
2
2Le
2Le
n1  L(, n) 
(41)
Where,
Bn () 2 2 2 2 2 2
Bn () 2 2 2 2 2 2
2n 2 2 e e (1)n e i e i
e e (n 1) (n 1)
(42)
Following are the thermal modulations considered: Case A: In Phase modulation
When the oscillating temperature field is symmetric so that the wall temperatures are modulated in phase (with = 0). In this case n is even or odd.
Case B: Outof Phase modulation
When the wall temperature field is asymmetric, it corresponds to outofphase modulation (with = ). In this case n is odd.
Case C: Only lower wall modulation
When the temperature of only the lower wall is modulated, the upper plate being held at a constant temperature, it corresponds to lower wall modulation (with = i).
In this case n takes both even and odd values.
9. RESULTS AND DISCUSSIONS
We now comprehend the effect of temperature modulation on the onset of double diffusive convection in a horizontal layer of an OldroydB liquid for the relevant parameters. The linear stability problem is solved based on the method proposed
by Venezian [17].The parameters of the system are Le, Ra, Rs, Pr,
1 , 2 , ,
which influence the convection. The first six
parameters are related to the fluid layer and the remaining are the external measures of controlling the convection.
1 0.1,2 0.1, pr 10
Le = 100, 200, 300
1 0.1,2 0.1, pr 10
Le = 100, 200, 300
800 800
Le 100, 2 0.1, pr 10
600
400
600
400
1 0.8, 0.5, 0.1
200 200
0 0
40 60
40 60
R2c
R2c
0 20 80 100
R2c
5 0 5 10 15 20 25 30 35 40
R2c
1
1
Fig 2: Case A: Graph of R2c versus for different values of Le Fig 3: Case A: Graph of R2c versus for different values of
800 1 0.1, Le 100, pr 10
800
600
2 0.05, Le 100, pr 10
600
400
200
2 0.01,
0.05,
0.1
400
200
1 1,0.5,0.8
0 0
20 0 20 40 60 80 100 120 140 160 180 200
R2c
2
2
Fig 4: Case A: Graph of R2c versus for different values of
20 0 20 40 60 80 100 120 140 160 180 200
R2c
1
1
Fig 5: Case B: Graph of R2c versus for different values of
800
1
0.1, Le 100, pr 10
800
1 0.1,2
0.1, pr 10
600
400
200
2 0.1,
0.05,
0.01
600
400
200
Le = 300, 200, 100
0 0
50 0 50 100 150 200 250 300 350 400 0 50 100 150 200
R2c
2
2
Fig 6: Case B: Graph of R2c versus for different values of
R2c
Fig 7: Case B: Graph of R2c versus for different values of Le
2 0.05, Le 100, pr 10
800
1 0.1, Le 100, pr 10
600
400
1 1,0.5,0.8
800
600
400
600
400
2 0.1, 0.05, 0.01
2 0.1, 0.05, 0.01
200
200
0
0 50 100 150 200 250 300 0
R2c
Fig 8: Case C: Graph of R2c versus for different values of 1
0 100 200 300 400 500 600
R2c
Fig 9: Case C: Graph of R2c versus for different values of 2
800
600
400
200
0
1 0.1,2 0.1, pr 10
Le = 300, 200, 100
1 0.1,2 0.1, pr 10
Le = 300, 200, 100
0 50 100 150 200 250 300 350
R2c
Fig 10: Case C: Graph of R2c versus for different values of Le
Figs (2) (4) show the correction Rayleigh
10. CONCLUSIONS
number, R2c, versus frequency of modulation, , under Case A (InPhase modulation) for varying values of the Lewis number, Le, stress relaxation parameter, 1 , and strain retardation parameter, 2 . The effect of Le is to stabilize the system. The stress relaxation parameter destabilizes the system whereas strain retardation parameter stabilizes. Figs (5) (7) are graphs showing case B (OutofPhase modulations) and figs (8) (10) are graphs showing case C (only lower wall modulation). It can be seen from the graphs that the values of R2c are larger in both of these cases. This is due to the fact that in the case of out of phase modulation the temperature has a linear gradient varying in time, o that the Rayleigh number is supercritical for half a cycle and subcritical during the other half cycle. These results on the various parameters do not change for the case of lower wall modulation.
The results of the study help in figuring out the effects of externally controlling the convection through modulations. They either advance or delay convection. The following conclusions are made:

In the case of inphase modulation the various parameters cause delay in convection.

The parameters have opposing effects in the case of in phase and outofphase modulations.

Lower wall modulations show the same results as that of outofphase modulations.

Modulation is an effective means of controlling convection to a large extent.

Lewis number and strain retardation decrease the heat/mass transfer for inphase modulation.
REFERENCES

P. G. Siddeshwar, G.N. Sekhar, and G. Jayalatha. Effect of time periodic vertical oscillations of the RayleighBÃ©nard system on nonlinear convection in viscoelastic liquids NonNewtonian Fluid Mech. Vol 165, pp. 14121418, 2010.

Zhenyu Li and Roger E. Khayat. Threedimensional thermal convection of viscoelastic fluids Phys. Rev. E, vol. 71,pp. 221251, 2005

Z. Li, R.E. Khayat, Finiteamplitude RayleighBenard convection and pattern selection for viscoelastic fluids, J. Fluid Mech. 529 (2005)

Pradeep G. Siddheshwar and C. V. Sri Krishna, RayleighBenard Convection in a viscoelastic fluidfilled highporosity medium with nonuniform basic temperature gradient Int. J. Math. Math.Sci., vol. 25, pp. 609619, 2001.

M. S. Malashetty, P. G. Siddheshwar, and Mahantesh Swamy. Effect of Thermal Modulation on the Onset of Convection in a Viscoelastic Fluid Saturated Porous Layer Transport Porous Med., vol 62, pp 5579, 2006.

R.C. Sharma .Effect of rotation on thermal instability of a viscoelastic fluid.Acta Physiol. Hung., Vol. 40, pp. 1117, 2006.

Mojtabi A., CharrierMojtabi M.C.: Double diffusive convection in porous media. In: Vafai, K. (eds) Hand Book of Porous Media, pp. 559603. Marcel Dekkes, New York (2000)

M. Stern.The Salt Fountain and Thermohaline Convection Tellus, vol 12, pp.172175,1960.

H. M. Stommel, A. B. Arons, Blanchard. An oceanography curiosity: the perpetual salt fountain DeepSea Res, vol 3.pp. 152 153, 1956.

S. Rosenblat and G. A. Tanaka, Modulation of thermal convection instability Phys. Fluids, vol. 14, pp. 1319, 1971.

P. G. Siddheshwar and S. Pranesh.Effect of temperature/gravity modulation on the onset of magnetoconvection in weak electrically conducting fluids with internal angular momentum. Int. J. Magn. Magn.Mater., vol. 192, pp. 159176, 1999.

P. G. Siddheshwar and .V. Sri Krishna, Rayleigh Benard Convection in a Viscoelastic Fluid Filled High Porosity Medium with Non Uniform Basic Temperature Gradient, Int. J of Mathematics and Mathematical Sciences, vol. 25, pp. 609619, 2001.

B. S. Bhadauria, Temperature Modulation of Double Diffusive Convection in a Horizontal Fluid Layer, Z. Naturforsch, vol. 61a, pp. 335 344, 2006.

M. Narayana, S.N. Gaikwad, P. Sibanda, R.B. Malge, Double Diffusive Magneto Convection in Viscoelastic fluids, Int. J. of Heat and Mass Transfer, vol. 6, pp. 19420, 2013.

Milo M. Jovanovi, Jelena D. Nikodijevi, Milica D. Nikodijevi, RayleighBÃ©nard convection instability in the presence of spatial temperature modulation on both plates. Int. J. Non Linear Mech., vol. 73, pp. 6477, 2015.

Pranesh and Sangeetha. Effect of imposed timeperiodic boundary temperature on the onset of RayleighBenard convection in a dielectric couple stress fluid. Int. J. Applied Math. An Comp., vol. 5, pp. 113, 2014.

Venezian G. Effect of modulation on the onset of thermal convection. J. Fluid Mech., vol 35, pp. 243254. 1969.