Dufour and Soret Effects on the Thermosolutal Instability of Rivlin–
Ericksen Elastico–Viscous Fluid in Porous Medium
Ramesh Chandaand Gian Chand Ranab
aDepartment of Mathematics, Government P. G. College Dhaliara 177103, Himachal Pradesh, India
bDepartment of Mathematics, NSCBM Govt. P. G. College Hamirpur 177005, Himachal Pradesh, India
Reprint requests to R. C.; E-mail:rameshnahan@yahoo.com
Z. Naturforsch.67a,685 – 691 (2012) / DOI: 10.5560/ZNA.2012-0074
Received February 13, 2012 / revised July 26, 2012 / published online September 19, 2012
Dufour and Soret effects on the convection in a horizontal layer of Rivlin–Ericksen elastico–
viscous fluid in porous medium are considered. For the porous medium, the Darcy model is used.
A linear stability analysis based upon normal mode analysis is employed to find a solution of the fluid layer confined between two free boundaries. The onset criterion for stationary and oscillatory convection has been derived analytically, and graphs have been plotted, giving various numerical val- ues to various parameters, to depict the stability characteristics. The effects of the Dufour parameter, Soret parameter, solutal Rayleigh number, and Lewis number on stationary convection have been investigated.
Key words:Thermosolutal Instability; Rivlin-Ericksen; Dufour; Soret Parameter.
1. Introduction
The onset of convection in a Newtonian fluid un- der varying assumptions of hydrodynamics and hy- dromagnetics has been given by Chandrasekhar [1].
Lapwood [2] has studied the stability of convec- tive flow in hydromagnetics in a porous medium using Rayleigh’s procedure. The Rayleigh instabil- ity of a thermal boundary layer in flow through a porous medium has been considered by Wooding [3].
McDonnel [4] suggested the importance of poros- ity in the astrophysical context. The onset of dou- ble diffusive convection in a fluid saturated porous medium heated from below is now regarded as a classical problem due to its wide range of applica- tions in saline geothermal fields, agricultural prod- uct storage, soil sciences, enhanced oil recovery, packed-bed catalytic reactors, and the pollutant trans- port in underground. A detailed review of the liter- ature concerning double diffusive convection in bi- nary fluids in porous media is given by Nield and Bejan [5,6], Trevisan and Bejan [7], Mojtabi and Charrier-Mojtabi [8,9], Malashetty and Kollur [10].
Thermal convection in a binary fluid driven by the Soret and Dufour effect has been investigated by Knobloch [11]. He has shown that the equations are
identical to the thermosolutal problem except the re- lation between the thermal and solutal Rayleigh num- bers.
The above literature deals with Newtonian fluids.
But in technological fields, there exists an important class of fluids, called non-Newtonian fluids, which are also studied extensively because of their practical ap- plications, such as fluid film lubrication, analysis of polymers in chemical engineering etc. An experimen- tal demonstration by Toms and Trawbridge [12] has revealed that a dilute solution of methyl methacry- late in n-butyl acetate agrees well with the theoretical model of Oldroyd [13]. There are many visco–elastic fluids which cannot be characterized by Maxwell’s constitutive relations. One such fluid is the Rivlin–
Ericksen fluid [14]. Parkash and Kumar [15] and Sharma and Kumar [16] studied the thermal insta- bility of the Rivlin–Ericksen elastico–viscous fluid in a porous medium. While Prakash and Chand [17] ex- amined the effect of kinematic visco–elastic instability of a Rivlin–Ericksen elastico–viscous fluid in porous medium and found that the kinematic visco–elasticity stabilizes the fluid layer.
In the present study, we investigated the Dufour and Soret effects on the thermal instability of a Rivlin–
Ericksen elastico–viscous fluid in a porous medium.
© 2012 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com
Fig. 1. Physical model.
2. Mathematical Formulations
Consider an infinite horizontal layer of a Rivlin–
Ericksen elastico–viscous fluid of thicknessdbounded by plane z=0 and z=d in a porous medium of porosityεand medium permeabilityk1which is acted upon by gravity g(0,0,−g) as shown in Figure1.
This layer is heated and soluted from below such that a constant temperature and concentration distri- bution is prescribed at the boundaries of the fluid layer. The temperatureTand concentrationCare taken to be T0 and C0 at z=0 and T1 and C1 at z=d, (T0>T1,C0>C1). Let∆T and∆Cbe the differences in temperature and concentration across the bound- aries.
Let q(u,v,w), p, ρ, T, C, α, α0, µ, µ0, κ, and κ0 be the Darcy velocity vector, hydrostatic pres- sure, density, temperature, solute concentration, ther- mal coefficient of expansion and an analogous solvent coefficient of expansion, viscosity, kinematic visco–
elasticity, thermal diffusivity, and solute diffusivity of fluid respectively.
We assume that the medium is homogenous, isotropic, and Darcy’s law is valid. Applying the Oberbeck–Boussineque approximation, the govern- ing equations for the Rivlin–Ericksen elastico–viscous fluid in a porous medium are
∇q=0, (1)
0=−∇p−1 k
µ+µ0∂
∂t
q
+ρ0 1−α(T−T0)−α0(C−C0) g,
(2)
σ∂T
∂t +q∇T =κ∇2T+DTC∇2C, (3) ε∂C
∂t +q∇C=κ0∇2C+DCT∇2T. (4)
whereDTC andDCT are the Dufour and Soret coeffi- cients;σ=(ρc(ρcp)m
p)f is the thermal capacity ratio,cpthe specific heat, and the subscripts m and f refer to the porous medium and the fluid, respectively.
We assume that temperature and concentration are constant at the boundaries of the fluid layer. Therefore boundaries conditions are
w=0, T=T0, C=C0 at z=0
and w=0, T =T1, C=C1 at z=d. (5) 3. Steady State and its Solutions
The steady state is given by u=v=w=0, p=p(z),
T =Ts(z), C=Cs(z). (6) The solution of the steady state is
Ts=T0−∆T
d z, Cs=C0−∆C d z, ps=p0−ρ0g
z+α∆T
2dz2+α0∆C 2dz2
, where the subscript 0 denotes the value of the variable at the boundaryz=0.
4. Perturbation Equations
Let the initial steady state as described by above equation be slightly perturbed so that the perturbed state is given by
q=q0, T =Ts+T0,
C=Cs+C0, p=ps+p0, (7) where the prime denotes the perturbed quantities. Sub- stituting (7) into (1) – (4) and neglecting higher-order terms of the perturbed quantities, we get
∇q0=0, (8)
0=−∇p0− 1 k1
µ+µ0∂
∂t
q0 +ρ0 αT0+α0C0
g,
(9)
σ∂T0
∂t −w0∆T
d =κ∇2T0+DTC∇2C0, (10) ε∂C0
∂t −w0∆C
d =κ0∇2C0+DCT∇2T0. (11)
Now introducing dimensionless variables as (x00,y00,z00) =
x0,y0,z0 d
, (u00,v00,w00) =
u0,v0,w0 κ
d,
t00= κ
σd2t, p00=k1d2 µ κ p0, T00= T0
∆T , C00= C0
∆C.
There after dropping the dashes(00)for simplicity.
Now (8) – (11) can be written in non-dimensional form as
∇q=0, (12)
0=−∇p−
1+F ∂
∂t
q+RaT+RsC, (13)
∂T
∂t −w=∇2T+Df∇2C, (14) ε
σ
∂C
∂t −w= 1 Le
∇2C+Sr∇2T, (15) where the non-dimensional parameters are
Ra=gρ αk∆T d
µ κ (thermal Rayleigh number), Rs=gρ α0k∆Cd
µ κ0 (solutal Rayleigh number), Le= κ
κ0 (Lewis number), F= µ0κ
µ σd2 (kinematic visco–elasticity parameter), Df=DTC∆C
κ∆T (Dufour parameter), Sr=DCT∆T
κ∆C (Soret parameter),
and the non-dimensional boundary conditions are w=T=C=0 at z=0 and z=1. (16) 5. Normal Modes
Analyzing the disturbances in normal modes and as- suming that the perturbed quantities are of the form
[w,T,C] = [W(z),Θ(z),Γ(z)]
·exp(ikxx+ikyy+nt), (17)
wherekx,ky are wave numbers inx- and y-direction, andnis the growth rate of the disturbances.
Using (17), (12) – (15) become
1+F∂
∂t
D2−a2 W +Raa2Θ+Rsa2Γ =0,
(18) W+ D2−a2−n
Θ +Df D2−a2
Γ =0, (19)
W+Sr D2−a2 Θ
+ 1
Le D2−a2
−ε σn
Γ=0, (20) whereD= dzd anda2=k2x+k2y is the dimensionless resultant wave number.
The boundary conditions for free–free boundary surfaces are thus
W =0, D2W=0, Θ=0, Γ=0 at z=0
and W=0, D2W =0, Θ=0, Γ=0 at z=1.
(21)
We assume the solution toW,Θ, andΓ is of the form
W =W0sinπz, Θ=Θ0sinπz,
Γ=Γ0sinπz, (22)
which satisfies the boundary conditions (21).
Substituting solution (22) into (18) – (20), integrat- ing each equation fromz=0 to z=1, and perform- ing some integrations by parts, we obtain the following matrix equation:
J(1+nF) −a2Ra −a2Rs
−1 (J+n) DfJ
−1 SrJ J
Le+εn σ
W Θ0
Γ0
=
" 0 0 0
# ,
whereJ=π2+a2.
The non-trivial solution of the above matrix requires that
Ra=
(1+nF)
J(J+n)
J Le+εnσ
−SrDfJ2 a2J
1 Le−Df
+εn
σ
+ SrJ−(J+n) J
1 Le−Df
+εnσ
Rs. (23)
For neutral instabilityn=iω, (whereωis a real and dimensional frequency), (23) reduces to
Ra=∆1+iω∆2, (24) where∆1and∆2are given in theAppendix.
Since Ra is a physical quantity, it must be real.
Hence, it follows from (24) that eitherω=0 (exchange of stability, steady state) or∆2=0 (ω6=0 overstability or oscillatory onset).
6. Stationary Convection
For stationary convection ω =0 (n=0), (23) re- duces to
(Ra)s=J2 a2
DfSrLe−1 DfLe−1
+(Sr−1)Le
1−DfLe Rs. (25) We find that for the stationary convection the kine- matic visco–elasticity parameterFvanishes withnand the Rivlin–Ericksen elastico–viscous fluid behaves like an ordinary Newtonian fluid. This is the same result as obtained by Motsa [18].
The critical cell size at the onset of instability is ob- tained from the condition
∂Ra
∂a
a=ac
=0, which gives ac=π. This is the same result as obtained by Lapwood [2]
for a Newtonian fluid.
The corresponding critical Rayleigh number Racfor steady onset is
(Rac)s=4π2
DfSrLe−1 DfLe−1
+(Sr−1)Le
1−DfLe Rs. (26) This is also the same result as obtained by Motsa [18].
IfRs=Df=Sr=0 then (Rac)s=4π2.
This is the exactly the same result as obtained by Nield and Bejan [5].
7. Oscillatory Convection
For oscillatory convection ω 6=0, we must have
∆2=0, which gives ω2=
"
J4F 1
Le−Df 2
+J2F 1
Le−Df 2
+ε σJ2F
1 Le−Df
−J3ε σ
1 Le−Df
+J3(Sr−1)Dfε
σ−a2 J 1
Le−Df
(27) +Jε
σ(Sr−1)
! Rs
#
· (
ε σJ2F
1 Le−Df
−J2F 1
Le−Df
ε σ −ε2
σ2J(1+JF)−J2Df
ε σ
)−1
. Equation (27) indicates the frequency of the oscilla- tory mode. If there is no positiveω2then an oscillatory instability is not possible. If there exist positive values ofω2, then the thermal oscillatory Rayleigh number is obtain by inserting the positive values ofω2in (24).
The thermal oscillatory Rayleigh number is given by
(Ra)osc= 1 a2
"
J4 1
Le−Df 2
−J4(Sr−1)Df
· 1
Le
−Df
−ω2 (
ε σJ2
1 Le
−Df
+J3F 1
Le
−Df 2
+J3FDf 1
Le
−Df
(28) +ω2JFε2
σ2−ε σJ
1 Le−Df
−Jε2 σ
)#
·
"
J2 1
Le−Df 2
+ω2ε2 σ2
#−1
+
J2(Sr−1)
1 Le−Df
−ω2ε
σ
J2
1 Le−Df2
+ω2ε2
σ2
R.
From (28), it is clear that oscillatory convection de- pends on the visco–elasticity parameterF.
8. Results and Discussion
The expression for the stationary thermal Rayleigh number is given in (25) and the oscillatory thermal
Rayleigh number is given in (28). We discuss our re- sults analytically and graphically.
Figure2 shows the variation of the Rayleigh num- ber with the wave number for different values of the Dufour parameter; it has been found that the Rayleigh number decreases with an increase in the value of Du- four parameter, thus the Dufour parameter has a desta- bilizing effect on stationary convection. It is the same result as obtained by Motsa [18]
Figure3shows the variation of the Rayleigh number with the wave number for different values of the Soret parameter; it has been found that the Rayleigh num- ber first increases then decreases and finally increases again with an increase in the value of Soret parame- ter, thus the Soret parameter has both stabilizing and
70 120 170 220 270 320
0 2 4 6 8 10
Ra
a
Df=0.3 Df=0.2 Df=0.1 Le=500, Sr=0.5, Rs=200.
Fig. 2 (colour online). Variation of Rayleigh numberRawith wave numberafor different values of Dufour parameter.
50 100 150 200 250 300 350
0 5 10 15 20 25
Ra
a
Sr=0.6
Sr=0.4 Sr=0.2 Le=500, Df=0.2, Rs=200.
Fig. 3 (colour online). Variation of Rayleigh numberRawith wave numberafor different values of Soret parameter.
destabilizing effect on the stationary convection. It is the same result as obtained by Motsa [18]
Figure4shows the variation of the Rayleigh num- ber with the wave number for different values of the Lewis number; it has been found that the Rayleigh number first increases then decreases and finally in- creases again with an increase in the value of Lewis number, thus the Lewis number has both stabilizing and destabilizing effect on the stationary convection.
Figure5shows the variation of the Rayleigh num- ber with the wave number for different values of the solutal Rayleigh number; it has been found that the thermal Rayleigh number increases with an increase
110 115 120 125 130 135 140 145
0 5 10 15
Ra
a
Le =100 Le=200Le=300 Sr=0.5 , Df=0.2, Rs=200.
Fig. 4 (colour online). Variation of Rayleigh numberRawith wave numberafor different values Lewis number.
200 300 400 500 600 700 800 900
0 5 10 15 20 25 30
Ra
a
Rs=75
Rs=25 Rs=50 Le=500,Sr =0.5 , Df=0.2
Fig. 5 (colour online). Variation of Rayleigh numberRawith wave numberafor different values solutal Rayleigh number.
in the value of solutal Rayleigh number, thus the solu- tal Rayleigh number stabilizes the stationary convec- tion.
9. Conclusions
We used linear instability analysis to study Soret and Dufour effects in double diffusive convection of a Rivlin–Erickson elastico–viscous fluid in a porous medium. An expression for the Rayleigh number for stationary and oscillatory convection is obtained. We draw following conclusions:
(i) In stationary convection, the Rivlin–Ericksen elastico–viscous fluid behaves like an ordinary New- tonian fluid.
(ii) The Dufour parameter destabilizes the stationary convection.
(iii) The Soret parameter and the Lewis number have both stabilizing and destabilizing effect on the station- ary convection.
(iv) The solutal Rayleigh number stabilizes the sta- tionary convection.
(v) In the limiting case whenRs=Sr=Df=0 the critical thermal Rayleigh number obtained is the same as reported by Nield and Bejan [5].
Acknowledgement
The authors are very grateful to the reviewers for their valuable comments and suggestions for the im- provement of the paper. The first author is thankful to the University Grants Commission of India for their fi- nancial support during this work.
Appendix
∆1and∆2appearing in (24) are derived as
∆1= 1 a2
"
J4 1
Le−Df 2
−J4(Sr−1)Df 1
Le−Df
−ω2 (
ε σJ2
1 Le−Df
+J3F
1 Le−Df
2
+J3FDf 1
Le−Df
+ω2JFε2 σ2−ε
σJ 1
Le−Df
−Jε2 σ
) +a2
J2(Sr−1)f 1
Le−Df
−ω2ε σ
Rs
#
·
"
J2 1
Le
−Df 2
+ω2ε2 σ2
#−1
,
∆2= 1 a2
"
J4F 1
Le−Df 2
+J2F 1
Le−Df 2
+ε σJ2F
1 Le−Df
−J3ε σ
1 Le−Df
+J3(Sr−1)Dfε σ−a2
J
1 Le−Df
+Jε
σ(Sr−1) Rs−ω2
ε σJ2F
1 Le−Df
−J2F 1
Le−Df ε
σ −ε2
σ2J(1+JF)−J2Dfε σ
#
·
"
J2 1
Le−Df
2
+ω2ε2 σ2
#−1
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