Transp Porous Med (2012) 94:339–357 DOI 10.1007/s11242-012-0008-y
Onset of Darcy–Brinkman convection in a binary viscoelastic fluid saturated porous layer Mahantesh S. Swamy · N. B. Naduvinamani · W. Sidram
Received: 2 February 2012 / Accepted: 5 April 2012 / Published online: 2 May 2012 © Springer Science+Business Media B.V. 2012
Abstract The onset of Darcy–Brinkman double-diffusive convection in a binary viscoelastic fluid-saturated porous layer is studied using both linear and weakly nonlinear stability analyses. The Oldroyd-B model is employed to describe the rheological behavior of the fluid. An extended form of Darcy–Oldroyd law incorporating the Brinkman’s correction and time derivative is used to describe the fluid flow and the Oberbeck–Boussinesq approximation is invoked. The onset criterion for stationary and oscillatory convection is derived analytically. The effects of rheological parameters, Darcy number, normalized porosity, Lewis number, solute Rayleigh number, and Darcy–Prandtl number on the stability of the system is investigated. The results indicated that there is a competition among the processes of thermal, solute diffusions and viscoelasticity that causes the convection to set in through the oscillatory modes rather than the stationary. The Darcy–Prandtl number has a dual effect on the threshold of oscillatory convection. The nonlinear theory based on the method of truncated representation of Fourier series is used to find the transient heat and mass transfer. Some existing results are reproduced as the particular cases of present study. Keywords Double-diffusive convection · Viscoelastic fluid · Porous layer · Heat mass transfers
M. S. Swamy (B) Department of Mathematics, Government College, Gulbarga 585105, India e-mail:
[email protected] N. B. Naduvinamani · W. Sidram Department of Mathematics, Gulbarga University, Jnana Ganga, Gulbarga 585106, India e-mail:
[email protected] W. Sidram e-mail:
[email protected]
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List of Symbols a Overall horizontal wavenumber c Specific heat Da Darcy number, μe K /μd 2 d Height of the porous layer g Gravitational acceleration Le Lewis number, κT /κS l, m Horizontal wavenumbers Nu Nusselt number p Pressure q Velocity vector, (u, v, w) RaT Darcy–Rayleigh number, βT gT d K /νκT RaS Solute Rayleigh number, βS gSd K /νκT S Solute concentration Sh Sherwood number T Temperature t Time Pr D Darcy–Prandtl number, φγ νd 2 /κT K x, y, z Space coordinates
Greek Symbols β Coefficient of expansion φ Porosity γ Ratio of specific heats, (ρc)m / ρc p f χ Normalized porosity, φ/γ κ Diffusivity λ Stress-relaxation time ε¯ Strain-retardation time λ Relaxation parameter, κT /γ d 2 λ ε Retardation parameter, κT /γ d 2 ε¯ μ Dynamic viscosity μe Effective viscosity ν Kinematic viscosity, μ/ρ0 ρ Fluid density ω Growth rate, τ + iσ ψ Stream function Subscripts/Superscripts b Basic state c Critical f Fluid phase T Thermal S Solutal Osc Oscillatory St Stationary 0 Reference state
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Non-dimensional Perturbed quantity
1 Introduction Double-diffusive convection (DDC) in porous media occurs in many systems, and this problem has attracted considerable interest over the years due to its numerous fundamental and industrial applications, such as high-quality crystal production, liquid gas storage, migration of moisture in fibrous insulation, transport of contaminants in saturated soil, solidification of molten alloys, and geothermally heated lakes and magmas. Extensive reviews of the literature on this subject can be found in the books by Ingham and Pop (1998, 2005), Vafai (2000, 2005), Nield and Bejan (2006), and Vadasz (2008). With the growing importance of non-Newtonian fluids in modern technology and also due to their natural occurrence, the investigations on such fluids are quite desirable. In particular, the flow of viscoelastic fluid is of considerable importance in several fields of applications such as material processing, petroleum, chemical, and nuclear industries, CO2 -geologic sequestration, bioengineering, and reservoir engineering. Although the problem of Rayleigh– Benard convection (RBC) has been extensively investigated for Newtonian fluids, relatively little attention has been devoted to the thermal convection of viscoelastic fluids (see e.g., Li and Khayat 2005 and references therein). The study of RBC in viscoelastic fluid may be important from a rheological point of view because the observation of the onset of convection provides potentially useful techniques to investigate the suitability of a constitutive model adopted for certain viscoelastic fluids. Some oil sands contain waxy crudes at shallow depth of the reservoirs which are considered to be viscoelastic fluid. In these situations, a viscoelastic model of fluid serves to be more realistic than the Newtonian model. Besides this viscoelastic fluids exhibit unique patterns of instabilities such as the overstability that is not predicted or observed in Newtonian flow. The related problem in the case of porous medium has not received much attention until recently. The published work on thermal convection of viscoelastic fluids in porous media is fairly limited. Herbert (1963) and Green (1968) were the first to analyze the problem of oscillatory convection in an ordinary viscoelastic fluid of the Oldroyd type under the condition of infinitesimal disturbances. Rudraiah et al. (1989, 1990) studied the stability of viscoelastic fluid saturated porous layer using Darcy and Brinkman models. A theoretical analysis of thermal instability in a porous layer saturated with viscoelastic fluid is carried out by Kim et al. (2003). Later on, Yoon et al. (2003, 2004) analyzed the onset of thermal convection in a horizontal porous layer saturated with viscoelastic liquid using a linear theory. Malashetty et al. (2006) and Shivakumara et al. (2006) studied the linear stability of Darcy and Brinkman porous layer respectively, saturated with Oldroyd-B fluid using a thermal nonequilibrium model. The instability of viscoelastic fluid saturating a horizontal porous layer is studied theoretically by Bertola and Cafaro (2006) by using a dynamical system approach. Sheu et al. (2008) derived, a novel, unified system with six-order dynamics of chaotic convection to investigate thermal convection in both pure Newtonian/viscoelastic fluid layers and the saturated porous media. Bertola and Cafaro (2008) have also studied the chaotic convection of a viscoelastic fluid in a porous medium. Recently, based on the modified Darcy–Brinkman– Oldroyd model Zhang et al. (2008) analyzed the buoyancy-driven motion of an Oldroyd-B fluid saturated in a horizontal porous layer. More recently, a stability analysis of rotating viscoelastic fluid saturated porous layer heated from below is performed by Malashetty et al. (2010).
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However, there are few studies available on the RBC in binary viscoelastic fluids (Martinez-Mardones et al. 2000, 2003; Laroze et al. 2007a,b) and the corresponding problem for the case of porous medium has also not been given much attention. Wang and Tan (2008) made the stability analysis of DDC of Maxwell fluid in a porous medium. Recently, Malashetty et al. (2009a,b) studied the DDC in a viscoelastic fluid saturated isotropic and anisotropic porous layer, respectively. More recently, Malashetty et al. (2011) performed linear and weakly nonlinear stability analysis of DDC in a rotating anisotropic porous layer saturated with viscoelastic fluid. For the low porosity media, the viscous effects near the boundary are negligible. In such situations Darcy’s law is a good approximation for the momentum equation. The main advantage of Darcy’s flow model is that it linearizes the momentum equation and thus reduces a significant amount of difficulty in solving the governing equations. Further, the classical Darcy model is valid for flow through regular structures over the whole spectrum of the porosity. This model is silent about the flow structure near the bounding surfaces where close packing of the porous material is not possible. It is well known that many applications in engineering disciplines as well as in circumstances linked to modern porous media involve high permeability porous layer. For instance, in biomedical hydrodynamic studies, a thin fibrous surface layer coating blood vessels (endothelial surface layer) is found to be a highly permeable, high porosity porous medium (Khaled and Vafai 2003). In such circumstances the use of a non-Darcy model, which takes care of boundary and/or inertia effects is of fundamental and practical interest to obtain accurate results. Further, it is believed that the results of this study are useful in bridging the gap between a non-porous case in which Da→∞ and a dense porous medium in which Da→0. A better understanding of the characteristics of the Darcy–Brinkman equation is therefore an important part of more practical problems and thus forms a motivation of the present report. We shall employ Brinkman’s model which has a Laplacian term analogous to that appearing in the Navier–Stokes equations. It removes some of the deficiencies and gives preferable results for flows in high porosity porous media. An extensive literature is available on the use of non-Darcian models to study flow and heat transfer in porous media in the recent past (Vafai 2000, 2005; Nield and Bejan 2006). Recently, the combined influence of horizontal and vertical heterogeneity on the DDC in a Brinkman porous layer has been studied by Kuznetsov and Nield (2008). More recently, Kuznetsov and Nield (2010, 2011) have performed a linear stability analysis of thermal convection in a porous layer saturated with a nanofluid using local thermal equilibrium and local thermal non equilibrium cases respectively. However, the works on thermal convection in a binary viscoelastic fluid saturated porous layer based upon the non-Darcian models are very sparse and are in much-to-be desired state. Therefore, in the present study, we intend to perform linear and weakly nonlinear stability analyses of a Darcy–Brinkman binary viscoelastic fluid saturated porous layer. Our objective is to study how the onset criterion for oscillatory convection is affected by the viscoelastic and other parameters, and also to know their effect on heat and mass transfer in a more general porous medium. In the limiting cases, some previously published results can be recovered as the particular cases of our results.
2 Mathematical Formulation We consider an infinite horizontal sparsely packed porous layer saturated with Oldroyd-B fluid and confined between the planes z = 0 and z = d, with vertically downward gravity g acting on it. A uniform adverse temperature gradient T = Tl − Tu and a stabilizing
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concentration gradient S = Sl − Su (Tl > Tu and Sl > Su ) are maintained between the lower and upper surfaces. A Cartesian frame of reference is chosen with origin in the lower boundary and z-axis vertically upwards. The interaction between heat and mass transfer, known as Soret and Dufour effects, are supposed to have no influence on the convective flow, so they are ignored. Homogeneity and local thermal equilibrium in the porous medium are assumed. The modified Darcy–Brinkman–Oldroyd model, which includes the time derivative, is employed as a momentum equation (see Zhang et al. 2008). The velocities are assumed to be small so that the advective and Forchheimer inertia effects are ignored. The equations governing the above system under the Boussinesq approximation are ∇ · q = 0, ∂ ρ0 ∂q ∂ μ 1 + λ¯ − q + μe ∇ 2 q , + ∇ p − ρg = 1 + ε¯ ∂t φ ∂t ∂t K ∂T + (q · ∇) T = κT ∇ 2 T, γ ∂t ∂S φ + (q · ∇) S = κS ∇ 2 S, ∂t ρ = ρ0 [1 − βT (T − T0 ) + βS (S − S0 )] .
(1) (2) (3) (4) (5)
The constants and variables in the above equations have their usual meaning, as given in the Nomenclature. Furthermore, γ = (ρc)m /(ρc p )f , where(ρc)m = (1 − φ)(ρc)s + φ(ρc p )f . The basic state is assumed to be quiescent and the variables except qb are functions of z alone. To study the stability of the system we superpose infinitesimal perturbation on the basic state so that the equations governing the perturbed quantities are given by ∇ · q = 0, (6) ∂ ρ0 ∂q μ ∂ 1 + λ¯ + ∇ p − ρ0 βT T − βS S g = 1 + ε¯ − q + μe ∇ 2 q , ∂t φ ∂t ∂t K (7) T ∂T (8) + q · ∇ T − w = κT ∇ 2 T , γ ∂t d S ∂ S (9) φ + q · ∇ S − w = κS ∇ 2 S , ∂t d (10) ρ = −ρ0 βT T − βS S . By operating curl twice on Eq. (7) we eliminate p and then use the scalings x , y , z = (x ∗ , y ∗ , z ∗ )d, t = γ d 2 /κT t ∗ , q = (κT /d) q∗ , T = (T ) T ∗ , S = (S) S ∗ , (11) to non-dimensionalize Eqs. (6)–(10) in the form (on dropping the asterisks), 1 ∂ 2 ∂ 2 2 1+λ ∇ w − RaT ∇1 T + RaS ∇1 S ∂t PrD ∂t ∂ − 1+ε Da∇ 4 w − ∇ 2 w = 0, ∂t ∂T + (q · ∇) T − w = ∇ 2 T, ∂t ∂S χ + (q · ∇) S − w = Le−1 ∇ 2 S. ∂t
(12) (13) (14)
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All the non-dimensional parameters are as defined in Nomenclature. The buoyancy due to thermal gradient is characterized by Rayleigh number RaT and that due to solute gradient is indicated by solute Rayleigh number RaS , the viscoelastic character of the liquid mixture appears in the relaxation parameter λ (which is also known as the Deborah number) and the retardation parameter ε. The Deborah number is a non-dimensional number used in rheology to characterize how fluid a material is. The smaller the Deborah number, the more fluid the material appears. The parameter ε is zero for Maxwell fluid while ε = λ for Newtonian fluid. The parameter λ that relates to the relaxation time to the thermal diffusion time is of the order one for most viscoelastic fluids. For dilute polymeric solutions the value of Deborah number is most likely in the range [0.1, 2] and ε in the range [0.1, 1]. It is worth mentioning here that the Darcy–Prandtl number PrD includes the Prandtl number, Darcy number, porosity and the specific heat ratio. The PrD depends on the properties of the fluid and on the nature of porous matrix. The Prandtl number affects the stability of the porous system through this combined dimensionless group. This is also known as Vadasz number. For the sparse porous media, Da ∈ [10−2 , 1], φ ∼ = 0.5 and typical value for Prandtl number for viscoelastic fluid is Pr = 10. Since PrD is magnified by a factor φ Da −1 the reasonable range for PrD will be [5, 500]. The ratio between thermal and solutal diffusivities is characterized by the Lewis number. The normalized porosity χ, is expressed in terms of the porosity of the porous medium, φ, and the solid to fluid heat capacity ratio, γ . Since 0 < φ < 1, it is clear that 0 < χ < 1. Since the boundaries are assumed to be stress free, isothermal, and isohaline the Eqs. (12)– (14) are to be solved for the boundary conditions w=
∂ 2w =T =S=0 ∂z 2
at z = 0, 1
(15)
3 Linear Stability Analysis In this section we predict the thresholds of both marginal and oscillatory convections. The eigenvalue problem defined by Eqs. (12)–(14) and subjected to the boundary conditions (15) is solved using the time-dependent periodic disturbances in a horizontal plane. Assuming that the amplitudes of the perturbations are very small, we write (w, T, S) = (W (z) , (z) , (z)) exp [i (lx + my) + ωt] ,
(16)
Infinitesimal perturbations of the rest state may either damp or grow depending on the value of the parameter ω. Substituting (16) into the linearized version of Eqs. (12)–(14) we obtain ω 2 D − a 2 W + RaT a 2 − RaS a 2 (1 + λω) PrD
− (1 + εω) D 2 − a 2 Da D 2 − a 2 − 1 W = 0, (17)
2 2 − W = 0, (18) ω− D −a 2
−1 2 D −a − W = 0, (19) χω − Le where D ≡ d/dz and a 2 = l 2 + m 2 . The boundary conditions (15) now become W = D 2 W = = = 0 at z = 0, 1.
(20)
We assume the solutions in the form (W (z) , (z) , (z)) = (W0 , 0 , 0 ) sin nπ z, (n = 1, 2, 3, . . . . . .).
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The most unstable mode corresponds to n = 1 (the fundamental mode). Therefore, substituting Eq. (21) with n = 1 into Eqs. (17)–(19), and using the solvability condition we obtain ω + δ2 ω δ2 RaS , RaT = 2 + Λ(Daδ 2 + 1) ω + δ 2 + (22) a PrD χω + Le−1 δ 2 1+εω where δ 2 = π 2 + a 2 , Λ = 1+λω . The growth rate ω is in general a complex quantity such that ω = τ + iσ . The system with τ < 0 is always stable, while for τ > 0 it will become unstable. For neutral stability state we have τ = 0.
3.1 Stationary Convection For the validity of principle of exchange of stabilities (i.e., steady case), we have ω = 0 (i.e., τ = σ = 0) at the margin of stability. Therefore, for marginally stable steady modes Eq. (22) reads, δ4 1 + Daδ 2 + Le RaS , 2 a √ attains critical value corresponding to a = acSt = r , which satisfies RaTSt =
The RaTSt
2Dar 3 + (3Daπ 2 + 1)r 2 − π 4 (Daπ 2 + 1) = 0.
(23)
(24)
Equations (23) and (24) coincide with the results of Poulikakos (1986) for the DDC in a sparsely packed porous layer. When Da → 0, Eq. (23) reduces to a classical result RaTSt = δ4 a2
+ Le RaS for the DDC in a Darcy porous medium (see Nield 1968). Furthermore, for 4 RaS = 0, Eq. (23) yields RaTSt = aδ 2 1 + Daδ 2 , which is the one obtained by Zhang et al. (2008) for the case of single component convection in a porous layer. As a special case when 4 Da → 0, this gives RaTSt = aδ 2 , with the critical value 4π 2 corresponding to the wavenumber π (Horton and Rogers 1945 and Lapwood 1948). 3.2 Oscillatory Convection We now set ω = iσ (i.e., τ = 0) in Eq. (22) to obtain RaT = 1 + iσ 2 ,
(25)
where 1 and 2 are as given in appendix. Since RaT is a physical quantity, it must be real. Hence, Eq. (25) requires that either σ = 0 (i.e., steady onset) or 2 = 0 (i.e., σ = 0, oscillatory onset). On setting 2 = 0, one can obtain an expression for the frequency of oscillations in the form 2 a0 σ 2 + a1 σ 2 + a2 = 0, (26) where ai s are as given in Appendix. Now Eq. (25), reads 1 + λεσ 2 1 δ2 ε−λ 2 2 − σ RaTOsc = 2 δ 2 Daδ 2 + 1 + Daδ + 1 a 1 + λ2 σ 2 PrD 1 + λ2 σ 2 +
Le−1 δ 4 + χσ 2 RaS . 2 Le−1 δ 2 + χ 2 σ 2
(27)
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The analytical expression for oscillatory Rayleigh number given by Eq. (27) is minimized with respect to the wave number numerically, after substituting for σ 2 (>0) from Eq. (26).
4 Finite-Amplitude Analysis with Limited Representation We perform the nonlinear analysis using a truncated representation of Fourier series containing only two terms. Although the linear stability analysis is sufficient for obtaining the stability condition of the motionless solution and the corresponding eigenfunctions describing qualitatively the convective flow, it cannot provide information about the values of the convection amplitudes and hence about the rate of heat and mass transfer. To obtain this additional information, we perform the nonlinear analysis, which is useful to understand the physical mechanism with minimum amount of mathematics and it is a step forward toward understanding the full nonlinear problem. For simplicity of analysis, we confine ourselves to the two-dimensional rolls, so that all the physical quantities are independent of y. We introduce the stream function ψ such that u = ∂ψ/∂z, w = −∂ψ/∂ x into the Eqs. (7)–(9) and non-dimensionalize with the scalings (11) to obtain ∂ ∂T ∂S 1 ∂ 2 1+λ ∇ ψ + RaT − RaS ∂t PrD ∂t ∂x ∂x ∂ Da∇ 4 ψ − ∇ 2 ψ , (28) = 1+ε ∂t ∂ ∂ (ψ, T ) ∂ψ − ∇2 T − + = 0, (29) ∂t ∂ (x, z) ∂x ∂ (ψ, S) ∂ψ ∂ = 0. (30) χ − Le−1 ∇ 2 S − + ∂t ∂ (x, z) ∂x A minimal double Fourier series which describes the finite-amplitude convection is ψ = A (t) sin(ax) sin(π z),
(31)
T = B (t) cos (ax) sin (π z) + C (t) sin (2π z) ,
(32)
S = D (t) cos (ax) sin (π z) + E (t) sin (2π z) ,
(33)
where the coefficients A−E are the time dependent amplitudes and are to be determined from the dynamics of the system. Substituting (31)–(33) into the coupled nonlinear system of partial differential equations (28)–(30) and equating the coefficients of like terms we obtain dX = G, dt
(34)
where X = (A, B, C, D, E, F)T , G = (G 1 , G 2 , G 3 , G 4 , G 5 , G 6 )T with G i s as given in the Appendix. The nonlinear system of autonomous differential equations (34) is not suitable to analytical treatment for the general time-dependent variable and we have to solve it using a numerical method. However, one can make qualitative predictions as discussed below. The system (34) is uniformly bounded in time and possesses many properties of the full problem. Thus, volume in the phase space must contract. In order to prove volume contraction, we must show that velocity field has a constant negative divergence. Indeed,
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∂ ∂M
dM dt
=−
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δ + 4π 2
2
Le−1 1+ χ
1
2 1 + ε PrD 1 + Daδ + , (35) λ
which is always negative and therefore the system is bounded. As a result, the trajectories are attracted to a set of measure zero in the phase space; in particular they may be attracted to a fixed point, a limit cycle or, perhaps, a strange attractor. From Eq. (35) we conclude that if a set of initial points in phase space occupies a region V (0) at time t = 0, then after some time t, the end points of the corresponding trajectories will fill a volume 2 1
Le−1 2 2 V (t) = V (0) exp − δ + 4π + 1 + ε PrD 1 + Daδ 1+ , (36) χ λ Therefore, the volume decreases exponentially with time. We can also infer that retardation parameter, Darcy number, Darcy–Prandtl number, reciprocals of Lewis number and the normalized porosity tend to enhance the dissipation. 4.1 Steady Finite Amplitude Motions From qualitative predictions we now look into the possibility of an analytical solution. In the case of steady motions, Eq. (34) can be solved in the closed form. The steady state solutions are useful because they predict that a finite-amplitude solution to the system is possible for subcritical values of the Rayleigh number and that the minimum values of RaT for which a steady solution is possible lies below the critical values for instability to either a marginal state or an overstable infinitesimal perturbation. Upon setting the left-hand side of Eq. (34) equal to zero and eliminating all the amplitudes, except A one can obtain A1 s 2 + A2 s + A3 = 0, where s = A2 /8 and Ai s are as in the Appendix. The required root of Eq. (37) is, 1/2 1 s= −A2 + A22 − 4 A1 A3 . 2 A1
(37)
(38)
When we let the radical in the above equation to vanish, we obtain the expression for finiteamplitude Rayleigh number RaTF , which characterizes the onset of finite-amplitude steady motions: 1/2 1 −B2 + B22 − 4B1 B3 , RaTF = 2B1 The values of Bi s are as given in Appendix. 4.2 Heat and Mass Transport The quantification of heat and mass transport is important in the study of thermal convection. This is because the onset of convection, as 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 alone. We now proceed to find the Nusselt number and Sherwood number. If H and J are the rate of heat and mass transport per unit area respectively, then ∂ Ttotal ∂ Stotal H = −κT and J = −κS , (39) ∂z z=0 ∂z z=0 where the angular bracket corresponds to a horizontal average and
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Ttotal = T0 − T
z z + T (x, z, t) and Stotal = S0 − S + S(x, z, t). d d
(40)
Substituting (32)–(33) into Eq. (40) and using the resultant equations in (39), we get H=
κS S κT T (1 − 2πC) and J = (1 − 2π E) . d d
The Nusselt number and Sherwood number are defined by Nu =
H J = 1 − 2πC and Sh = = 1 − 2π E. κT T /d κS S/d
Writing C and E in terms of A, we obtain 2s , s + δ 2 /a 2 2s Sh = 1 + . 2 s + δ /a 2 Le2
Nu = 1 +
(41) (42)
The second term on the right-hand side of Eq. (41) and (42) represents the convective contribution to the heat and mass transport, respectively.
5 Results and Discussion Linear and weakly nonlinear stability analysis of a Darcy–Brinkman DDC in a binary viscoelastic fluid saturated porous layer is carried out. In the linear theory, which is based on usual normal mode technique, we have derived for stationary and oscillatory the onset criteria convection. The marginal stability curves in RaTOsc − a plane for the oscillatory mode are displayed through Fig. 1a–d for fixed values of the parameters, viz., λ = 0.7, ε = 0.1, Da = 0.1, χ = 0.4, Le = 10, RaS = 100, and PrD = 10 except the varying parameter. The expressions for both stationary and oscillatory critical Rayleigh number, which characterize the stability of the system, are obtained analytically. The stationary critical Rayleigh number is found to be independent of the viscoelastic parameters and coincides with the result of Poulikakos (1986) for the case of sparsely packed porous layer saturated with Newtonian fluid. The stationary critical Rayleigh number and wavenumber are independent of viscoelastic parameters because of the absence of base flow in the present case. The critical Rayleigh number for oscillatory mode is derived as a function of viscoelastic parameters, Darcy number, normalized porosity, Lewis number, solute Rayleigh number, and Darcy– Prandtl number. It is found that due to competition between the processes of viscoelasticity, thermal and solute diffusions the overstable motions are possible significantly earlier than the stationary motions. Figure 1a depicts the effect of Darcy number Da on the oscillatory neutral curves. It is worth mentioning here that the oscillatory Rayleigh number is normalized with respect to the Osc = Ra Osc /Da which corresponds to that for the clear fluid. Darcy number, in the form RaD T Osc decreases with Da. This is because, with the increasing We find from this figure that RaD Darcy number the permeability increases and thus the viscous drag decreases. Consequently, the critical Rayleigh number decreases. Furthermore, the wavenumber at which the miniOsc occurs decreases with Da, indicating that the wavelength increases with Da. mum of RaD The effect of normalized porosity χ on RaTOsc is illustrated in the Fig. 1b. We observe that with the increase in χ, the minimum of the Rayleigh number for oscillatory mode decreases. This indicates that, the effect of χ is to advance the onset of oscillatory convection. As the
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Fig. 1 Oscillatory neutral stability curves for different values of a Da, b χ , c λ, and d ε
normalized porosity increases, the thermal “lag” effect (double-advective behavior in the terminology of Phillips 2009) is reduced. This makes advective heat transfer more effective and so makes it easier for the destabilizing thermal buoyancy gradient to produce convection. The influence of relaxation parameter λ on the oscillatory neutral curves is revealed in Fig. 1c. We observe that the critical value of RaTOsc decreases with increasing values of λ. Therefore, the effect of the relaxation parameter λ is to advance the oscillatory convection. It is important to note that when λ = ε the oscillatory neutral curve coincides with that for the viscous Newtonian case. That is, the stability of the system in the oscillatory mode becomes independent of the viscoelastic parameters when their values become identical. In Fig. 1d, the effect of retardation parameter ε on the oscillatory neutral curves is displayed. It is observed that increasing ε results in the increase of oscillatory critical Rayleigh number. Therefore, the retardation parameter delays the onset of oscillatory convection. This figure also indicates that the neutral curve is identical with that of the Newtonian case when ε = λ. Further, we observe from Fig. 1c, d that the oscillatory neutral curves of viscoelastic fluids lie below that of Newtonian fluid, therefore, we infer that the effect of viscoelasticity is to advance the oscillatory convection as compared to the case of viscous Newtonian fluid saturated sparsely packed porous layer. The influence of retardation parameter ε on the critical value of oscillatory RayOsc for different values of various governing parameters is revealed leigh number RaTc through Fig. 2a–f. The region left to each curve of the oscillatory mode represents the unstable region. It is important to note that initially the convection sets in, in the form of oscillatory mode and as soon as the value of retardation parameter ε attains a critical value (say ε ∗ ) overstable motions lose their dominance in favor of stationary
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Ra S = 100, Pr D = 10
0.0
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ε
(c) 1150 0.3
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O sc illat
or y re gi
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(f) 1150
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1100
100
350
1000
ry r illato
600
n egio
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λ = 0.7, Da = 0.1, χ = 0.4
2100 2000
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λ = 0.7, Da = 0.1, χ = 0.4
600 Stationary Oscillatory Oscillatory for Newtonian case Da = 0.1, χ = 0.4, Le = 10 Ra S = 100, Pr D = 10
0.0
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cill ato ry r
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6000 4500 3000
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egi on
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100
λ = 0.7, Da = 0.1, χ = 0.4
Le = 10, RaS = 100
0
50 0.0
0.2
0.4
0.6
ε
0.8
1.0
0.0
0.2
0.4
0.6
0.8
ε
Fig. 2 Variation of critical Rayleigh number with ε for different values of a Da, b χ , c λ, d Le, e RaS , and f PrD
convection i.e., the convection ceases to be oscillatory and steady convection occurs as the first bifurcation. The Rayleigh number then takes the value given by Eq. (23), which characterizes the stationary convection and consequently becomes independent of ε. Figure 2a indicates the influence of Darcy number Da, which characterizes the sparseness of the porous layer, on the normalized Rayleigh number RaDc . We find that RaDc decreases significantly with Da and the corresponding value of ε ∗ is shifted toward the smaller values. Therefore, the Darcy number has a destabilizing effect toward both stationary and oscillatory modes. The effect of normalized porosity on critical oscillatory Rayleigh number is shown in Fig. 2b for fixed values of the other parameters. We find that with an increase in the value of normalized porosity the region of oscillatory mode is enlarged. Thus, the normalized porosity
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exhibits a destabilizing effect. The value of ε ∗ is shifted toward the larger values with the increasing χ. Osc . For a fixed value of ε, In Fig. 2c we display the effect of relaxation parameter λ on RaTc the critical oscillatory Rayleigh number decreases with λ. Therefore the relaxation parameter advances the oscillatory convection. The range of values of ε, within which the overstable motions are possible, increases with λ. Further, the dash-dot curve which corresponds to Osc for the Newtonian case lies above the oscillatory region for viscoelastic fluid. This RaTc justifies the fact that the viscoelasticity destabilizes the system toward the oscillatory mode, as compared to the clear viscous fluid case. Osc is depicted through Fig. 2d. We find that the critThe effect of Lewis number Le on RaTc ical oscillatory Rayleigh number decreases with increasing Le. On the other hand the critical stationary Rayleigh number increases with Le. Therefore, the effect of Le is to advance the onset of oscillatory convection where as it reinforces the stability toward stationary mode. Figure 2e indicates the influence of solute Rayleigh number on the stability of the system. The critical Rayleigh number RaTc increases with an increase in the value of RaS , for both stationary and oscillatory modes. This result is similar to the one with the Newtonian fluid. Thus, the effect of increasing the solute Rayleigh number is to stabilize the system. In Fig. 2f the effect of Darcy–Prandtl number on RaTc is displayed. We find that an Osc to decrease for small values of ε and the trend increase in the value of PrD leads RaTc reverses when the value of ε is considerably large. It is also noticed that with the increase of number there is a significant increase in the threshold values of both ε ∗ and Darcy–Prandtl Osc RaTc max (the critical oscillatory Rayleigh number at ε = ε ∗ ). The detailed behavior of oscillatory critical Rayleigh number with respect to the Darcy– Osc Prandtl number is analyzed in the RaTc − PrD plane through the Fig. 3a–f. In each case Osc attains minimum. For Pr < Pr ∗ , there is a threshold value for PrD , say PrD∗ at which RaTc D D the critical oscillatory Rayleigh number is a decreasing function of PrD while for PrD > PrD∗ it is increasing. The effect of Darcy number on the critical oscillatory Rayleigh number is revealed in Fig. 3a. The critical oscillatory Rayleigh number decreases with Da. Further, it is important to note that the effect of PrD on the critical oscillatory Rayleigh number becomes less significant for the values of Da ≥ 0.05. From, Fig. 3b we observe that the critical Osc decreases with χ. Thus, the effect of χ is to destabilize oscillatory Rayleigh number RaTc the system. Figure 3c shows the destabilizing effect of λ and also a shift of PrD∗ toward a higher value with λ. The retardation parameter ε exhibits a similar but reciprocal pattern of influence (Fig. 3d). The effect of Lewis number and solute Rayleigh number on the critical oscillatory RayOsc decreases with leigh number is shown in the Fig. 3e, f, respectively. It is observed that RaTc the Le, while increases with RaS . Thus, the effect of Lewis number is to advance the onset of oscillatory convection, and that of solute Rayleigh number is to inhibit the same. In the realm of weakly non-linear theory we have computed the steady finite-amplitude Rayleigh number RaTF and the quantified the heat and mass transports in terms of Nusselt and Sherwood number respectively. From the Eqs. (41)–(42) one can infer that Sh > N u and both Nusselt number and Sherwood number start with the conduction state value (i.e., one) at the point of onset of steady finite-amplitude convection. When RaT is increased beyond F there is a sharp increase in the value of both Nu and Sh. However, further increase in RaTc RaT will not change Nu and Sh significantly. It is important to note that the finite- amplitude steady convection is independent of the viscoelastic parameters and the results are same as those of the Newtonian fluid case. To know the transient behavior of Nusselt and Sherwood numbers the autonomous system of unsteady finite-amplitude equations (34) is solved numerically using Runge–Kutta method
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M. S. Swamy et al. 40
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λ = 0.7, ε = 0.1, χ = 0.4
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Le = 10, RaS = 100
Le = 10, RaS = 100
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RaDc x 10
-3
450 Da = 0.005
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0.1, 0.05
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Le = 10, RaS = 100
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ε = 0.1, Da = 0.1, χ = 0.4
1
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100 1000
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λ = 0.7, ε = 0.1, Da = 0.1 χ = 0.4, Le = 10
300 Osc
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400 350
300 Le = 5
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Fig. 3 Variation of critical Rayleigh number with PrD for different values of a Da, b χ , c λ, d ε, e Le, and f RaS
with the suitable initial conditions. Then Nu and Sh are evaluated as a function of time t. The unsteady transient behavior of Nu and Sh is shown graphically through the Fig. 4a–g. It is found that both Nu and Sh start with a conduction state value (i.e., 1) at t = 0 and then oscillate periodically about their steady state value (i.e., close to 3) for t > 0. This periodic variation of Nu and Sh is very short lived and decays as time progresses. The values of Nu and Sh then tend toward their steady state value 3. Figure 4a, e, g respectively show that Da, PrD and RaS enhance the heat and mass transports. From Fig. 4b it is clear that χ has no significant influence on unsteady heat transport while it enhances the mass transport. The effect of the relaxation parameter λ on transient heat and mass transfer is shown in Fig. 4c. We find that an increase in the relaxation parameter increases both heat and mass transfer
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(a) 5
353 5
λ = 0.7, ε = 0.1, χ = 0.4, Le = 2 F RaS = 100, PrD = 10, RaT= 8 x RaTc
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3
3
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2
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1
1 Da = 0.001 Da = 0.1
0 0.00
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t 5
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RaS = 100, PrD = 10, RaT= 8 x RaTc
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2
1
1 χ = 0.2 χ = 0.8
0 0.00
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t
t 5
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λ = 0.9, 0.3
4
3
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2 Newtonian Case Viscoelastic Case
1
1
ε = 0.1, Da = 0.1, χ = 0.4, Le = 2
Newtonian Case Viscoelastic Case
F
RaS = 100, PrD = 10, RaT= 8 x RaTc
0 0.00
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1 0 0.00
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0.05
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0.15
t
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1 Newtonian Case Viscoelastic Case
0 0.00
0.05
0.10
0.15
0.20
t
Fig. 4 Variation of Nu and Sh with time for different values of a Da, b χ , c λ, d ε, e PrD , f Le, and g RaS
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(e) 5
5
λ = 0.7, ε = 0.1, Da = 0.1, χ = 0.4 F
Le = 2, RaS = 100, RaT= 8 x RaTc
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Nu
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1 0 0.00
PrD = 5 PrD = 20
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Nu
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1 0 0.00
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0.05
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t
(g) 5
0.10
0.15
0.20
0.25
t 5
λ = 0.7, ε = 0.1, Da = 0.1, χ = 0.4 F
Le = 2, PrD = 10, RaT= 8 x RaTc
4
3
3
Sh
Nu
4
2
2
1 0 0.00
1 RaS = 10 RaS = 100
0.05
0.10
0.15
t
0.20
0.25
0.30
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0.05
0.10
0.15
0.20
t
Fig. 4 continued
marginally. On the other hand the effect of increasing retardation parameter ε is to suppress the heat and mass transfer (Fig. 4d). The effect of Lewis number on Nu and Sh is shown in Fig. 4f. We find that an increase in the value of Lewis number decreases the amplitude of the heat transfer while increases the amplitude of the mass transfer.
6 Conclusions The problem of DDC in a viscoelastic fluid saturated sparsely packed porous layer is investigated using both linear and weakly non-linear stability analyses. The linear theory provides
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the onset criteria for both stationary and oscillatory convection. The nonlinear theory which is based on the truncated Fourier series technique provides a mean to measure the convection amplitudes and the rate of heat and mass transfer. The main conclusions of the present study are as follows: 1.
2.
3. 4.
5. 6. 7.
Due to the competition among the processes of thermal diffusion, solute diffusion, and the rheological properties the convection sets in through oscillatory modes rather than stationary. It is found that initially the convection sets in, in the form of oscillatory mode and as soon as the retardation parameter ε attains a critical value say ε ∗ (which depends on the values of other governing parameters), the convection ceases to be oscillatory and steady convection occurs as the first bifurcation. The stationary mode is independent of the viscoelastic parameters because of the absence of base flow. The effect of the relaxation parameter is to advance the onset of oscillatory convection where as the retardation parameter delays the same. The solute Rayleigh number strengthens the stabilizing effect of ε while the Lewis number reinforces the destabilizing effect of λ toward oscillatory modes. The threshold value of retardation parameter ε ∗ upto which the overstable motions are possible, increases with λ, χ, Le, RaS , and PrD where as decreases with Da. The Darcy–Prandtl number has a dual effect on the oscillatory mode. The convective heat and mass transfer are suppressed by ε where as enhanced by λ, Da, PrD and RaS . Lewis number enhances mass transfer while suppresses heat transfer. The normalized porosity χ has no influence on heat transfer while it enhances the mass transfer. It is interesting to note both heat and mass transfer approach the steady state value as time progresses.
Acknowledgements This study is supported by University Grants Commission, New Delhi, under the Special Assistance Programme DRS Phase-II. The authors (MSS & WS) emotionally dedicate the article to the memory of Late Prof. M.S. Malashetty. The authors are grateful to the reviewers for their constructive comments and valuable suggestions.
Appendix
1 =
1 + λεσ 2 1 δ2 2 ε−λ 2 2 2 Daδ + 1 + Daδ + 1 δ − σ a2 1 + λ2 σ 2 PrD 1 + λ2 σ 2
Le−1 δ 4 + χσ 2 RaS , 2 Le−1 δ 2 + χ 2 σ 2 1 + λεσ 2 1 δ2 ε−λ 2 + Daδ + Daδ 2 + 1 + 1 2 = 2 δ 2 a PrD 1 + λ2 σ 2 1 + λ2 σ 2 +
Le−1 δ 2 − χδ 2 RaS . 2 Le−1 δ 2 + χ 2 σ 2 a0 = χ 2 λLe2 δ 2 λ + PrD Daδ 2 + 1 ε , a1 = δ 6 + a 2 Le PrD RaS (1 − Leχ) λ2 + PrD δ 2 Daδ 2 + 1 λ δ 2 ε − Le2 χ 2 + Le2 χ 2 PrD + δ 2 + Da PrD δ 2 + PrD δ 2 Daδ 2 + 1 ε , +
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a2 = δ 4 PrD + δ 2 + Da PrD δ 2 + a 2 Le PrD RaS (1 − Leχ) − PrD δ 6 Daδ 2 + 1 (λ − ε) . πa G 1 = F, G 2 = − a A + δ 2 B + aπ AC , G 3 = −4π 2 C + AB, 2 1 1 2 −1 πa a A + Le−1 δ 2 D + πa AE , G 5 = − 4π Le E − G4 = − AD , χ χ 2 2 δ PrD + ε Daδ 4 + δ 2 ε G 1 + δ 2 1 + Daδ 2 A + a B RaT G6 = − 2 λδ PrD − a D RaS + aλRaT G 2 − aλRaS G 4 ] A1 = a 4 Le2 δ 2 1 + Daδ 2 , A2 = a 2 δ 4 1 + Le2 1 + Daδ 2 + a 2 Le (RaS − Le RaT ) , A3 = δ 2 δ 4 1 + Daδ 2 + a 2 (Le RaS − RaT ) . B1 = a 4 Le4 , B2 = 2a 2 Le2 δ 4 1 − Le2 1 + Daδ 2 − a 2 Le RaS , 2 B3 = δ 4 Le2 − 1 1 + Daδ 2 − a 2 Le RaS .
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