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Linear and nonlinear double-diffusive convection in a saturated anisotropic porous layer with Soret effect and internal heat source
International Journal of Heat and Mass Transfer 59 (2013) 103–111
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Linear and nonlinear double-diffusive convection in a saturated anisotropic porous layer with Soret effect and internal heat source A.A. Altawallbeh a, B.S. Bhadauria b, I. Hashim a,c,⇑ a
School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi Selangor, Malaysia Department of Applied Mathematics & Statistics, School of Physical Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow 226025, UP, India c Solar Energy Research Institute, Universiti Kebangsaan Malaysia, 43600 Bangi Selangor, Malaysia b
a r t i c l e
i n f o
Article history: Received 3 August 2012 Received in revised form 19 October 2012 Accepted 3 December 2012 Available online 5 January 2013 Keywords: Double diffusion Porous layer Soret effect Internal heat
a b s t r a c t Double-diffusive convection in an anisotropic porous layer heated and salted from below with an internal heat source and Soret effect is studied analytically using both linear and nonlinear stability analyses. The generalized Darcy model including the time derivative term is employed for the momentum equation. The critical Rayleigh number and wavenumber for stationary and oscillatory modes and frequency of oscillations are obtained analytically using linear theory. The effects of the anisotropy parameters, concentration Rayleigh number, internal heat source, Soret parameter, Vadasz number and Lewis number on the stationary, oscillatory, and heat and mass transfer are shown graphically. Heat and mass transfer have been obtained in terms of the Nusselt number and Sherwood number. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The problem of double-diffusive convection in porous media has received considerable attention in recent years, on account of its wide applications in, for example, the effect of contaminants on lakes and under ground water, atmospheric pollution, chemical processes, food processing, energy storage, and materials processing. In double diffusion, the buoyancy force is affected not only by the difference of temperature, but also by the difference of concentration of the fluid. A detailed discussion on double diffusion natural convection can be found in the books by Nield and Bejan [1], Ingham and Pop [2] and Vafai [3]. The onset of thermal instability in a horizontal porous layer saturated with Newtonian fluid was first studied by Horton and Rojers [4] and Lapwood [5]. Other studies on linear and nonlinear double diffusive convection in a porous layer were given by Haajizadeh et al. [6], Gaikwad et al. [7], Charrier-Mojtabi et al. [8], Malashetty and Swamy [9] and Capone et al. [10]. Early studies on convection in a porous medium have usually been concerned with homogeneous isotropic porous structures. In the last one decade, the effects of non-homogeneity and anisotropy of the porous medium have been studied. The geological and pedagogical processes rarely form isotropic media as is usually as⇑ Corresponding author at: Solar Energy Research Institute, Universiti Kebangsaan Malaysia, 43600 Bangi Selangor, Malaysia. Tel.: +60 3 8921 5758; fax: +60 3 8925 4519. E-mail address: [email protected] (I. Hashim). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.12.005
sumed in transport studies. In geothermal system with a ground structure composed of many strata of different permeability, the overall horizontal permeability may be up to 10 times as large as the vertical component. Processes such as sedimentation, compaction, frost action, and reorientation of the solid matrix are responsible for the creation of anisotropic natural porous media. Anisotropy can also be a characteristic of artificial porous material like pelleting used in chemical engineering process, fiber materials used in insulating purposes. Anisotropy finds also application in mathematical modeling in geothermal systems such as fractured rocks [11]. The first study in a fluid layer saturated anisotropic porous medium was conducted by Castinel and Combarnous [12]. Many other studies were conducted in a fluid layer saturated anisotropic porous medium [13,14]. Recently, many authors have studied the effect of anisotropy on the onset of convection in a porous layer [15,9,16,11,17]. A situation in which a porous medium has its own source of heat can occur through radioactive decay or through, in the present perspective, a relatively weak exothermic reaction which can take place within the porous material. Internal heat generation becomes very important in geophysics, reactor safety analysis, metal waste form development, fire and combustion studies and storage of radioactive materials. The onset of convection in a horizontal layer of an anisotropic porous medium with an internal heat source subjected to an inclined temperature gradient was studied by Parthiban and Parthiban [18]. An analytical solution for small Rayleigh number in a finite container with isothermal walls and uniform heat generation within the porous medium was given by
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Nomenclature d D Da g K Le Nu P Pr q RaT Ras Ri Sh Sr
hight of the porous layer cross diffusion due to T component Darcy number, Kz/d2 gravitational acceleration ^kÞ ^ permeability of the porous medium, K x ð^i^i þ ^j^jÞ þ K z ðk Lewis number jT z =js Nusselt number pressure Prandtl number velocity vector, (u, v, w) DTdK z thermal Rayleigh number, bT gmj Tz DCdK z concentration Rayleigh number, bs gmj T z
internal Rayleigh number Sherwood number s Soret parameter, jDb T z bT
Joshi et al. [19]. Recently, Bhadauria et al. [16] studied the natural convection in a rotating anisotropic porous layer with internal heat generation using a weak nonlinear analysis. In [17], Bhadauria studied double-diffusive natural convection in an anisotropic porous layer in the presence of an internal heat source. If the cross diffusion terms are included in the species transport equations, then the situation will be quite different. Due to the cross diffusion effect, each property gradient has a significant influence on the flux of the other property. A flux of salt caused by a spatial gradient of temperature is called the Soret effect. Similarly, a flux of heat caused by a spatial gradient of concentration is called the DuFour effect. The DuFour coefficient is of order of magnitude smaller than the Soret coefficient in liquids, and the corresponding contribution to the heat flux may be neglected. Many studies can be found in the literature concerning the Soret and DuFour effects. A study by Rudaraiah and Malashetty [20] discussed the doublediffusive convection in a porous medium in the presence of Soret and DuFour effects. In another study, Rudraiah and Siddheshwar [21] investigated a weak nonlinear stability analysis of doublediffusive convection with cross-diffusion in a fluid saturated porous medium. Bahloul et al. [22] studied analytically and numerically a double-diffusive and Soret-induced convection in a shallow horizontal porous layer. Mansour et al. [23] investigated the multiplicity of solutions induced by thermosolutal convection in a square porous cavity heated from below and subject to horizontal solute gradient in the presence of Soret effect. Recently, Malashetty and Biradar [24] carried out an analytical study of linear and nonlinear double-diffusive convection in a fluid-saturated porous layer with Soret and DuFour effects. Also, a study by Malashetty et al. [25] investigated Soret effect on double diffusive convection in a Darcy porous medium saturated with a couple stress fluid. The aim of this work is to study the effects of an internal heat source and Soret effect on the convection in a binary fluid-saturated anisotropic porous layer. The onset criteria for stationary and oscillatory double-diffusive convection are obtained using linear stability analysis. Heat and mass transfer have been studied using nonlinear stability analysis, and the results were presented graphically in terms of the Nusselt number Nu and Sherwood number Sh.
T t Va x, y, z
temperature time Vadasz number, Pr/Da space coordinates
Greek symbols wavenumber bs concentration expansion coefficient bT thermal expansion coefficient g thermal anisotropy parameter jT z thermal diffusivity js concentration diffusivity l dynamic viscosity m kinematic viscosity w stream function q density n mechanical anisotropy parameter
a
tween two parallel horizontal planes at z = 0 and z = d with a distance d apart. A Cartesian frame of reference is chosen in such a way that the origin lies on the lower plane and the z-axis as vertically upward, where the gravity force g is acting vertically downward. Adverse temperature and concentration gradients are applied across the porous layer, and the lower and upper planes are kept, respectively, at temperatures T0 + DT and T0, and concentrations S0 + DS and S0, where DT and DS are temperature difference and concentration difference between the walls, respectively. The Soret effect is considered and the Oberbeck–Boussinesq approximation is assumed to account for the effect of density variations. Under these assumptions, the governing equations are given by Gaikwad et al. [11]
rq¼0
q0 @q / @t
ð1Þ
¼ rP
l K
q þ qg
ð2Þ
r
@T þ ðq rÞT ¼ r ðjT z rTÞ þ QðT T 0 Þ @t
ð3Þ
/
@C þ ðq rÞC ¼ ðjs r2 CÞ þ Dr2 T @t
ð4Þ
q ¼ q0 ½1 bT ðT T 0 Þ bs ðC C 0 Þ
ð5Þ
subject to the following boundary conditions
T ¼ T 0 þ MT; T ¼ T0;
C ¼ C 0 þ MC
C ¼ C0
at z ¼ 0
at z ¼ d
ð6Þ ð7Þ
where q is velocity (u, v, w), l is the dynamic viscosity, Q is internal ^kÞ; ^ jT is heat source, K is the permeability tensor K x ð^i^i þ ^j^jÞ þ K z ðk z the thermal diffusivity, T is temperature, bT is the thermal expansion coefficient, bs is the concentration expansion coefficient, js is concentration diffusivity of the fluid, r is the ratio of heat capacities, D is the cross diffusion due to T components, q is the density, g = (0, 0, g) is the gravitational acceleration and q0 is the reference density. 2.1. Basic state
2. Mathematical formulation The physical model under consideration is an infinite horizontal anisotropic porous layer with internal heat source, confined be-
The basic state of the fluid is assumed to be quiescent, and is given by
qb ¼ ð0;0; 0Þ; P ¼ Pb ðzÞ; T ¼ T b ðzÞ; C ¼ C b ðzÞ; q ¼ qb ðzÞ
ð8Þ
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"
From Eqs. (1)–(8), we obtain
dPb ¼ qg dz
@ 1 @2 @2 þ @t Le @z2 @x2
ð9Þ
!# S ¼ Sr þ
2
d T jT z 2b þ Q ðT b T 0 Þ ¼ 0 dz 2
ð10Þ
2
d C d T js 2b þ D 2b ¼ 0 dz dz
ð11Þ
where the subscript b refers to the basic state. The conduction state solutions are given by
T b ðzÞ ¼ T 0 þ DT
C b ðzÞ ¼
sin
DDT sin
js
pffiffiffiffiffiffiffiffiffiffiffiffiffi Q=jT z 1 dz pffiffiffiffiffiffiffiffiffiffiffiffiffi sin Q =jT z
ð12Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi Q =jT z 1 dz z DDT pffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 DC þ d js sin Q=jT z
þ C0 :
ð13Þ
T ¼ Tb þ T0;
C ¼ Cb þ C0;
P ¼ Pb þ P0 ;
¼ qb þ q
ð14Þ
where the primes denote infinitesimally small quantities. Now, substitute Eq. (14) into Eqs. (1)–(5), and using the basic states (9)–(11) we obtain
r q0 ¼ 0
r
jT z
¼ rP
l
0
0
0
q þ ðq0 gbT T q0 gbs C Þ
K
ð16Þ
@T 0 dT b þ ðq0 rÞT 0 þ w0 ¼ r ðjT rT 0 Þ þ QT 0 @t dz
ð17Þ
@C 0 dC b / þ ðq0 rÞT 0 þ w0 ¼ js r2 C 0 þ Dr2 T 0 @t dz
ð18Þ
Eliminating the pressure from Eq. (16) we obtain
/ @t @z
@w0 l @u0 @w0 @T 0 @C 0 ¼ q0 g bT bs @x K @z @x @x @x
ð19Þ
For simplicity, the case of two-dimensional rolls have been considered, and thus all physical quantities are made as independent of y. Now, we introduce the stream functions u = @ w/@y, v = @ w/@x, and use the following transformations
ðx ; z Þ ¼ ¼
x z ; ; d d
t ¼
t jT z 2
d
w ¼
;
w
jT z
;
T ¼
T0 ; MT
C0 MC
ð20Þ
2
2
!
2
2
1 @ @ @ @ 1 @ þ þ þ Va @t @z2 @x2 @x2 n @z2
@T @S ¼ RaT þ Ras @x @x
"
2
Va ¼
Pr ; Da
n¼
Kx ; Kz
g¼
and f ðz Þ ¼ dT b ðz Þ=dz and bðz Þ ¼ dSb ðz Þ=dz . T b and Sb in dimensionless forms are given by
T b ðzÞ ¼
pffiffiffiffi sin Ri ð1 z Þ pffiffiffiffi sin Ri
ð24Þ
Sb ðzÞ ¼
pffiffiffiffi Sr LeRaT sin Ri ð1 z Þ Sr LeRaT pffiffiffiffi þ 1 ð1 z Þ Ras sin Ri Ras
ð25Þ
@ 2 w ¼ T ¼ S ¼ 0 at z ¼ 0 and z ¼ 1: @z2
ð26Þ
For the next steps the asterisks are removed, r and / are set equal to unity for simplicity.
3. Linear stability theory
2
#
In this section, the linear stability analysis is discussed. To make the linear stability study, we neglect the Jacobian in Eqs. (21)–(23), and assume the solutions to be periodic waves of the form
2 3 2 3 w W sin ax 6 7 6 7 4 T 5 ¼ eet 4 H cos ax 5 sin pz S U cos ax
ð27Þ
where (W, H, U) are the amplitudes of (w, T, S), e = er + iei is the growth rate and in general a complex quantity and a is a horizontal wave number. Now, substitute (27) into (21)–(23) to obtain
e
Va
a2 þ a21 W ¼ RaT aH þ Ras aU
ð28Þ
e þ a22 Ri H ¼ 2aWF
eþ
S
ð29Þ
a2 Ra U ¼ Sr T Ha2 þ 2aWB Le Ras
ð30Þ
2 where a2 ¼ ðp2 þ a2 Þ; a21 ¼ pn þ a2 and a22 ¼ ðp2 þ ga2 Þ. Now, write Eqs. (28)–(30) in a matrix form
into Eqs. (15) and (17)–(19), we obtain
"
;
ð15Þ 0
q0 @ @u0
2
jT x b gK MTd ; RaT ¼ T z ; jT z mjT z b gK MCd jT Dbs ; m ¼ l=q0 ; Le ¼ z ; Sr ¼ Ras ¼ s z mjT z js jT z bT Qd
Ri ¼
w ¼
q
0
/ @t
ð23Þ
where
On the basic state we superpose perturbation in the form
q0 @q0
@ðw ; S Þ @ðx ; z Þ
In the above equations r and / are set equal to unity for simplicity. Now, Eqs. (21)–(23) are solved for stress-free, isothermal and isohaline conditions
2.2. Perturbed state
q ¼ qb þ q0 ;
RaT 2 @w r T þ bðz Þ Ras @x
2
!#
e Va
a2 þ a21
6 2aF 4 2aB
w ð21Þ
@ @ @ @w @ðw ; T Þ g 2 Ri T ¼ f ðz Þ þ @t @z2 @ðx ; z Þ @x @x
ð22Þ
Ras a
e þ a22 Ri
0
T Sr Ra a2 Ras
e þ aLe2
32
2 3
3
0 W 76 7 6 7 54 H 5 ¼ 4 0 5 0 U
ð31Þ
where
F¼
RaT a
Z 0
1
2
f ðzÞ sin ðpzÞdz;
B¼
Z
1
2
bðzÞ sin ðpzÞdz
0
For the non-trivial solution of (31), it is found that
ð32Þ
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RaT ¼ ðRi 4p2 Þðe a22 þ Ri Þðe2 a2 Le þ ea4 þ a21 VaeLe þ
a21 Vaa2
2
2
2
þ a VaLeRas Þ=½a VaðLe
2
þ 4p eLe þ Le
2
Sr Ri a22
2
Sr R2i
2
2
þ 4p Sr a Le
2 2
þ Le Sr Ri e þ 4p a Þ
ð33Þ
3.1. Marginal stationary state For the existence of marginal stability, the real part of e must be zero at marginal state. For stationary convection to occur, e must be zero. This means that the real and imaginary part of e are equal to zero. So, from Eq. (33) the stationary Rayleigh number equals to
RastT ¼ ðRi þ 4p2 Þðga2 þ p2 Ri Þða2 LeRas n þ a4 n þ a2 np2 h þ a2 p2 þ p4 Þ= na2 Le2 Sr R2i þ 4a2 p2 Sr Le þ 4p4 Sr Le i þ4a2 p2 þ 4p4 þ Le2 Sr Ri ga2 þ Le2 Sr Ri p2
2
1 p 1 ðga2 þ 1ÞLeRas 2 2 þ ð ga þ 1Þ a þ ðSr Le þ 1Þ a2 n ða2 þ 1Þ
ð34Þ
p2 2 1 ðga2 þ 1Þ þ 2 ðga þ 1Þ a2 þ LeRas 2 n a a þ1
ð36Þ
In the case of one single component (Ras = 0) the result of Storesletten [27] is recovered
RastT ¼
p2 2 1 ðga þ 1Þ a2 þ n a2
ð37Þ
Finally, in the case of an isotropic porous medium, putting g = n = 1, then
RastT ¼
p2 ð1 þ a2 Þ2 a2
ð38Þ
2 2 which has the critical value Rast T ¼ 4p at ac ¼ 1 as obtained previously by Horton and Rojers [4] and Lapwood [5].
3.2. Marginal oscillatory state For marginal oscillatory state, the real part of e must be equal to zero. So, putting e = ie in Eq. (33), the expression for Raosc is obT tained as 2 2 2 2 2 2 2 2 Raosc T ¼ x=½a VaLe ðSr LeRi þ 4p Þ e þ a VaðSr Le Ri a2
4p2 Sr a2 Le 4a2 p2 þ
Sr Le2 R2i Þ2
ð39Þ
where e2 is given by
e2 ¼
" # " ## 1 @ @2 @2 @2 1 @2 @T @S þ w þ RaT Ras ¼0 þ þ Va @t @z2 @x2 @x @x @x2 n @z2 # @ @2 @2 @w @T @w @T @w þ ¼ f ðzÞ g 2 Ri T @t @z2 @x @z @z @x @x @x
"
@ 1 @2 @2 þ @t Le @z @x2 ¼
ð35Þ
In the case of no Soret effect and internal heat source, we recover the result given by Malashetty and Swamy [26]
RastT ¼
"
ð41Þ
"
Now, rescale a2 = a2p2, and so when the internal heat source is absent we recover the result of Gaikwad et al. [11]
RastT ¼
temperature, and concentration. This study will help in understanding the physics of the problem with minimum mathematical expressions. Further, the results can be used as a starting point to generalize it for the full nonlinear problem. Also it is to be noted that the effect of nonlinearity is to distort the temperature and concentration fields through the interaction of w and T, and w and S, respectively. As a result, a component of the form sin (2pz) will be generated. Dropping the asterisks and rearranging the terms, Eqs. (21)–(23) become
X
: Le2 Lea21 VaSr Ri 4a4 p2 Sr þ 4a22 a2 p2 4Ri a2 p2 þ 4a21 Vap2 þ a4 Sr Ri ð40Þ
The expressions for x and X which appear in (39) and (40) are given in Appendix A. 4. A weak nonlinear theory For flows with Ra > Rast T , the linear stability analysis is not valid, because of the missing nonlinear effects. In this section, a local nonlinear stability analysis shall be performed by using a severely truncated representation of Fourier series for the stream function,
!# S
ð42Þ
@w @S @w @S RaT 2 þ Sr r T @x @z @z @x Ras
@w bðzÞ: @x
ð43Þ
Now consider a minimal Fourier series with one term in the stream function, and to get some effects of non-linearity, take two terms in the temperature and concentration fields as given below
w ¼ A1 ðtÞ sinðaxÞ sinðpzÞ
ð44Þ
T ¼ B1 ðtÞ cosðaxÞ sinðpzÞ þ B2 ðtÞ sinð2pzÞ
ð45Þ
S ¼ C 1 ðtÞ cosðaxÞ sinðpzÞ þ C 2 ðtÞ sinð2pzÞ
ð46Þ
where the amplitudes A1(t), B1(t), B2(t), C1(t) and C2(t) are functions of time and are to be determined. Substituting (44)–(46) in (41)– (43) and equating the coefficients of like terms of the resulting equations, we obtain the following system of nonlinear differential equations
dA1 ðtÞ Vaa21 A1 ðtÞ VaaRaT B1 ðtÞ VaaRas C 1 ðtÞ ¼ þ dt a2 a2 a2
ð47Þ
dB1 ðtÞ ¼ 2aFA1 ðtÞ þ Ri a22 B1 ðtÞ apA1 ðtÞB2 ðtÞ dt
ð48Þ
dB2 ðtÞ 1 ¼ apA1 ðtÞB1 ðtÞ þ ðRi 4p2 ÞB2 ðtÞ dt 2
ð49Þ
dC 1 ðtÞ a2 a2 Sr RaT ¼ 2aBA1 ðtÞ C 1 ðtÞ B1 ðtÞ apA1 ðtÞC 2 ðtÞ dt Le Ras
ð50Þ
dC 2 ðtÞ a 4p2 4p2 Sr RaT B2 ðtÞ ¼ pA1 ðtÞC 1 ðtÞ C 2 ðtÞ dt 2 Le Ras
ð51Þ
The nonlinear system of autonomous differential equations is not suitable to analytical treatment for the general time-dependent variables and we have to solve it numerically. After determining the numerical values of the amplitude functions A1(t), B1(t), B2(t), C1(t) and C2(t), the Nusselt number and Sherwood number can be obtained as functions of time. The steady analysis is performed by setting the left hand side of Eqs. (47)–(51) equal to zero,
Vaa21 A1 ðtÞ VaaRaT VaaRas B1 ðtÞ þ C 1 ðtÞ ¼ 0 a2 a2 a2
2aFA1 ðtÞ þ ðRi a22 ÞB1 ðtÞ apA1 ðtÞB2 ðtÞ ¼ 0
ð52Þ ð53Þ
A.A. Altawallbeh et al. / International Journal of Heat and Mass Transfer 59 (2013) 103–111
107
(a)
(b)
Fig. 3. Effect of Ri on the marginal stability curves for stationary and oscillatory convection for the case n = 0.5, g = 0.8, Ras = 25, Sr = 0.05, Le = 20 and Va = 12.
Fig. 1. Effects of (a) g when n = 0.3 and (b) n when g = 0.8 on the marginal stability curves for stationary and oscillatory convection for the case Ri = 3, Sr = 0.05, Le = 20, Ras = 25, Va = 12. Fig. 4. Effect of Va on the marginal stability curves for stationary and oscillatory convection for the case n = 0.5, g = 0.8, Ri = 3, Sr = 0.05, Le = 20 and Ras = 25.
(a)
(b)
Fig. 5. Effect of Sr on the marginal stability curves for stationary and oscillatory convection for the case n = 0.5, g = 0.8, Ri = 3, Va = 12, Le = 10 and Ras = 50.
1 apA1 ðtÞB1 ðtÞ þ ðRi 4p2 ÞB2 ðtÞ ¼ 0 2
2aBA1 ðtÞ
a 2 Fig. 2. Effects of (a) Ras when Le = 10 and (b) Le when Ras = 50 on the marginal stability curves for stationary and oscillatory convection for the case n = 0.5, g = 0.8, Ri = 3, Sr = 0.05 and Va = 12.
a2 a2 Sr RaT B1 ðtÞ apA1 ðtÞC 2 ðtÞ ¼ 0 C 1 ðtÞ Le Ras
pA1 ðtÞC 1 ðtÞ
4p2 4p2 Sr RaT B2 ðtÞ ¼ 0 C 2 ðtÞ Le Ras
ð54Þ
ð55Þ
ð56Þ
with the consideration that all the amplitudes are constants. The Eqs. (52)–(56) have been solved for B1, B2, C1 and C2 in terms of A1 as given in Appendix A.
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4.1. Steady heat and mass transport
z Stotal ¼ S0 DS þ Sðx; z; tÞ d
Heat and mass transport is a very important issue in studying convection in fluids. This is because the onset of convection, as the Rayleigh number is increased, is more readily detected by its effect on the heat and mass transport. In the basic state, heat and mass transport is by conduction only. If H and J are the rates of heat and mass transport per unit area, respectively, then
Substituting Eqs. (45) and (46) into Eqs. (59) and (60) respectively and using the resultant equations in Eqs. (57) and (58), we get
H ¼ jT
@T total @z
Sh ¼ 1
ð57Þ z¼0
pffiffiffiffi 2pC 2 Ras sinð Ri Þ pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi ; Sr RaT cosð Ri Þ Ri þ sinð Ri ÞRas þ sinð Ri ÞSr RaT
ð61Þ
ð62Þ
where the expressions for B2 and C2 are given in Appendix A.
@Stotal @T total J ¼ js D @z z¼0 @z z¼0
ð58Þ
where the angular bracket represents the horizontal average and
T total
2pB2 pffiffiffiffi ; Nu ¼ 1 pffiffiffiffi Ri cotð Ri Þ
ð60Þ
z ¼ T 0 DT þ Tðx; z; tÞ d
ð59Þ
5. Results and discussion The onset of double-diffusive convection in a binary fluid-saturated anisotropic porous layer in the presence of Soret effect and an internal heat source is studied analytically using both linear
(a)
(b)
(c)
(d)
(e)
Fig. 6. Variations of Nu with RaT for steady nonlinear stability: effects of (a) n for the case Ri = 1, Sr = 0.005, g = 0.5, Le = 1, Ras = 100, (b) Ras for the case Ri = 1, Sr = 0.005, g = 0.5, Le = 1, n = 0.5, (c) Sr for the case Ri = 1, g = 0.5, n = 0.5, Le = 1, Ras = 100, (d) Ri for the case Sr = 0.005, g = 0.5, n = 0.5, Le = 1, Ras = 100, and (e) g for the case Ri = 1, Sr = 0.005, n = 0.5, Le = 1, Ras = 100.
A.A. Altawallbeh et al. / International Journal of Heat and Mass Transfer 59 (2013) 103–111
and nonlinear analyses. The expressions for the onset of stationary and oscillatory convection are calculated using the linear stability theory in terms of the governing parameters Ras, Le, Sr, Ri, g, n and Va. In the nonlinear theory, the heat and mass transfer are discussed in terms of the Nusselt number and sherwood number. The marginal stability curves for steady convection (Eq. (34)) and oscillatory convection (Eq. (39)) have been presented graphically in Figs. 1–5. It is found that, in all the cases investigated, the instability sets in via the oscillatory mode. The effects of the anisotropy parameters on the marginal stability curves are shown in Fig. 1. The effect of increasing g is to increase both the oscillatory and steady marginal curves, and hence the system is stabilized (see Fig. 1(a)). In Fig. 1(b), it is observed that increasing n destabilizes the system for both the stationary and oscillatory modes. Fig. 2 indicates the effect of the concentration Rayleigh number Ras and Lewis number Le on the stationary and oscillatory convection. The effect of Ras on the oscillatory marginal curve is quite small, but quite large on the stationary curve as shown in Fig. 2(a). The
109
effect of increasing Ras is stabilizing. While the opposite is observed for the effect of the Lewis number Le. The destabilizing effect of the internal Rayleigh number Ri is depicted in Fig. 3. Fig. 4 shows that the Vadasz number Va has no effect on the steady curve and a negligible effect on the oscillatory curve. The effect of the Soret parameter, Sr, is depicted in Fig. 5. Both negative and positive values of Sr were chosen. It is observed that the Soret parameter has a negligible effect on the oscillatory curves. On the other hand, the negative (positive) Soret parameter has a stabilizing (destabilizing) effect on the stationary curves. In the study of double-diffusive convection, heat and mass transport are important. Heat and mass transport are given in terms of the Nusselt number Nu and Sherwood number Sh respectively, which represent the ratio of heat or mass transported across the layer to the heat and mass transported by conduction alone. Fig. 6 shows the results obtained from Eq. (61). Fig. 6(a) shows that an increase of n increases Nu, which enhances the heat transfer. Fig. 6(d) shows that Ri has the effect of increasing the heat transfer. On the other hand, increasing
(a)
(b)
(c)
(d)
(e)
Fig. 7. Variations of Sh with RaT for steady nonlinear stability: effects of (a) n for the case Ri = 1, Sr = 0.005, g = 0.5, Le = 1, Ras = 100, (b) Ras for the case Ri = 1, Sr = 0.005, g = 0.5, Le = 1, n = 0.5, (c) Sr for the case Ri = 1, g = 0.8, n = 0.3, Le = 1, Ras = 100, (d) Ri for the case Sr = 0.005, g = 0.5, n = 0.5, Le = 1, Ras = 100, and (e) g for the case Ri = 1, Sr = 0.005, n = 0.5, Le = 1, Ras = 100.
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Ras and g decrease the heat transfer as depicted in Fig. 6(b) and (e). It is noticeable that increasing Sr increases the heat transfer (see Fig. 6(c)). However, for RaT > 180 the effect of varying Sr diminishes. The effects of the various parameters on heat mass transfer that can be determined from Eq. (62) are shown graphically in Fig. 7. Fig. 7(a) clearly shows that the effect of the mechanical anisotropy parameter n is to enhance the mass transfer. However, increasing Ras has the opposite effect as depicted in Fig. 7(b). Fig. 7(c) depicts the effect of the Soret parameter on the Sherwood number. It is found that there is a critical value of RaT below which increasing Sr enhances the mass transfer and above which increasing Sr decreases the mass transfer. The effect of Ri on the mass transfer is small for low values of RaT and then becomes insignificant for RaT (see Fig. 7(d)). The effect of increasing the thermal anisotropy parameter g is initially to decrease the mass transfer, but as RaT is increased further the opposite effect is observed. 6. Conclusion The onset of double-diffusive convection in a binary fluid-saturated anisotropic porous layer in the presence of Soret effect and internal heat source is studied analytically using linear and nonlinear stability analysis. The porous layer is heated and salted from below. For the stationary and oscillatory modes, it is observed that increasing the values of mechanical anisotropy parameter n, Lewis number Le, and internal Rayleigh number Ri destabilizes the system. On the other hand, increasing the values of thermal anisotropy parameter g and concentration Rayleigh number Ras is to stabilize the system. The positive values of Sr decreases the marginal curves for the stationary convection. While, the negative ones increases them. The effect of the Soret parameter on the oscillatory curves is negligible. The effects of the mechanical anisotropy parameter n and internal Rayleigh number Ri are to enhance the heat and mass transfer as they are increased. On the other hand, increasing the concentration Rayleigh number Ras and thermal anisotropy parameter g has the effect of decreasing the heat and mass transfer. Increasing the Soret parameter enhances the heat transfer. We also found that there is a critical value of RaT below which increasing Sr enhances the mass transfer and above which increasing Sr decreases the mass transfer.
Part of this study was done during the visit of the author B.S. Bhadauria (BSB) to the Universiti Kebangsaan Malaysia (UKM), in June 2012 as a visiting Professor of Mathematics. The authors gratefully acknowledge the Grant provided by UKM out of the University Research Fund OUP-2012-61 and DIP-2012-12. The author BSB is also grateful to Banaras Hindu University, Varanasi for sanctioning the lien to work as Professor of Mathematics at Department of Applied Mathematics, BB Ambedkar University, Lucknow, India. Appendix A. hn
x ¼ðRi þ 4p2 Þ Le3 a2 Ri e4 Le
and
a2 Le
h i h þ Sr a4 þ a21 VaLe Le2 R3i 2Le2 a22 R2i þ LeRi Lea22 4a2 p2 i þ4Lea2 p2 a22 a4 þ a21 Vaa2 þ a22 a21 VaLe þ 4p2 a2 Ri a2 þ a21 Va þ a2 a22 : ð64Þ The Eqs. (52)–(56) have been solved for A21 , with x ¼ A21 , we get
x1 x2 þ x2 x þ x3 ¼ 0;
ð65Þ
where
x1 ¼ p2 a21 a4 Le2 ; x2 ¼ 4Le2 a4 RaT ð4p2 Sr Ri þ 4p2 ÞF 16a4 BRas p2 2a21 a2 4p2 a2 R2i Le2 þ Ri a22 Le2 þ 4p2 Ri Le2 4p2 a22 Le2 ; x3 ¼32RaT a2 a2 ð1 þ LeSr ÞðRi þ 4p2 ÞF þ 32LeRas a2 Ri a22 ðRi þ 4p2 ÞB 6a2 a21 Ri a22 ðRi þ 4p2 Þ: The required root of Eq. (65) is
x2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 x2 þ x22 4x1 x3 : 2x1
ð66Þ
Also the expressions for B2 and C2 which appear in Eqs. (61) and (62) are
B2 ¼
2a2 p2 A21 F 2R2i 8Ri p2 2a22 Ri þ 8a22 p2 þ a2 p3 A21
ð67Þ
and C2 ¼
2A21 a2 Le2 B
pðLe2 a2 A21 þ 8a2 Þ 4a2 Sr RaT ðRi Le þ 4p3 þ 4Lep2 ÞLea2 A21 F : 2 2Ri 8Ri p2 2a22 Ri þ 8a22 p2 þ a2 p3 A21 Le2 a2 A21 þ 8a2 pRas
ð68Þ
References
Acknowledgments
4a22 a4 Lep2 þ R3i a2 Le2 þ a42 a2 Le2 Ri 4Ri a4 Lep2 þ 4a6 p2 2a22 a2 Le2 R2i þ 4a21 VaLep2 a2 Lea21 Vaa2 Ri Le2 a2 VaRas Ri e2 o þVaLe a22 þ Ri LeRas a2 þ a21 a2 LeR2i a22 LeRi 4a2 p2 Sr 4Le2 a2 p2 e4 4p2 Le2 Ri a21 Va þ a6 Le2 a2 VaRas Le2 a22 a21 Va e2 i 4Vaa2 p2 a22 þ Ri LeRas a2 þ a21 a2
X ¼4a2 p2 VaLe2 Ras a2 Sr þ Ri a22 þ
ð63Þ
[1] D.A. Nield, A. Bejan, Convection in Porous Media, third ed., Springer-Verlag, New York, 2006. [2] D.B. Ingham, I. Pop, Transport Phenomena in Porous Media, vol. III, Pergamon, Oxford, 2005. [3] K. Vafai, Handbook of Porous Media, second ed., Taylor & Francis Group, LLC, US, 2005. [4] C.W. Horton, F.T. Rogers, Convection currents in a porous medium, J. Appl. Phys. 16 (1945) 367–370. [5] E.R. Lapwood, Convection of a fluid in a porous medium, Proc. Camb. Philos. Soc. 44 (1948) 508–521. [6] M. Haajizadeh, A.F. Ozguc, C.L. Tien, Natural convection in a vertical porous enclosure with internal heat generation, Int. J. Heat Mass Transfer 27 (1984) 1893–1902. [7] S. N Gaikwad, M.S. Malashetty, K.R. Prasad, An analytical study of linear and non-linear double diffusive convection with Soret and Dufour effects in couple stress fluid, Int. J. Non-Linear Mech. 42 (2007) 903–913. [8] M.C. Charrier-Mojtabi, B. Elhajjar, A. Mojtabi, Analytical and numerical stability analysis of Soret-driven convection in a horizontal porous layer, Phys. Fluids 19 (2007) 124104. [9] M.S. Malashetty, M. Swamy, The effect of rotation on the onset of convection in a horizontal anisotropic porous layer, Int. J. Therm. Sci. 46 (2007) 1023–1032. [10] F. Capone, M. Gentile, A.A. Hill, Double-diffusive penetrative convection simulated via internal heating in an anisotropic porous layer with throughflow, Int. J. Heat Mass Transfer 54 (2011) 1622–1626. [11] S.N. Gaikwad, M.S. Malashetty, K.R. Prasad, An analytical study of linear and nonlinear double diffusive convection in a fluid saturated anisotropic porous layer with Soret effect, Appl. Math. Model. 33 (2009) 3617–3635. [12] G. Castinel, M. Combarnous, Critere d apparition de la convection naturelle dans une couche poreuse anisotrope horizontale, C.R. Acad. Sci. B 278 (1974) 701–704.
A.A. Altawallbeh et al. / International Journal of Heat and Mass Transfer 59 (2013) 103–111 [13] O. Kvernvold, P.A. Tyvand, Nonlinear thermal convection in anisotropic porous media, J. Fluid Mech. 90 (1979) 609–624. [14] P.A. Tyvand, L. Storesletten, Onset of convection in an anisotropic porous medium with oblique principal axes, J. Fluid Mech. 226 (1991) 371– 382. [15] S. Govender, On the effect of anisotropy on the stability of convection in a rotating porous media, Transp. Porous Med. 64 (2006) 413–422. [16] B.S. Bhadauria, A. Kumar, J. Kumar, N.C. Sacheti, P. Chandran, Natural convection in a rotating anisotropic porous layer with internal heat generation, Transp. Porous Med. 90 (2011) 687–705. [17] B.S. Bhadauria, Double-diffusive convection in a saturated anisotropic porous layer with internal heat source, Transp. Porous Med. 92 (2012) 299–320. [18] C. Parthiban, P.R. Parthiban, Thermal instability in an anisotropic porous medium with internal heat source and inclined temperature gradient, Int. Commun. Heat Mass Transfer 24 (7) (1997) 1049–1058. [19] M.V. Joshi, U.N. Gaitonde, S.K. Mitra, Analytical study of natural convection in a cavity with volumetric heat generation, ASME J. Heat Transfer 128 (2006) 176– 182. [20] N. Rudraiah, M.S. Malashetty, The influence of coupled molecular diffusion on double diffusive convection in a porous medium, ASME J. Heat Transfer 108 (1986) 872–878.
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[21] N. Rudraiah, P.G. Siddheshwar, A weak nonlinear stability analysis of double diffusive convection with cross-diffusion in a fluid saturated porous medium, Heat Mass Transfer 33 (4) (1998) 287–293. [22] A. Bahloul, N. Boutana, P. Vasseur, Double-diffusive and Soret-induced convection in a shallow horizontal porous layer, J. Fluid Mech. 491 (2003) 325–352. [23] A. Mansour, A. Amahmid, M. Hasnaoui, M. Bourich, Multiplicity of solutions induced by thermosolute convection in a square porous cavity heated from below and subjected to horizontal concentration gradient in the presence of Soret effect, Numer. Heat Transfer Part A 49 (2006) 69–94. [24] M.S. Malashetty, B.S. Biradar, Linear and nonlinear double-diffusive convection in a fluid-saturated porous layer with cross-diffusion effects, Transp. Porous Med. 91 (2012) 649–675. [25] M.S. Malashetty, I. Pop, P. Kollur, W. Sidrama, Soret effect on double diffusive convection in a Darcy porous medium saturated with a couple stress fluid, Int. J. Therm. Sci. 53 (2011) 130–140. [26] M.S. Malashetty, M. Swamy, Linear and non-linear double convection in a fluid saturated anisotropic porous layer, in: Proceedings of the International Conference on Advances in Applied Mathematics, 2005, pp. 253–264. [27] L. Storesletten, Effects of anisotropy on convective flow through porous media, in: Derek B. Ingham, I. Pop (Eds.), Transport Phenomena in Porous Media, Pergamon Press, Oxford, 1998, p. 261283.
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Linear and nonlinear double-diffusive convection in a saturated anisotropic porous layer with Soret effect and internal heat source A.A. Altawallbeh a, B.S. Bhadauria b, I. Hashim a,c,⇑ a
School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi Selangor, Malaysia Department of Applied Mathematics & Statistics, School of Physical Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow 226025, UP, India c Solar Energy Research Institute, Universiti Kebangsaan Malaysia, 43600 Bangi Selangor, Malaysia b
a r t i c l e
i n f o
Article history: Received 3 August 2012 Received in revised form 19 October 2012 Accepted 3 December 2012 Available online 5 January 2013 Keywords: Double diffusion Porous layer Soret effect Internal heat
a b s t r a c t Double-diffusive convection in an anisotropic porous layer heated and salted from below with an internal heat source and Soret effect is studied analytically using both linear and nonlinear stability analyses. The generalized Darcy model including the time derivative term is employed for the momentum equation. The critical Rayleigh number and wavenumber for stationary and oscillatory modes and frequency of oscillations are obtained analytically using linear theory. The effects of the anisotropy parameters, concentration Rayleigh number, internal heat source, Soret parameter, Vadasz number and Lewis number on the stationary, oscillatory, and heat and mass transfer are shown graphically. Heat and mass transfer have been obtained in terms of the Nusselt number and Sherwood number. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The problem of double-diffusive convection in porous media has received considerable attention in recent years, on account of its wide applications in, for example, the effect of contaminants on lakes and under ground water, atmospheric pollution, chemical processes, food processing, energy storage, and materials processing. In double diffusion, the buoyancy force is affected not only by the difference of temperature, but also by the difference of concentration of the fluid. A detailed discussion on double diffusion natural convection can be found in the books by Nield and Bejan [1], Ingham and Pop [2] and Vafai [3]. The onset of thermal instability in a horizontal porous layer saturated with Newtonian fluid was first studied by Horton and Rojers [4] and Lapwood [5]. Other studies on linear and nonlinear double diffusive convection in a porous layer were given by Haajizadeh et al. [6], Gaikwad et al. [7], Charrier-Mojtabi et al. [8], Malashetty and Swamy [9] and Capone et al. [10]. Early studies on convection in a porous medium have usually been concerned with homogeneous isotropic porous structures. In the last one decade, the effects of non-homogeneity and anisotropy of the porous medium have been studied. The geological and pedagogical processes rarely form isotropic media as is usually as⇑ Corresponding author at: Solar Energy Research Institute, Universiti Kebangsaan Malaysia, 43600 Bangi Selangor, Malaysia. Tel.: +60 3 8921 5758; fax: +60 3 8925 4519. E-mail address: [email protected] (I. Hashim). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.12.005
sumed in transport studies. In geothermal system with a ground structure composed of many strata of different permeability, the overall horizontal permeability may be up to 10 times as large as the vertical component. Processes such as sedimentation, compaction, frost action, and reorientation of the solid matrix are responsible for the creation of anisotropic natural porous media. Anisotropy can also be a characteristic of artificial porous material like pelleting used in chemical engineering process, fiber materials used in insulating purposes. Anisotropy finds also application in mathematical modeling in geothermal systems such as fractured rocks [11]. The first study in a fluid layer saturated anisotropic porous medium was conducted by Castinel and Combarnous [12]. Many other studies were conducted in a fluid layer saturated anisotropic porous medium [13,14]. Recently, many authors have studied the effect of anisotropy on the onset of convection in a porous layer [15,9,16,11,17]. A situation in which a porous medium has its own source of heat can occur through radioactive decay or through, in the present perspective, a relatively weak exothermic reaction which can take place within the porous material. Internal heat generation becomes very important in geophysics, reactor safety analysis, metal waste form development, fire and combustion studies and storage of radioactive materials. The onset of convection in a horizontal layer of an anisotropic porous medium with an internal heat source subjected to an inclined temperature gradient was studied by Parthiban and Parthiban [18]. An analytical solution for small Rayleigh number in a finite container with isothermal walls and uniform heat generation within the porous medium was given by
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Nomenclature d D Da g K Le Nu P Pr q RaT Ras Ri Sh Sr
hight of the porous layer cross diffusion due to T component Darcy number, Kz/d2 gravitational acceleration ^kÞ ^ permeability of the porous medium, K x ð^i^i þ ^j^jÞ þ K z ðk Lewis number jT z =js Nusselt number pressure Prandtl number velocity vector, (u, v, w) DTdK z thermal Rayleigh number, bT gmj Tz DCdK z concentration Rayleigh number, bs gmj T z
internal Rayleigh number Sherwood number s Soret parameter, jDb T z bT
Joshi et al. [19]. Recently, Bhadauria et al. [16] studied the natural convection in a rotating anisotropic porous layer with internal heat generation using a weak nonlinear analysis. In [17], Bhadauria studied double-diffusive natural convection in an anisotropic porous layer in the presence of an internal heat source. If the cross diffusion terms are included in the species transport equations, then the situation will be quite different. Due to the cross diffusion effect, each property gradient has a significant influence on the flux of the other property. A flux of salt caused by a spatial gradient of temperature is called the Soret effect. Similarly, a flux of heat caused by a spatial gradient of concentration is called the DuFour effect. The DuFour coefficient is of order of magnitude smaller than the Soret coefficient in liquids, and the corresponding contribution to the heat flux may be neglected. Many studies can be found in the literature concerning the Soret and DuFour effects. A study by Rudaraiah and Malashetty [20] discussed the doublediffusive convection in a porous medium in the presence of Soret and DuFour effects. In another study, Rudraiah and Siddheshwar [21] investigated a weak nonlinear stability analysis of doublediffusive convection with cross-diffusion in a fluid saturated porous medium. Bahloul et al. [22] studied analytically and numerically a double-diffusive and Soret-induced convection in a shallow horizontal porous layer. Mansour et al. [23] investigated the multiplicity of solutions induced by thermosolutal convection in a square porous cavity heated from below and subject to horizontal solute gradient in the presence of Soret effect. Recently, Malashetty and Biradar [24] carried out an analytical study of linear and nonlinear double-diffusive convection in a fluid-saturated porous layer with Soret and DuFour effects. Also, a study by Malashetty et al. [25] investigated Soret effect on double diffusive convection in a Darcy porous medium saturated with a couple stress fluid. The aim of this work is to study the effects of an internal heat source and Soret effect on the convection in a binary fluid-saturated anisotropic porous layer. The onset criteria for stationary and oscillatory double-diffusive convection are obtained using linear stability analysis. Heat and mass transfer have been studied using nonlinear stability analysis, and the results were presented graphically in terms of the Nusselt number Nu and Sherwood number Sh.
T t Va x, y, z
temperature time Vadasz number, Pr/Da space coordinates
Greek symbols wavenumber bs concentration expansion coefficient bT thermal expansion coefficient g thermal anisotropy parameter jT z thermal diffusivity js concentration diffusivity l dynamic viscosity m kinematic viscosity w stream function q density n mechanical anisotropy parameter
a
tween two parallel horizontal planes at z = 0 and z = d with a distance d apart. A Cartesian frame of reference is chosen in such a way that the origin lies on the lower plane and the z-axis as vertically upward, where the gravity force g is acting vertically downward. Adverse temperature and concentration gradients are applied across the porous layer, and the lower and upper planes are kept, respectively, at temperatures T0 + DT and T0, and concentrations S0 + DS and S0, where DT and DS are temperature difference and concentration difference between the walls, respectively. The Soret effect is considered and the Oberbeck–Boussinesq approximation is assumed to account for the effect of density variations. Under these assumptions, the governing equations are given by Gaikwad et al. [11]
rq¼0
q0 @q / @t
ð1Þ
¼ rP
l K
q þ qg
ð2Þ
r
@T þ ðq rÞT ¼ r ðjT z rTÞ þ QðT T 0 Þ @t
ð3Þ
/
@C þ ðq rÞC ¼ ðjs r2 CÞ þ Dr2 T @t
ð4Þ
q ¼ q0 ½1 bT ðT T 0 Þ bs ðC C 0 Þ
ð5Þ
subject to the following boundary conditions
T ¼ T 0 þ MT; T ¼ T0;
C ¼ C 0 þ MC
C ¼ C0
at z ¼ 0
at z ¼ d
ð6Þ ð7Þ
where q is velocity (u, v, w), l is the dynamic viscosity, Q is internal ^kÞ; ^ jT is heat source, K is the permeability tensor K x ð^i^i þ ^j^jÞ þ K z ðk z the thermal diffusivity, T is temperature, bT is the thermal expansion coefficient, bs is the concentration expansion coefficient, js is concentration diffusivity of the fluid, r is the ratio of heat capacities, D is the cross diffusion due to T components, q is the density, g = (0, 0, g) is the gravitational acceleration and q0 is the reference density. 2.1. Basic state
2. Mathematical formulation The physical model under consideration is an infinite horizontal anisotropic porous layer with internal heat source, confined be-
The basic state of the fluid is assumed to be quiescent, and is given by
qb ¼ ð0;0; 0Þ; P ¼ Pb ðzÞ; T ¼ T b ðzÞ; C ¼ C b ðzÞ; q ¼ qb ðzÞ
ð8Þ
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A.A. Altawallbeh et al. / International Journal of Heat and Mass Transfer 59 (2013) 103–111
"
From Eqs. (1)–(8), we obtain
dPb ¼ qg dz
@ 1 @2 @2 þ @t Le @z2 @x2
ð9Þ
!# S ¼ Sr þ
2
d T jT z 2b þ Q ðT b T 0 Þ ¼ 0 dz 2
ð10Þ
2
d C d T js 2b þ D 2b ¼ 0 dz dz
ð11Þ
where the subscript b refers to the basic state. The conduction state solutions are given by
T b ðzÞ ¼ T 0 þ DT
C b ðzÞ ¼
sin
DDT sin
js
pffiffiffiffiffiffiffiffiffiffiffiffiffi Q=jT z 1 dz pffiffiffiffiffiffiffiffiffiffiffiffiffi sin Q =jT z
ð12Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi Q =jT z 1 dz z DDT pffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 DC þ d js sin Q=jT z
þ C0 :
ð13Þ
T ¼ Tb þ T0;
C ¼ Cb þ C0;
P ¼ Pb þ P0 ;
¼ qb þ q
ð14Þ
where the primes denote infinitesimally small quantities. Now, substitute Eq. (14) into Eqs. (1)–(5), and using the basic states (9)–(11) we obtain
r q0 ¼ 0
r
jT z
¼ rP
l
0
0
0
q þ ðq0 gbT T q0 gbs C Þ
K
ð16Þ
@T 0 dT b þ ðq0 rÞT 0 þ w0 ¼ r ðjT rT 0 Þ þ QT 0 @t dz
ð17Þ
@C 0 dC b / þ ðq0 rÞT 0 þ w0 ¼ js r2 C 0 þ Dr2 T 0 @t dz
ð18Þ
Eliminating the pressure from Eq. (16) we obtain
/ @t @z
@w0 l @u0 @w0 @T 0 @C 0 ¼ q0 g bT bs @x K @z @x @x @x
ð19Þ
For simplicity, the case of two-dimensional rolls have been considered, and thus all physical quantities are made as independent of y. Now, we introduce the stream functions u = @ w/@y, v = @ w/@x, and use the following transformations
ðx ; z Þ ¼ ¼
x z ; ; d d
t ¼
t jT z 2
d
w ¼
;
w
jT z
;
T ¼
T0 ; MT
C0 MC
ð20Þ
2
2
!
2
2
1 @ @ @ @ 1 @ þ þ þ Va @t @z2 @x2 @x2 n @z2
@T @S ¼ RaT þ Ras @x @x
"
2
Va ¼
Pr ; Da
n¼
Kx ; Kz
g¼
and f ðz Þ ¼ dT b ðz Þ=dz and bðz Þ ¼ dSb ðz Þ=dz . T b and Sb in dimensionless forms are given by
T b ðzÞ ¼
pffiffiffiffi sin Ri ð1 z Þ pffiffiffiffi sin Ri
ð24Þ
Sb ðzÞ ¼
pffiffiffiffi Sr LeRaT sin Ri ð1 z Þ Sr LeRaT pffiffiffiffi þ 1 ð1 z Þ Ras sin Ri Ras
ð25Þ
@ 2 w ¼ T ¼ S ¼ 0 at z ¼ 0 and z ¼ 1: @z2
ð26Þ
For the next steps the asterisks are removed, r and / are set equal to unity for simplicity.
3. Linear stability theory
2
#
In this section, the linear stability analysis is discussed. To make the linear stability study, we neglect the Jacobian in Eqs. (21)–(23), and assume the solutions to be periodic waves of the form
2 3 2 3 w W sin ax 6 7 6 7 4 T 5 ¼ eet 4 H cos ax 5 sin pz S U cos ax
ð27Þ
where (W, H, U) are the amplitudes of (w, T, S), e = er + iei is the growth rate and in general a complex quantity and a is a horizontal wave number. Now, substitute (27) into (21)–(23) to obtain
e
Va
a2 þ a21 W ¼ RaT aH þ Ras aU
ð28Þ
e þ a22 Ri H ¼ 2aWF
eþ
S
ð29Þ
a2 Ra U ¼ Sr T Ha2 þ 2aWB Le Ras
ð30Þ
2 where a2 ¼ ðp2 þ a2 Þ; a21 ¼ pn þ a2 and a22 ¼ ðp2 þ ga2 Þ. Now, write Eqs. (28)–(30) in a matrix form
into Eqs. (15) and (17)–(19), we obtain
"
;
ð15Þ 0
q0 @ @u0
2
jT x b gK MTd ; RaT ¼ T z ; jT z mjT z b gK MCd jT Dbs ; m ¼ l=q0 ; Le ¼ z ; Sr ¼ Ras ¼ s z mjT z js jT z bT Qd
Ri ¼
w ¼
q
0
/ @t
ð23Þ
where
On the basic state we superpose perturbation in the form
q0 @q0
@ðw ; S Þ @ðx ; z Þ
In the above equations r and / are set equal to unity for simplicity. Now, Eqs. (21)–(23) are solved for stress-free, isothermal and isohaline conditions
2.2. Perturbed state
q ¼ qb þ q0 ;
RaT 2 @w r T þ bðz Þ Ras @x
2
!#
e Va
a2 þ a21
6 2aF 4 2aB
w ð21Þ
@ @ @ @w @ðw ; T Þ g 2 Ri T ¼ f ðz Þ þ @t @z2 @ðx ; z Þ @x @x
ð22Þ
Ras a
e þ a22 Ri
0
T Sr Ra a2 Ras
e þ aLe2
32
2 3
3
0 W 76 7 6 7 54 H 5 ¼ 4 0 5 0 U
ð31Þ
where
F¼
RaT a
Z 0
1
2
f ðzÞ sin ðpzÞdz;
B¼
Z
1
2
bðzÞ sin ðpzÞdz
0
For the non-trivial solution of (31), it is found that
ð32Þ
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A.A. Altawallbeh et al. / International Journal of Heat and Mass Transfer 59 (2013) 103–111
RaT ¼ ðRi 4p2 Þðe a22 þ Ri Þðe2 a2 Le þ ea4 þ a21 VaeLe þ
a21 Vaa2
2
2
2
þ a VaLeRas Þ=½a VaðLe
2
þ 4p eLe þ Le
2
Sr Ri a22
2
Sr R2i
2
2
þ 4p Sr a Le
2 2
þ Le Sr Ri e þ 4p a Þ
ð33Þ
3.1. Marginal stationary state For the existence of marginal stability, the real part of e must be zero at marginal state. For stationary convection to occur, e must be zero. This means that the real and imaginary part of e are equal to zero. So, from Eq. (33) the stationary Rayleigh number equals to
RastT ¼ ðRi þ 4p2 Þðga2 þ p2 Ri Þða2 LeRas n þ a4 n þ a2 np2 h þ a2 p2 þ p4 Þ= na2 Le2 Sr R2i þ 4a2 p2 Sr Le þ 4p4 Sr Le i þ4a2 p2 þ 4p4 þ Le2 Sr Ri ga2 þ Le2 Sr Ri p2
2
1 p 1 ðga2 þ 1ÞLeRas 2 2 þ ð ga þ 1Þ a þ ðSr Le þ 1Þ a2 n ða2 þ 1Þ
ð34Þ
p2 2 1 ðga2 þ 1Þ þ 2 ðga þ 1Þ a2 þ LeRas 2 n a a þ1
ð36Þ
In the case of one single component (Ras = 0) the result of Storesletten [27] is recovered
RastT ¼
p2 2 1 ðga þ 1Þ a2 þ n a2
ð37Þ
Finally, in the case of an isotropic porous medium, putting g = n = 1, then
RastT ¼
p2 ð1 þ a2 Þ2 a2
ð38Þ
2 2 which has the critical value Rast T ¼ 4p at ac ¼ 1 as obtained previously by Horton and Rojers [4] and Lapwood [5].
3.2. Marginal oscillatory state For marginal oscillatory state, the real part of e must be equal to zero. So, putting e = ie in Eq. (33), the expression for Raosc is obT tained as 2 2 2 2 2 2 2 2 Raosc T ¼ x=½a VaLe ðSr LeRi þ 4p Þ e þ a VaðSr Le Ri a2
4p2 Sr a2 Le 4a2 p2 þ
Sr Le2 R2i Þ2
ð39Þ
where e2 is given by
e2 ¼
" # " ## 1 @ @2 @2 @2 1 @2 @T @S þ w þ RaT Ras ¼0 þ þ Va @t @z2 @x2 @x @x @x2 n @z2 # @ @2 @2 @w @T @w @T @w þ ¼ f ðzÞ g 2 Ri T @t @z2 @x @z @z @x @x @x
"
@ 1 @2 @2 þ @t Le @z @x2 ¼
ð35Þ
In the case of no Soret effect and internal heat source, we recover the result given by Malashetty and Swamy [26]
RastT ¼
"
ð41Þ
"
Now, rescale a2 = a2p2, and so when the internal heat source is absent we recover the result of Gaikwad et al. [11]
RastT ¼
temperature, and concentration. This study will help in understanding the physics of the problem with minimum mathematical expressions. Further, the results can be used as a starting point to generalize it for the full nonlinear problem. Also it is to be noted that the effect of nonlinearity is to distort the temperature and concentration fields through the interaction of w and T, and w and S, respectively. As a result, a component of the form sin (2pz) will be generated. Dropping the asterisks and rearranging the terms, Eqs. (21)–(23) become
X
: Le2 Lea21 VaSr Ri 4a4 p2 Sr þ 4a22 a2 p2 4Ri a2 p2 þ 4a21 Vap2 þ a4 Sr Ri ð40Þ
The expressions for x and X which appear in (39) and (40) are given in Appendix A. 4. A weak nonlinear theory For flows with Ra > Rast T , the linear stability analysis is not valid, because of the missing nonlinear effects. In this section, a local nonlinear stability analysis shall be performed by using a severely truncated representation of Fourier series for the stream function,
!# S
ð42Þ
@w @S @w @S RaT 2 þ Sr r T @x @z @z @x Ras
@w bðzÞ: @x
ð43Þ
Now consider a minimal Fourier series with one term in the stream function, and to get some effects of non-linearity, take two terms in the temperature and concentration fields as given below
w ¼ A1 ðtÞ sinðaxÞ sinðpzÞ
ð44Þ
T ¼ B1 ðtÞ cosðaxÞ sinðpzÞ þ B2 ðtÞ sinð2pzÞ
ð45Þ
S ¼ C 1 ðtÞ cosðaxÞ sinðpzÞ þ C 2 ðtÞ sinð2pzÞ
ð46Þ
where the amplitudes A1(t), B1(t), B2(t), C1(t) and C2(t) are functions of time and are to be determined. Substituting (44)–(46) in (41)– (43) and equating the coefficients of like terms of the resulting equations, we obtain the following system of nonlinear differential equations
dA1 ðtÞ Vaa21 A1 ðtÞ VaaRaT B1 ðtÞ VaaRas C 1 ðtÞ ¼ þ dt a2 a2 a2
ð47Þ
dB1 ðtÞ ¼ 2aFA1 ðtÞ þ Ri a22 B1 ðtÞ apA1 ðtÞB2 ðtÞ dt
ð48Þ
dB2 ðtÞ 1 ¼ apA1 ðtÞB1 ðtÞ þ ðRi 4p2 ÞB2 ðtÞ dt 2
ð49Þ
dC 1 ðtÞ a2 a2 Sr RaT ¼ 2aBA1 ðtÞ C 1 ðtÞ B1 ðtÞ apA1 ðtÞC 2 ðtÞ dt Le Ras
ð50Þ
dC 2 ðtÞ a 4p2 4p2 Sr RaT B2 ðtÞ ¼ pA1 ðtÞC 1 ðtÞ C 2 ðtÞ dt 2 Le Ras
ð51Þ
The nonlinear system of autonomous differential equations is not suitable to analytical treatment for the general time-dependent variables and we have to solve it numerically. After determining the numerical values of the amplitude functions A1(t), B1(t), B2(t), C1(t) and C2(t), the Nusselt number and Sherwood number can be obtained as functions of time. The steady analysis is performed by setting the left hand side of Eqs. (47)–(51) equal to zero,
Vaa21 A1 ðtÞ VaaRaT VaaRas B1 ðtÞ þ C 1 ðtÞ ¼ 0 a2 a2 a2
2aFA1 ðtÞ þ ðRi a22 ÞB1 ðtÞ apA1 ðtÞB2 ðtÞ ¼ 0
ð52Þ ð53Þ
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107
(a)
(b)
Fig. 3. Effect of Ri on the marginal stability curves for stationary and oscillatory convection for the case n = 0.5, g = 0.8, Ras = 25, Sr = 0.05, Le = 20 and Va = 12.
Fig. 1. Effects of (a) g when n = 0.3 and (b) n when g = 0.8 on the marginal stability curves for stationary and oscillatory convection for the case Ri = 3, Sr = 0.05, Le = 20, Ras = 25, Va = 12. Fig. 4. Effect of Va on the marginal stability curves for stationary and oscillatory convection for the case n = 0.5, g = 0.8, Ri = 3, Sr = 0.05, Le = 20 and Ras = 25.
(a)
(b)
Fig. 5. Effect of Sr on the marginal stability curves for stationary and oscillatory convection for the case n = 0.5, g = 0.8, Ri = 3, Va = 12, Le = 10 and Ras = 50.
1 apA1 ðtÞB1 ðtÞ þ ðRi 4p2 ÞB2 ðtÞ ¼ 0 2
2aBA1 ðtÞ
a 2 Fig. 2. Effects of (a) Ras when Le = 10 and (b) Le when Ras = 50 on the marginal stability curves for stationary and oscillatory convection for the case n = 0.5, g = 0.8, Ri = 3, Sr = 0.05 and Va = 12.
a2 a2 Sr RaT B1 ðtÞ apA1 ðtÞC 2 ðtÞ ¼ 0 C 1 ðtÞ Le Ras
pA1 ðtÞC 1 ðtÞ
4p2 4p2 Sr RaT B2 ðtÞ ¼ 0 C 2 ðtÞ Le Ras
ð54Þ
ð55Þ
ð56Þ
with the consideration that all the amplitudes are constants. The Eqs. (52)–(56) have been solved for B1, B2, C1 and C2 in terms of A1 as given in Appendix A.
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4.1. Steady heat and mass transport
z Stotal ¼ S0 DS þ Sðx; z; tÞ d
Heat and mass transport is a very important issue in studying convection in fluids. This is because the onset of convection, as the Rayleigh number is increased, is more readily detected by its effect on the heat and mass transport. In the basic state, heat and mass transport is by conduction only. If H and J are the rates of heat and mass transport per unit area, respectively, then
Substituting Eqs. (45) and (46) into Eqs. (59) and (60) respectively and using the resultant equations in Eqs. (57) and (58), we get
H ¼ jT
@T total @z
Sh ¼ 1
ð57Þ z¼0
pffiffiffiffi 2pC 2 Ras sinð Ri Þ pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi ; Sr RaT cosð Ri Þ Ri þ sinð Ri ÞRas þ sinð Ri ÞSr RaT
ð61Þ
ð62Þ
where the expressions for B2 and C2 are given in Appendix A.
@Stotal @T total J ¼ js D @z z¼0 @z z¼0
ð58Þ
where the angular bracket represents the horizontal average and
T total
2pB2 pffiffiffiffi ; Nu ¼ 1 pffiffiffiffi Ri cotð Ri Þ
ð60Þ
z ¼ T 0 DT þ Tðx; z; tÞ d
ð59Þ
5. Results and discussion The onset of double-diffusive convection in a binary fluid-saturated anisotropic porous layer in the presence of Soret effect and an internal heat source is studied analytically using both linear
(a)
(b)
(c)
(d)
(e)
Fig. 6. Variations of Nu with RaT for steady nonlinear stability: effects of (a) n for the case Ri = 1, Sr = 0.005, g = 0.5, Le = 1, Ras = 100, (b) Ras for the case Ri = 1, Sr = 0.005, g = 0.5, Le = 1, n = 0.5, (c) Sr for the case Ri = 1, g = 0.5, n = 0.5, Le = 1, Ras = 100, (d) Ri for the case Sr = 0.005, g = 0.5, n = 0.5, Le = 1, Ras = 100, and (e) g for the case Ri = 1, Sr = 0.005, n = 0.5, Le = 1, Ras = 100.
A.A. Altawallbeh et al. / International Journal of Heat and Mass Transfer 59 (2013) 103–111
and nonlinear analyses. The expressions for the onset of stationary and oscillatory convection are calculated using the linear stability theory in terms of the governing parameters Ras, Le, Sr, Ri, g, n and Va. In the nonlinear theory, the heat and mass transfer are discussed in terms of the Nusselt number and sherwood number. The marginal stability curves for steady convection (Eq. (34)) and oscillatory convection (Eq. (39)) have been presented graphically in Figs. 1–5. It is found that, in all the cases investigated, the instability sets in via the oscillatory mode. The effects of the anisotropy parameters on the marginal stability curves are shown in Fig. 1. The effect of increasing g is to increase both the oscillatory and steady marginal curves, and hence the system is stabilized (see Fig. 1(a)). In Fig. 1(b), it is observed that increasing n destabilizes the system for both the stationary and oscillatory modes. Fig. 2 indicates the effect of the concentration Rayleigh number Ras and Lewis number Le on the stationary and oscillatory convection. The effect of Ras on the oscillatory marginal curve is quite small, but quite large on the stationary curve as shown in Fig. 2(a). The
109
effect of increasing Ras is stabilizing. While the opposite is observed for the effect of the Lewis number Le. The destabilizing effect of the internal Rayleigh number Ri is depicted in Fig. 3. Fig. 4 shows that the Vadasz number Va has no effect on the steady curve and a negligible effect on the oscillatory curve. The effect of the Soret parameter, Sr, is depicted in Fig. 5. Both negative and positive values of Sr were chosen. It is observed that the Soret parameter has a negligible effect on the oscillatory curves. On the other hand, the negative (positive) Soret parameter has a stabilizing (destabilizing) effect on the stationary curves. In the study of double-diffusive convection, heat and mass transport are important. Heat and mass transport are given in terms of the Nusselt number Nu and Sherwood number Sh respectively, which represent the ratio of heat or mass transported across the layer to the heat and mass transported by conduction alone. Fig. 6 shows the results obtained from Eq. (61). Fig. 6(a) shows that an increase of n increases Nu, which enhances the heat transfer. Fig. 6(d) shows that Ri has the effect of increasing the heat transfer. On the other hand, increasing
(a)
(b)
(c)
(d)
(e)
Fig. 7. Variations of Sh with RaT for steady nonlinear stability: effects of (a) n for the case Ri = 1, Sr = 0.005, g = 0.5, Le = 1, Ras = 100, (b) Ras for the case Ri = 1, Sr = 0.005, g = 0.5, Le = 1, n = 0.5, (c) Sr for the case Ri = 1, g = 0.8, n = 0.3, Le = 1, Ras = 100, (d) Ri for the case Sr = 0.005, g = 0.5, n = 0.5, Le = 1, Ras = 100, and (e) g for the case Ri = 1, Sr = 0.005, n = 0.5, Le = 1, Ras = 100.
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Ras and g decrease the heat transfer as depicted in Fig. 6(b) and (e). It is noticeable that increasing Sr increases the heat transfer (see Fig. 6(c)). However, for RaT > 180 the effect of varying Sr diminishes. The effects of the various parameters on heat mass transfer that can be determined from Eq. (62) are shown graphically in Fig. 7. Fig. 7(a) clearly shows that the effect of the mechanical anisotropy parameter n is to enhance the mass transfer. However, increasing Ras has the opposite effect as depicted in Fig. 7(b). Fig. 7(c) depicts the effect of the Soret parameter on the Sherwood number. It is found that there is a critical value of RaT below which increasing Sr enhances the mass transfer and above which increasing Sr decreases the mass transfer. The effect of Ri on the mass transfer is small for low values of RaT and then becomes insignificant for RaT (see Fig. 7(d)). The effect of increasing the thermal anisotropy parameter g is initially to decrease the mass transfer, but as RaT is increased further the opposite effect is observed. 6. Conclusion The onset of double-diffusive convection in a binary fluid-saturated anisotropic porous layer in the presence of Soret effect and internal heat source is studied analytically using linear and nonlinear stability analysis. The porous layer is heated and salted from below. For the stationary and oscillatory modes, it is observed that increasing the values of mechanical anisotropy parameter n, Lewis number Le, and internal Rayleigh number Ri destabilizes the system. On the other hand, increasing the values of thermal anisotropy parameter g and concentration Rayleigh number Ras is to stabilize the system. The positive values of Sr decreases the marginal curves for the stationary convection. While, the negative ones increases them. The effect of the Soret parameter on the oscillatory curves is negligible. The effects of the mechanical anisotropy parameter n and internal Rayleigh number Ri are to enhance the heat and mass transfer as they are increased. On the other hand, increasing the concentration Rayleigh number Ras and thermal anisotropy parameter g has the effect of decreasing the heat and mass transfer. Increasing the Soret parameter enhances the heat transfer. We also found that there is a critical value of RaT below which increasing Sr enhances the mass transfer and above which increasing Sr decreases the mass transfer.
Part of this study was done during the visit of the author B.S. Bhadauria (BSB) to the Universiti Kebangsaan Malaysia (UKM), in June 2012 as a visiting Professor of Mathematics. The authors gratefully acknowledge the Grant provided by UKM out of the University Research Fund OUP-2012-61 and DIP-2012-12. The author BSB is also grateful to Banaras Hindu University, Varanasi for sanctioning the lien to work as Professor of Mathematics at Department of Applied Mathematics, BB Ambedkar University, Lucknow, India. Appendix A. hn
x ¼ðRi þ 4p2 Þ Le3 a2 Ri e4 Le
and
a2 Le
h i h þ Sr a4 þ a21 VaLe Le2 R3i 2Le2 a22 R2i þ LeRi Lea22 4a2 p2 i þ4Lea2 p2 a22 a4 þ a21 Vaa2 þ a22 a21 VaLe þ 4p2 a2 Ri a2 þ a21 Va þ a2 a22 : ð64Þ The Eqs. (52)–(56) have been solved for A21 , with x ¼ A21 , we get
x1 x2 þ x2 x þ x3 ¼ 0;
ð65Þ
where
x1 ¼ p2 a21 a4 Le2 ; x2 ¼ 4Le2 a4 RaT ð4p2 Sr Ri þ 4p2 ÞF 16a4 BRas p2 2a21 a2 4p2 a2 R2i Le2 þ Ri a22 Le2 þ 4p2 Ri Le2 4p2 a22 Le2 ; x3 ¼32RaT a2 a2 ð1 þ LeSr ÞðRi þ 4p2 ÞF þ 32LeRas a2 Ri a22 ðRi þ 4p2 ÞB 6a2 a21 Ri a22 ðRi þ 4p2 Þ: The required root of Eq. (65) is
x2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 x2 þ x22 4x1 x3 : 2x1
ð66Þ
Also the expressions for B2 and C2 which appear in Eqs. (61) and (62) are
B2 ¼
2a2 p2 A21 F 2R2i 8Ri p2 2a22 Ri þ 8a22 p2 þ a2 p3 A21
ð67Þ
and C2 ¼
2A21 a2 Le2 B
pðLe2 a2 A21 þ 8a2 Þ 4a2 Sr RaT ðRi Le þ 4p3 þ 4Lep2 ÞLea2 A21 F : 2 2Ri 8Ri p2 2a22 Ri þ 8a22 p2 þ a2 p3 A21 Le2 a2 A21 þ 8a2 pRas
ð68Þ
References
Acknowledgments
4a22 a4 Lep2 þ R3i a2 Le2 þ a42 a2 Le2 Ri 4Ri a4 Lep2 þ 4a6 p2 2a22 a2 Le2 R2i þ 4a21 VaLep2 a2 Lea21 Vaa2 Ri Le2 a2 VaRas Ri e2 o þVaLe a22 þ Ri LeRas a2 þ a21 a2 LeR2i a22 LeRi 4a2 p2 Sr 4Le2 a2 p2 e4 4p2 Le2 Ri a21 Va þ a6 Le2 a2 VaRas Le2 a22 a21 Va e2 i 4Vaa2 p2 a22 þ Ri LeRas a2 þ a21 a2
X ¼4a2 p2 VaLe2 Ras a2 Sr þ Ri a22 þ
ð63Þ
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