Transp Porous Med (2011) 88:315–345 DOI 10.1007/s11242-011-9741-x
The Effect of Rotation on the Onset of Double Diffusive Convection in a Sparsely Packed Anisotropic Porous Layer M. S. Malashetty · Irfana Begum
Received: 26 November 2010 / Accepted: 1 February 2011 / Published online: 19 February 2011 © Springer Science+Business Media B.V. 2011
Abstract The effect of rotation on the onset of double diffusive convection in a sparsely packed anisotropic porous layer, which is heated and salted from below, is investigated analytically using the linear and nonlinear theories. The Brinkman model that includes the Coriolis term is employed for the momentum equation. The critical Rayleigh number, wavenumber for stationary and oscillatory modes and a dispersion relation are obtained analytically using linear theory. The effect of anisotropy parameters, Taylor number, Darcy number, solute Rayleigh number, Lewis number, Darcy–Prandtl number, and normalized porosity on the stationary, oscillatory and finite amplitude convection is shown graphically. It is found that contrary to its usual influence on the onset of convection in the absence of rotation, the mechanical anisotropy parameter show contrasting effect on the onset criterion at moderate and high rotation rates. The nonlinear theory based on the truncated representation of Fourier series method is used to find the heat and mass transfers. The effect of various parameters on heat and mass transfer is shown graphically. Some of the convection systems previously reported in the literature is shown to be special cases of the system presented in this study. Keywords Double diffusive convection · Rotation · Brinkman model · Porous layer · Anisotropy · Heat mass transfer List of Symbols a Wavenumber c Specific heat of solid cp Specific heat of fluid e Kz Da Darcy number, μ μf d2 d Height of the porous layer g Gravitational acceleration, (0, 0, −g) K Inverse anisotropic permeability tensor, K x−1 ii + K y−1 jj + K z−1 kk Le Lewis number, κT z /κS M. S. Malashetty (B) · I. Begum Department of Mathematics, Gulbarga University, Jnana Ganga Campus, Gulbarga 585106, India e-mail:
[email protected]
123
316
M. S. Malashetty, I. Begum
l, m Nu PrD p q RaT RaS S Sh S Ta t T T x, y, z
Horizontal wavenumbers Nusselt number Darcy–Prandtl number, γ ενd2 /K z κT z Pressure Velocity vector, (u, v, w) Thermal Rayleigh number, βT gT d K z /νκT z Solute Rayleigh number, βS gSd K z /νκT z Solute concentration Sherwood number Salinity difference between the walls 2 z Taylor number, 2K εν Time Temperature Temperature difference between the walls Space coordinates
Greek symbols βT Thermal expansion coefficient βS Solute expansion coefficient ε Porosity Φ Dimensionless amplitude of concentration perturbation (ρc)m γ Ratio of specific heat, (ρc p )f η Thermal anisotropy parameter, κT x /κT z κT Anisotropic thermal diffusion tensor, κT x ii + κT y jj + κT z kk κS Solute diffusivity λ Normalized porosity, γε μf Fluid viscosity μe Effective viscosity ν Kinematic viscosity Θ Dimensionless amplitude of temperature perturbation ρ Density σ Growth rate Angular velocity of rotation, (0, 0, ) ξ Mechanical anisotropy parameter, K x /K z ψ Stream function Other symbols ∇12 ∇2
2 ∂2 + ∂∂y 2 ∂x2 ∂2 ∇12 + ∂z 2
Subscripts b Basic state c Critical f Fluid
123
The Effect of Rotation on the Onset Double Diffusive Convection
m 0 s
317
Porous medium Reference value Solid
Superscripts ∗ Dimensionless quantity Perturbed quantity F Finite amplitude Osc Oscillatory state St Stationary state
1 Introduction Recent interest in double diffusive convection in porous media has been motivated by its wide range of applications, from the solidification of binary mixtures to the heat transfer in geothermal reservoirs. Double diffusive convection has implications for many geological processes, such as crustal heat and solute transport, metamorphism, the diagenetic evolution of sedimentary basins and ore genesis. Besides its importance in the hydrogeological context, double diffusive convection in porous media has wide variety of geotechnical applications, among them contaminant transport in saturated soil, underground disposal of nuclear wastes, liquid re-injection, the migration of moisture in fibrous insulation, and electro-chemical and drying processes. The problem of double diffusive convection in porous medium has been extensively investigated and the growing volume of work devoted to this area is well documented by Nield and Bejan (2006), Ingham and Pop (1998, 2005), Vafai (2000, 2005), and Vadasz (2008). The study of the double diffusive generalization of the Horton–Rogers–Lapwood problem was first undertaken by Nield (1968) on the basis of linear stability theory for various thermal and solutal boundary conditions. The onset of double diffusive convection in a horizontal porous layer has been investigated by Rudraiah et al. (1982) using a weak nonlinear theory. Finite amplitude double diffusive convection near the threshold of both stationary and oscillatory instabilities in a binary mixture was investigated by Brand and Steinberg (1983). The linear stability analysis of the thermosolutal convection in a sparsely packed porous layer was made by Poulikakos (1986) using the Darcy–Brinkman model. Small amplitude nonlinear solutions in the form of standing and traveling waves and the transition to finite amplitude convection, as predicted by bifurcation theory, were studied by Knobloch (1986). The double diffusive convection in porous media in the presence of Soret and Dufour coefficients has been analyzed by Rudraiah and Malashetty (1986). Murray and Chen (1989) have extended the linear stability theory, taking into account effects of temperature-dependent viscosity and volumetric expansion coefficients and nonlinear basic salinity profile. Natural convection in porous layer, with two stratifying agencies, heated from below in a square cavity has been investigated numerically by Rosenberg and Spera (1992). Double diffusive fingering convection in a porous medium with horizontally periodic boundary conditions was studied by Chen and Chen (1993). Malashetty (1993) made a linear stability analysis to determine the effects of anisotropic thermo-convective currents on the double diffusive convection in a sparsely packed porous medium.
123
318
M. S. Malashetty, I. Begum
Straughan and Hutter (1999) have investigated the double diffusive convection with Soret effect in a porous layer using Darcy–Brinkman model and derived a priori bounds. An analytical and numerical study of double diffusive convection with parallel flow in a horizontal sparsely packed porous layer under the influence of constant heat and mass flux was performed using a Brinkman model by Amahmid et al. (1999). A double diffusive bifurcation phenomenon was studied by Mamou and Vasseur (1999) using linear and nonlinear stability analyses with uniform flux and uniform temperature boundary conditions. Bahloul et al. (2003) have carried out an analytical and numerical study of the double diffusive convection in a shallow horizontal porous layer under the influence of Soret effect. Hill (2005) performed linear and nonlinear stability analyses of double diffusive convection in a fluid saturated porous layer with a concentration-based internal heat source using Darcy’s law. Double diffusive natural convection within a multi layer anisotropic porous medium is studied numerically and analytically by Bennacer et al. (2005). Mansour et al. (2006) have investigated the multiplicity of solutions induced by thermosolutal convection in a square porous cavity heated from below and subjected to horizontal concentration gradient in the presence of Soret effect. The study of double diffusive convection in a rotating porous media is motivated both theoretically and by its practical applications in engineering. Some of the important areas of applications in engineering include the food and chemical process, solidification and centrifugal casting of metals, rotating machinery, petroleum industry, biomechanics and geophysical problems. There are only few studies available on double diffusive convection in a porous medium in the presence of rotation. Chakrabarti and Gupta (1981) have analyzed the nonlinear thermohaline convection in a rotating porous medium. The effect of rotation on linear and nonlinear double diffusive convection in a sparsely packed porous medium was studied by Rudraiah et al. (1986). The Lyapunov direct method is applied to study the nonlinear conditional stability problem of a rotating doubly diffusive convection in a sparsely packed porous layer by Guo and Kaloni (1995). The nonlinear stability of the conduction–diffusion solution of a fluid mixture heated and salted from below and saturating a porous medium in the presence of rotation is studied by Lombardo and Mulone (2002) using Lyapunov direct method. Due to the structure of a porous material there can be a pronounced anisotropy in properties such as permeability or thermal diffusivity. Anisotropy is generally a consequence of preferential orientation or asymmetric geometry of porous matrix or fibers encountered in numerous systems in industry and nature. In geological processes such as sedimentation, compaction, frost action and the reorientation of the solid matrix are responsible for the creation of an anisotropic porous medium. Anisotropy is particularly important in a geological context, since sedimentary rocks generally have a layered structure; the permeability in the vertical direction is often much less than in the horizontal direction. Anisotropy can also be a characteristic of artificial porous materials like pelleting used in chemical engineering process and fiber material used in insulating purpose. The review of research on convective flow through anisotropic porous media has been well documented by McKibbin (1985, 1992) and Storesletten (1998, 2004). Castinel and Combarnous (1974) have conducted an experimental and theoretical investigation on the Rayleigh–Benard convection in an anisotropic porous medium. Epherre (1977) extended the stability analysis to a porous medium with anisotropy in thermal diffusivity also. A theoretical analysis of nonlinear thermal convection in an anisotropic porous medium was performed by Kvernvold and Tyvand (1979). Nilsen and Storesletten (1990) have studied the problem of natural convection in both isotropic and anisotropic porous channels. Tyvand and Storesletten (1991) investigated the problem concerning the onset of convection in an anisotropic porous layer in which the principal axes were obliquely oriented to the gravity vector. Recently many authors have studied the effect
123
The Effect of Rotation on the Onset Double Diffusive Convection
319
of anisotropy and/or rotation on the onset of convection in a horizontal porous layer (see e.g., Govender 2006, 2007; Govender and Vadasz 2007; Malashetty and Swamy 2007). More recently Malashetty and Heera (2008) studied the effect of rotation on the onset of double diffusive convection in a horizontal anisotropic porous layer using a linear and nonlinear theory. The linear and nonlinear double diffusive convection in a rotating sparsely packed porous layer using a thermal non-equilibrium model was investigated by Malashetty et al. (2009). It is now well known that many applications in engineering disciplines as well as in circumstances linked to modern porous media involve high permeability porous media and in such situations the Darcy equation fails to give satisfactory results. Therefore use of nonDarcian models, which takes care of boundary and/or inertia effects, is of fundamental and practical interest to obtain accurate results for high permeability porous media. It may be noted that most of the previous investigators have assumed that the fluid viscosity is same as the effective viscosity in their study. However, Givler and Altobelli (1994) have determined experimentally that μe = (5 ∼ 12)μf where μe is the effective viscosity and μf is the fluid viscosity, for water flowing through high porosity porous media. Therefore, consideration of the ratio of effective viscosity to the fluid viscosity different from unity is warranted to know its influence on the critical stability. Although copious 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), the works on thermal convection in a rotating porous layer based upon the non-Darcian models are very sparse and it is in much-to-be desired state. The double diffusive convection thought to occur in certain casting and crystal growth operations. It is demonstrated that the dynamics occurring in the mushy layer are critical to the quality of the final product and suppression of convection is an important factor. The rotation is being used as a means to suppress convection. The earlier studies have modeled the mushy layer as isotropic porous medium. Realistically, the permeability and thermal diffusivity are anisotropic. In view of these, it is of interest to gain a general understanding of the manner in which rotation affects the hydrodynamic stability of a double diffusive anisotropic porous layer. Therefore, the objective of the present study is therefore to investigate the combined effect of rotation, mechanical and thermal anisotropy on the double diffusive convection in a horizontal sparsely packed porous layer using linear and nonlinear analyses.
2 Mathematical Formulation Consider a sparsely packed, anisotropic porous layer, saturated with Boussinesq fluid of infinite horizontal extent confined between the planes z = 0 and z = d, with the vertically downward gravity force g acting on it. A uniform adverse temperature difference T = (Tl − Tu ) and a stabilizing concentration difference S = (Sl − Su ) where Tl > Tu and Sl > Su are maintained between the lower and upper boundaries. A Cartesian frame of reference is chosen with the origin in the lower boundary and the z-axis vertically upward. The porous layer rotates uniformly about the z-axis with a constant angular velocity = (0, 0, ). The interaction between heat and mass transfer, known as Soret and Dufour effects, is supposed to have no influence on the convective flow, so they are ignored. It is also assumed that the fluid and solid phases are in local thermal equilibrium. The Darcy–Brinkman model that includes the Coriolis term is used for the momentum equation. The velocities are assumed to be small so that the advective and Forchheimer inertia effects are ignored. The Boussinesq approximation, which states that the variation in density is negligible everywhere in the conservations except in the buoyancy term, is assumed to hold. With these assumptions the basic
123
320
M. S. Malashetty, I. Begum
governing equations are ρ0
∇ · q = 0, 1 ∂q 2 + × q + μf K · q = −∇ p + ρg + μe ∇ 2 q, ε ∂t ε ∂T + (q · ∇) T = ∇ · (κT · ∇T ), γ ∂t ∂S + (q · ∇) S = κS ∇ 2 S, ε ∂t ρ = ρ0 [1 − βT (T − T0 ) + βS (S − S0 )] ,
(2.1) (2.2) (2.3) (2.4) (2.5)
where q = (u, v, w) is the velocity, p the pressure, T the temperature, S the concentration, ε the porosity, = (0, 0, ) constant angular velocity, K = K x−1 ii + K y−1 jj + K z−1 kk is the inverse of the anisotropic permeability tensor and κT = κT x ii + κT y jj + κT z kk is the anisotropic thermal diffusion tensor. We restrict consideration to horizontal isotropy in mechanical and thermal properties of the porous medium, i.e., K x = K y and κT x = κT y . The permeability and thermal diffusivity tensors of the porous medium are assumed to have principal axes aligned with the coordinate system. The quantities ρ0 , μf , μe , κS , βT and βS denote the density, fluid viscosity, effective viscosity, the mass diffusivity, thermal and solute (ρc)m expansion coefficients, respectively. Further, γ = (ρc , (ρc)m = (1−ε)(ρc)S +ε(ρcp )f , cp p )f is the specific heat of the fluid, at constant pressure, c is the specific heat of the solid, the subscripts f, s and m denotes fluid, solid and porous medium values, respectively. The basic state of the fluid is assumed to be quiescent and is given by, qb = (0, 0, 0) ,
p = pb (z), T = Tb (z), S = Sb (z), ρ = ρb (z).
(2.6)
The temperature Tb (z), solute concentration Sb (z), pressure pb (z), and density ρb (z), satisfy the following equations. d pb = −ρb g, dz d2 Tb = 0, dz 2 d2 Sb = 0, dz 2 ρb = ρ0 [1 − βT (Tb − T0 ) + βS (Sb − S0 )] .
(2.7) (2.8) (2.9) (2.10)
Then the conduction state temperature and concentration are given by Tb = −
T S z + Tl , Sb = − z + Sl . d d
(2.11)
We now superimpose infinitesimal perturbations on the quiescent basic state and study the stability of the system. Let the basic state be disturbed by an infinitesimal perturbation. We now have q = qb + q , T = Tb (z) + T , S = Sb (z) + S , p = pb (z) + p , ρ = ρb (z) + ρ , (2.12) where the prime indicates that the quantities are infinitesimal perturbations.
123
The Effect of Rotation on the Onset Double Diffusive Convection
321
Substituting Eq. 2.12 into 2.1–2.5, and using the basic state solutions, we get the linearized equations governing the perturbations in the form 1 ∂q 2 ρ0 + × q + μf K · q = −∇ p − ρ0 βT T − β S S g + μe ∇ 2 q , ε ∂t ε (2.13) T ∂T (2.14) γ ∂t + q · ∇ T − d w = ∇ · κT · ∇T , T ∂ S 2 ε ∂t + q · ∇ S − d w = κS ∇ S . (2.15) By operating curl twice on Eq. 2.13 we eliminate p from it and then render the resulting equation and the Eqs. 2.14–2.15 dimensionless by setting ∗ ∗ ∗ x , y , z = x , y , z d, t = t ∗ γ d2 /κT z , u , v , w = (κT z /d) u ∗ , v ∗ , w ∗ , T = (T ) T ∗ , S = (S) S ∗ ,
(2.16)
to obtain nondimensional equations as (on dropping the asterisks for simplicity)
1 ∂ 2 1 ∂ 1 2 + 1 ∂ 2 − Da∇ 4 2 + T a ∂2 w ∇ + ∇ + − Da∇ 2 2 1 PrD ∂t ξ ∂z PrD ∂t ξ ∂z 1 ∂ 1 2 2 2 RaT ∇1 T − RaS ∇1 S , = PrD ∂t + ξ − Da∇ ∂ ∂2 2 ∂t − η∇1 + ∂z 2 + q · ∇ T − w = 0,
∂ 1 ∇ 2 + q · ∇ S − w = 0, λ ∂t − Le where ∇12 =
∂2 ∂x2
+
∂2 ∂ y2
and ∇ 2 = ∇12 +
∂2 . ∂z 2
(2.17) (2.18) (2.19)
The dimensionless groups, that appear
μe K z are PrD = z κT z , the Darcy–Prandtl number, Da = μf d2 , the Darcy number, 2 z , the Taylor number, RaT = βT gT d K z /νκT z , the thermal Rayleigh numT a = 2K εν ber, RaS = βS gSd K z /νκT z , the solute Rayleigh number, Le = κT z /κS , the Lewis number, ξ = K x /K z , the mechanical anisotropy parameter, η = κT x /κT z , the thermal anisotropy parameter, λ = γε , normalized porosity. Equations 2.17–2.19 are to be solved for the stressfree, isothermal and isosolutal boundary conditions
γ ενd2 /K
w=
∂ 2w = T = S = 0, at z = 0, 1. ∂z 2
(2.20)
The stress-free boundary conditions are chosen for mathematical simplicity, without qualitatively important physical effect being lost. The use of stress-free boundary conditions is a useful mathematical simplification but is not physically sound. The correct boundary conditions for a viscous binary fluid are to impose rigid–rigid boundary conditions but then the problem is not tractable analytically.
3 Linear Stability Analysis In this section we predict the thresholds of both marginal and oscillatory convections using linear theory. The eigenvalue problem defined by Eqs. 2.17–2.19 subject to the boundary conditions (2.20) is solved using the time-dependent periodic disturbances in a horizontal
123
322
M. S. Malashetty, I. Begum
plane. Assuming that the amplitudes of the perturbations are very small, we write ⎛ ⎞ ⎛ ⎞ w W (z) ⎝ T ⎠ = ⎝ Θ(z) ⎠ exp [i (lx + my) + σ t], S Φ(z)
(3.1)
where l, m are horizontal wavenumbers and σ is the growth rate. Infinitesimal perturbations of the rest state may either damp or grow depending on the value of the parameter σ . Substituting Eq. 3.1 into the linearized version of Eqs. 2.17–2.19 we obtain
σ D 2 −a 2 PrD
+
D 2 − a 2 − Da D 2 − a 2 2 ξ
=
σ 1 2 2 PrD + ξ − Da D − a
σ 1 2 2 2 2 PrD + ξ − Da(D − a ) −RaT a Θ + RaS a Φ , σ − D 2 − ηa 2 Θ − W = 0, 1 D 2 − a 2 Φ − W = 0, λσ − Le
+ T a D2
w
(3.2) (3.3) (3.4)
where D = d/dz and a 2 = l 2 + m 2 . We assume the solutions of Eqs. 3.2–3.4 satisfying the boundary conditions (2.20) in the form ⎛ ⎞ ⎛ ⎞ W (z) W0 ⎝ Θ(z) ⎠ = ⎝ Θ0 ⎠ Sin nπ z, (n = 1, 2, 3, . . .). (3.5) Φ(z) Φ0 The most unstable mode corresponds to n = 1 (fundamental mode). Therefore, substituting Eq. 3.5 with n = 1 into Eqs. 3.2–3.4, we obtain a matrix equation ⎞⎛ ⎞ ⎛ ⎞ ⎛ a 2 RaS M11 −a 2 RaT W0 0 ⎠ ⎝ Θ0 ⎠ = ⎝ 0 ⎠ ⎝ −1 σ + δ 2 (3.6) 0 2 Φ0 0 −1 0 λσ + δ 2 Le−1 −1 , δ2 = π 2 + where M11 = δ 2 σ PrD−1 + δ12 + Daδ 4 + π 2 T a σ PrD−1 + ξ −1 + Daδ 2 a 2 , δ12 = π 2 ξ −1 + a 2 and δ22 = π 2 + ηa 2 . The condition of nontrivial solution of above system of homogeneous linear equations (3.6) yields an expression for thermal Rayleigh number in the form RaT = σ + δ22 σ π2T a a 2 RaS 2 2 2 + Daδ + δ1 + + δ . a2 PrD λσ + δ 2 Le−1 σ PrD−1 + ξ −1 + Daδ 2 (3.7)
3.1 Marginal State For the validity of principle of exchange of stabilities (i.e., steady case), we have σ = 0 (i.e., σr = σi = 0) at the margin of stability. Then the Rayleigh number at which marginally stable steady mode exists becomes
123
The Effect of Rotation on the Onset Double Diffusive Convection
RaTSt
323
2T a 2 π 1 2 ηa 2 + π 2 = 2 Da a + π 2 + a 2 + π 2 ξ −1 + −1 a ξ + Da a 2 + π 2 2 ηa + π 2 Le RaS . (3.8) + a2 + π 2
The minimum value of the Rayleigh number RaTSt occurs at the critical wavenumber √ a = acSt where acSt = h satisfies a polynomial equation of degree seven in h. In the limit as Da → 0, i.e., for a densely packed porous medium equation (3.8) reduces to 2 2 ηa + π 2 1 2 St 2 −1 2 2 + π ξ T a ηa + π + RaT = 2 a + π ξ Le RaS . (3.9) a a2 + π 2 This is exactly the one given by Malashetty and Heera (2008). When Da → 0 and T a = 0, i.e., for a densely packed porous medium in the absence of rotation, Eq. 3.8 reduces to 2 ηa + π 2 1 RaTSt = 2 a 2 + π 2 ξ −1 ηa 2 + π 2 + (3.10) Le RaS , a a2 + π 2 given by Malashetty and Swamy (2010). Further, for an isotropic porous medium, that is when ξ = η = 1, Eq. 3.8 gives 3 π 4 1 + α2 Ta RaS St RaT = , (3.11) η1 + 2 + α2 α η1 τ −1 2 2 1 where η1 = π14 π 4 Da + π 2 + a 2 , α = πa 2 , τ = Le , which is the one obtained by Rudraiah et al. (1986). 3.2 Oscillatory State We now set σ = iσi in Eq. 3.7 and clear the complex quantities from the denominator, to obtain RaT = 1 + iσi 2 , where,
(3.12)
RaS δ22 δ 2 Le−1 + σ 2 λ σ 2 δ2 − + 2 PrD δ 2 Le−1 + σ 2 λ2 π 2 T a σ 2 PrD−1 + δ22 Daδ 2 + ξ −1 + 2 , 2 a 2 Daδ 2 + ξ −1 + σ PrD−1 RaS δ 2 Le−1 − δ22 λ δ22 δ 2 1 4 2 2 = 2 Daδ + δ1 + + 2 a PrD δ 2 Le−1 + σ 2 λ2 π 2 T a Daδ 2 + ξ −1 − δ22 PrD−1 + 2 . 2 a 2 Daδ 2 + ξ −1 + σ PrD−1 1 1 = 2 a
Daδ 4 δ22
+ δ12 δ22
Since RaT is a physical quantity, it must be real. Hence, from Eq. 3.12 it follows that either σi = 0 (steady onset) or 2 = 0 (σi = 0, oscillatory onset).
123
324
M. S. Malashetty, I. Begum
For oscillatory onset 2 = 0 (σi = 0) and this gives a dispersion relation of the form (on dropping the subscript i) 2 a0 σ 2 + a1 σ 2 + a2 = 0.
(3.13)
Now Eq. 3.12 with 2 = 0, gives Ra Osc =
RaS δ 2 δ 2 Le−1 + σ 2 λ σ 2 δ2 + 2 2 PrD δ 2 Le−1 + σ 2 λ2 π 2 T a σ 2 PrD−1 + δ22 Daδ 2 + ξ −1 + 2 . 2 a 2 Daδ 2 + ξ −1 + σ PrD−1 1 a2
Daδ 4 δ22 + δ12 δ22 −
(3.14)
The expression for the oscillatory Rayleigh number given by Eq. 3.14 in the limit Da → 0 coincides exactly with the one given by Malashetty and Heera (2008) after making necessary rescaling. We find the oscillatory neutral solutions from Eq. 3.14. It proceeds as follows: First determine the number of positive solutions of Eq. 3.13. If there are none, then no oscillatory instability is possible. If there are two, then the minimum (over a 2 ) of Eq. 3.14 with σ 2 given by Eq. 3.13 gives the oscillatory neutral Rayleigh number. Since Eq. 3.13 is quadratic in σ 2 , it can give rise to more than one positive value of σ 2 for fixed values of the governing parameters. The analytical expression for oscillatory Rayleigh number given by Eq. 3.14 is minimized with respect to the wavenumber numerically, after substituting for σ 2 (> 0) from Eq. 3.13, for various values of physical parameters in order to know their effects on the onset of oscillatory convection. The critical values of Rayleigh number and corresponding wavenumbers for stationary and oscillatory mode are computed from the Eqs. 3.8 and 3.14 for different values of the anisotropic parameters, Darcy number and Taylor number, and are presented in Tables 1, 2 and 3.
4 Weakly Nonlinear Theory In this section we consider the nonlinear analysis using a truncated representation of Fourier series considering 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 can neither provide information about the values of the convection amplitudes, nor regarding 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 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 stream function ψ such that u = ∂ψ/∂z, w = −∂ψ/∂ x into the Eq. 2.13, eliminate pressure and nondimensionalize the resulting equation and Eqs. 2.14–2.15 using the transformations (2.16) to obtain
1 ∂ 2 ∂V ∂T ∂S ∂2 1 ∂2 4 − Da∇ ψ − T a 1/2 ∇ + 2 + + RaT ∇12 − RaS = 0, PrD ∂t ∂x ξ ∂z 2 ∂z ∂x ∂x (4.1)
123
2.64594
2.54971
198.789
194.151
181.598
229.356
1.5
2.0
10.0
Isotropic Case (ξ = 1, η = 1)
3.08237
1108.59
591.616
593.981
594.967
596.929
602.798
610.575
620.228
648.752
2.44969
5.74465
5.74465
5.74465
5.788
5.78800
5.83104
5.87376
6.04161
741.471
487.413
477.12
473.134
465.688
447.328
430.802
420.037
424.921
For fixed values of η = 0.3, Da = 0.1, RaS = 100, PrD = 10, Le = 10 and λ = 0.7
2.54971
2.34542
2.7388
228.885
207.267
0.5
3.3913
3.60569
4.12323
1.0
278.509
252.783
0.2
0.3
343.002
ac
Osc RaTc
ac
T a = 100
Osc RaTc
St RaTc
Ta = 0
0.1
ξ
5.33863
5.91616
5.78800
5.70096
5.61258
5.33863
5.09912
4.89908
4.74352
ac
1523.52
697.75
693.807
692.34
689.665
683.598
679.074
677.390
686.278
St RaTc
5.19625
6.78240
6.70828
6.67091
6.63332
6.51928
6.40320
6.32463
6.24508
ac
1665.28
914.178
895.908
888.536
874.256
834.831
789.908
744.767
664.729
Osc RaTc
T a = 500
7.1415
7.61584
7.48338
7.4499
7.34854
7.10641
6.81916
6.48082
5.91616
ac
2345.880
967.867
955.976
951.229
942.108
917.465
890.363
864.210
820.906
St RaTc
7.17642
8.27653
8.15481
8.1241
8.06232
7.84226
7.61584
7.38248
6.96427
ac
Table 1 Critical oscillatory and stationary Rayleigh number and corresponding wavenumber for different values of mechanical anisotropy parameter ξ and Taylor number T a
The Effect of Rotation on the Onset Double Diffusive Convection 325
123
123
645.141
10.0
3.24052
1.73234
2.12156
2.44969
2.64594
2.54971
3.00017
3.00017
3.16244
375.237
2355.34
1852.12
1416.96
1304.81
1108.59
790.089
700.208
496.03
7.28018
0.707814
1.0005
1.41457
1.73234
2.44969
4.89908
5.33863
6.4032
3634.93
2025.29
1029.95
861.324
691.201
518.182
482.940
411.244
374.518
4.24276
4.47225
4.74352
4.79594
4.94985
5.19625
5.2916
5.47732
5.61258
ac
For fixed values of ξ = 0.5, Da = 0.1, RaS = 100, PrD = 10, Le = 10 and λ = 0.7
308.612
439.69
285.682
1.5
2.0
229.356
1.0
5.0
233.728
238.571
0.3
0.5
219.157
224.034
0.1
ac
Osc RaTc
ac
T a = 100
Osc RaTc
St RaTc
Ta = 0
0.2
η
7595.28
4641.15
2438.12
1981.11
1482.11
929.135
808.912
551.065
406.381
St RaTc
1.58146
2.12156
3.53568
4.12323
4.84778
5.87376
6.1645
7.00007
7.77824
ac
10914.5
5741.45
2861.43
2242.12
1558.09
1145.58
1042.24
830.852
729.319
Osc RaTc
T a = 500
6.63332
6.74544
5.52277
5.56785
6.7824
5.61258
5.61258
5.65694
5.65694
ac
18438.9
9490.00
4081.89
3169.84
2248.82
1307.81
1114.38
714.969
501.756
St RaTc
5.56785
5.83104
6.36404
6.51928
6.81916
7.38248
7.58294
8.2159
8.86008
ac
Table 2 Critical oscillatory and stationary Rayleigh number and corresponding wavenumber for different values of thermal anisotropy parameter η and Taylor number T a
326 M. S. Malashetty, I. Begum
144.351
150.485
189.480
197.857
228.885
783.993
0.005
0.01
0.05
0.06
0.1
1.0
783.99
2288.85
3297.61
3789.60
15048.50
28870.20
138997.00
2.645
3.082
3.240
3.316
3.937
4.123
4.359
1188.39
602.798
552.580
537.708
453.247
434.425
412.968
1188.39
6027.98
9209.66
10754.16
45324.70
86885.00
412968.00
For fixed values of η = 0.3, ξ = 0.5, RaS = 100, PrD = 10, Le = 10 and λ = 0.7
138.997
3.464
5.788
6.442
6.708
8.831
9.695
11.202
ac
1135.10
834.83
850.97
858.63
957.78
1032.25
1193.70
Osc RaTc
ac
St /Da RaTc
T a = 500
St RaTc
Osc RaTc
Osc /Da RaTc
Ta = 0
0.001
Da
1135.1
8348.3
14182.8
17172.6
95778.0
206450.0
1193700.0
Osc /Da RaTc
3.605
7.106
8.093
8.485
13.153
15.984
21.436
ac
1375.32
917.46
906.86
907.41
983.31
1040.52
1169.33
St RaTc
18.533 98331.4
1375.3
9174.7
15114.5
18148.2
4.062
7.842
8.972
9.407
13.802
15.748
1.169 × 106 208104.0
ac
St /Da RaTc
Table 3 Critical oscillatory and stationary Rayleigh number and corresponding wavenumber for different values of Darcy number Da and Taylor number T a
The Effect of Rotation on the Onset Double Diffusive Convection 327
123
328
M. S. Malashetty, I. Begum
1 ∂ ∂ϑ ∂ 2ψ 1 + − Da∇ 2 + T a 1/2 2 = 0, PrD ∂t ξ ∂z ∂z 2 ∂T ∂ ∂ (ψ, T ) ∂ψ ∂2 − η 2 + 2 T− = 0, + ∂t ∂x ∂z ∂ (x, z) ∂x ∂ (ψ, S) ∂ψ ∂S 1 2 λ − ∇ S− + = 0. ∂t Le ∂ (x, z) ∂x
(4.2) (4.3) (4.4)
Here, ϑ is the z-component of the vorticity vector called the zonal velocity. The first effect of non-linearity is to distort the temperature and concentration fields through the interaction of ψ, T , and also ψ, S. The distortion of these fields will corresponds to a change in the horizontal mean, i.e., a component of the form sin(2π z) will be generated. Thus a minimal Fourier series which describes the finite amplitude free convection is given by, ψ = A1 (t) sin(ax) sin(π z),
(4.5)
T = A2 (t) cos(ax) sin(π z) + A3 (t) sin(2π z),
(4.6)
S = A4 (t) cos(ax) sin(π z) + A5 (t) sin(2π z),
(4.7)
ϑ = A6 (t) sin(ax) cos(π z) + A7 (t) sin(2π x),
(4.8)
where the amplitudes Ai (t), i = 1, 7, are to be determined from the dynamics of the system. Substituting equations (4.5)–(4.8) into equations (4.1)–(4.4) and equating the coefficients of like terms we obtain the following non-linear autonomous system dX = D, dt
(4.9)
where X = (Ai )T , D = (Di )T i = 1, 6 with PrD 2 δ1 A1 + A1 Daδ 4 − π T a 1/2 A6 + a RaT A2 − a RaS A4 , 2 δ πa D2 = −a A1 − δ22 A2 − πa A1 A3 , D3 = −4π 2 A3 + A1 A2 , 2 2 1 δ A4 D4 = − A1 a + + πa A1 A5 , λ Le 4π 2 A5 1 πa A1 A4 Pr A4 A6 π − , D6 = Da A6 δ 2 − − π 2 T a 1/2 A1 . D5 = λ 2 Le π ξ D1 = −
The non-linear system of autonomous differential equations 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 of equations (4.9) 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 flow field has a constant negative divergence. Indeed, 6 PrD δ12 2 ∂ dAi PrD π 1 2 2 2 =− + δ2 + 4π + (δ + 4π ) + , (4.10) ∂ Ai dt δ2 Leλ ξ i=1
which is always negative, and therefore, the system is bounded and dissipative. 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. 4.10
123
The Effect of Rotation on the Onset Double Diffusive Convection
329
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 V (t) = V (0) exp −
PrD π PrD δ12 2 1 2 δ +4π 2 + + δ2 +4π 2 + δ2 Le λ ξ
t .
(4.11)
This expression indicates that the volume decreases exponentially with time. We can also infer that, the large Darcy–Prandtl number and very small Lewis number (Le < 1) tend to enhance dissipation. 4.1 Steady Finite Amplitude Motions The simplified model represented by Eq. 4.9 has the great advantage that steady finite amplitude solutions can be obtained at once and their stability can be investigated analytically. From qualitative predictions we look into the possibility of an analytical solution. In the case of steady motions, setting the left-hand sides of Eq. 4.9 equal to zero, and writing all Ai in terms of A1 , we get a1 r 2 + a2 r + a3 = 0, where r = where
A21 8
(4.12)
and
1 Daδ 4 + δ12 Daδ 2 ξ − 1 , a1 = a 4 Le2 −π 2 T a + ξ 2 −a a 2 Le (Le RaT − RaS ) Daδ 2 ξ − 1 + δ 2 + Le2 δ22 δ12 + π 2 T aξ a2 = ξ −Daδ 2 Daδ 4 ξ − δ 2 + δ12 ξ , 1 2 −a RaT δ 2 − Le RaS δ22 Daδ 2 ξ − 1 + δ 2 δ22 Daδ 2 − δ12 a3 = ξ −π 2 T aξ Daδ 4 ξ + δ12 ξ − δ 2 .
The required root of Eq. 4.12 is r=
1/2 1 −a2 + a22 − 4a1 a3 . 2a1
(4.13)
When we let the radical in the above equation to vanish, we obtain the expression for finite amplitude Rayleigh number Ra F , which characterizes the onset of finite amplitude steady motions. The finite amplitude Rayleigh number can be obtained in the form Ra F =
1/2 1 −b2 + b22 − 4b1 b3 , 2b1
(4.14)
123
330
M. S. Malashetty, I. Begum
where 2 1 4 4 a Le Daδ 2 ξ − 1 , 2 ξ 1 b2 = 2 2a 2 Le2 Daδ 2 ξ − 1 a 2 Le RaS (1− Daδ 2 ξ ξ − δ 2 − Le2 δ22 Daδ 4 + δ12 − Daδ 2 δ12 ξ + π 2 T aξ − Daδ 6 ξ , 1 b3 = 2 a 2 Le RaS Daδ 2 ξ − 1 + δ 2 − Le2 δ22 (Daδ 2 ξ 2 − δ12 − π 2 T aξ Daδ 4 ξ − δ 2 + δ12 ξ . b1 =
4.2 Heat and Mass Transports In the study of convection in fluids, the quantification of heat and mass transport is important. 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. If H and J are the rate of heat and mass transport per unit area, respectively, then ∂ Ttotal ∂ Stotal H = −κT z , and J = −κS , (4.15) ∂z z=0 ∂z z=0 where the angular bracket corresponds to a horizontal average and z z Ttotal = T0 − T + T (x, z, t), and Stotal = S0 − S + S(x, z, t). (4.16) d d Substituting Eqs. 4.6 and 4.7 in Eq. 4.16 and using the resultant equations in Eq. 4.15, we get κS S κT z T (1 − 2π A3 ) , and J = (1 − 2π A5 ) . d d The Nusselt number and Sherwood number are defined by H=
Nu =
H J = 1 − 2π A3 , and Sh = = (1 − 2π A5 ). κT z T /d κS S/d
(4.17)
(4.18)
Writing A3 and A5 in terms of A1 , and substituting in Eq. 4.18, we obtain Nu = 1 +
2r 2r , Sh = 1 + . r + δ 2 /a 2 Le2 r + δ22 /a 2
(4.19)
The second term on the right-hand side of Eq. 4.19 represent the convective contribution to heat and mass transport, respectively.
5 Results and Discussion The effect of rotation on the onset of double diffusive convection in a sparsely packed anisotropic porous layer, which is heated and salted from below, is investigated analytically using the linear and nonlinear theories. In the linear stability theory the expressions for the stationary and oscillatory Rayleigh number are obtained analytically along with the dispersion
123
The Effect of Rotation on the Onset Double Diffusive Convection
331
800
0.5, 1.5, 2, 10 700
ξ = 0.1, 0.2
RaT
ξ = 0.1, 0.2, 0.5, 1.5, 2, 10 600 Stationary Oscillatory Osc Isotropic case
Ta = 100, RaS = 100, η = 0.3, PrD = 10 Le = 10, Da = 0.1 λ = 0.7
500
400 0
3
6
9
a Fig. 1 Neutral stability curves for different values of mechanical anisotropy parameter ξ
relation for frequency of oscillation. The nonlinear theory provides the quantification of heat and mass transports and also explains the possibility of the finite amplitude motions. The neutral stability curves in the RaT −a plane for various parameter values are as shown in Figs. 1, 2, 3, 4, 5, 6, 7, and 8. We fixed the values for the parameters as ξ = 0.5, η = 0.3, T a = 100, Da = 0.1, RaS = 100, Le = 10, λ = 0.7 and PrD = 10 except the varying parameter. From these figures it is clear that the neutral curves are connected in a topological sense. This connectedness allows the linear stability criteria to be expressed in terms of the critical Rayleigh number, RaTc below which the system is stable and unstable above. The effect of mechanical anisotropy parameter ξ on neutral curves for fixed values of other parameters is shown in Fig. 1. We observe from this figure that the convection sets in as oscillatory mode prior to the stationary mode. It can be observed that the critical value of Rayleigh number decreases with increasing ξ up to certain value, and with further increase in the value of the ξ , the critical Rayleigh number increases. The initial decreasing of the Rayleigh number with increasing ξ is found only for small and moderate values of the Taylor number. We also found that for small and moderate Taylor number, the critical Rayleigh number for stationary and oscillatory mode increases with increasing ξ when ξ exceeds certain value (Table 1). We find here a striking effect that for high rotation the effect of mechanical anisotropy parameter reverses. That is the minimum of the Rayleigh number for stationary and oscillatory mode increases with increasing ξ . However, for zero Taylor number that is in the absence of rotation, the critical Rayleigh number for stationary and oscillatory mode decreases monotonically with increasing ξ . The effect of mechanical anisotropy parameter in the absence of rotation can be interpreted as follows: For fixed vertical permeability K z , increasing ξ , decreases the minimum of the Rayleigh number for stationary and oscillatory modes, physically, this means that the basic conduction state in the porous medium becomes more unstable as the horizontal permeability increases. Larger horizontal permeability induces horizontal motion, and the conduction state solution is thus destabilized. The
123
332
M. S. Malashetty, I. Begum 1200
η=2
900
RaT
Ta = 100, Le = 10, ξ = 0.5, Da = 0.1, RaS = 100, PrD = 10, λ = 0.7
600
1.5
0.5
Stationary Oscillatory Osc Isotropic case
300
0
0.3 0.1
4
8
12
a Fig. 2 Neutral stability curves for different values of thermal anisotropy parameter η
smaller resistance to horizontal flow also leads to a lengthening of the horizontal wavelength at the onset. Figure 2 indicates the effect of thermal anisotropy parameter η on the neutral stability curves for the fixed values of other parameters. We find that when η > 1, i.e., the horizontal component of the thermal diffusivity is more dominant, the system becomes more stable. This is the case of insulating materials. On the other hand, when η < 1, i.e., the vertical component of the thermal diffusivity dominates, the system becomes more unstable. This is the case, in industry applications, for materials which facilitate heat transfer. The physical interpretation for this may be given as: Larger values of the horizontal thermal diffusivity correspond to stabilization of the basic state, and onset of convection at a larger wavelength. This can be explained by the fact that, as η increases, a heated fluid parcel loses more heat in the horizontal directions, and hence loses its buoyancy force. Therefore, the basic state becomes more stable, and the wave length is increased. The details of critical values Rayleigh number and corresponding wavenumbers for wide range of values of the thermal anisotropy parameter is presented in Table 2. Figure 3 depicts the effect of Taylor number T a on the neutral stability curves. We find that the effect of increasing T a is to increase the critical value of the Rayleigh number for stationary and oscillatory modes and the corresponding wavenumber. Thus the Taylor number T a has a stabilizing effect on the double diffusive convection in sparsely packed anisotropic porous medium. Figure 4 presents the effect of Darcy number Da on the neutral stability curves. It is worth mentioning here that the Rayleigh number is normalized with respect to the Darcy number in the sense that the Rayleigh number is modified for the clear fluid. We find from this figure that RaTc /Da decreases with increasing Da for both stationary and oscillatory modes. This is because, if the Darcy number is increased then the permeability is increased and thus the viscous drag is decreased. Hence the critical Rayleigh number decreases. Further, the wavenumber at which the minimum occur decreases with increasing Da, indicating that the wavelength increases with Da. In Table 3 we present the critical
123
The Effect of Rotation on the Onset Double Diffusive Convection
333
1000
RaT
800
500
600
100 400
Stationary Oscillatory
50 η = 0.3, ξ = 0.5, RaS = 100, PrD = 10, Le = 10, Da = 0.1, λ= 0.7
10 Ta = 0
200 0
2
4
6
8
10
a Fig. 3 Neutral stability curves for different values of Taylor number Ta 5
1x10
RaS = 100, η = 0.3 , PrD = 10 Le = 10, ξ = 0.5, λ = 0.7 Ta = 100
4
RaT /Da
8x10
Da = 0.01
4
4x10
0.05
Stationary Oscillatory
0.1 0 0
4
8
12
16
a Fig. 4 Neutral stability curves for different values of Darcy number Da
Rayleigh numbers and the corresponding wavenumbers for a range of values of the Darcy number. This table also indicates the normalized critical Rayleigh numbers, i.e., RaTc /Da. Figure 5 depicts the effect of solute Rayleigh number RaS on the neutral stability curves for stationary and oscillatory modes. We find that the effect of increasing RaS is to increase the critical value of the Rayleigh number for stationary and oscillatory modes and the corresponding wavenumber. Thus the solute Rayleigh number RaS has a stabilizing effect on the double diffusive convection in sparsely packed anisotropic porous medium. In Fig. 6 the marginal stability curves for different values of Lewis number Le are drawn. It is found that with
123
334
M. S. Malashetty, I. Begum 800
RaS = 200
600
RaT
150 100 50
400
23.020
η = 0.3, ξ = 0.5, Ta = 100 PrD = 10, Le = 10, Da = 0.1 λ = 0.7
Stationary Oscillatory
10
200 0
2
4
6
8
10
a Fig. 5 Neutral stability curves for different values of solute Rayleigh number Ras
the increase of Le the critical values of Rayleigh number and the corresponding wavenumber for the oscillatory mode decrease while those for stationary mode increase. Therefore, the effect of Le is to advance the onset of oscillatory convection where as its effect is to inhibit the stationary onset. The neutral stability curves for different values of Darcy–Prandtl number PrD are presented in Fig. 7. From this figure it is evident that for small and moderate values of PrD the critical value of oscillatory Rayleigh number decreases with the increase of PrD , however this trend is reversed for large values of PrD . The effect of normalized porosity λ is depicted in the Fig. 8. We observe that an increase in λ decreases the minimum of the Rayleigh number for oscillatory mode, indicating that, the effect of increasing λ is to advance the onset of oscillatory convection. The detailed behavior of stationary and oscillatory critical Rayleigh number with respect to the Taylor number is analyzed in the RaTc − T a plane through Figs. 9, 10, 11, 12, 13, 14, and 15. We observe that the critical Rayleigh number for stationary and oscillatory mode increases with Taylor number, indicating its stabilizing effect. The other striking effect of rotation found in this study is that it reverses the effect of mechanical anisotropy parameter and the Darcy number when its value exceeds certain threshold value. In Fig. 9, we display the variation of RaTc with Taylor number T a for different values of mechanical anisotropy parameter ξ for the fixed values of other parameters. It is important to note that the critical Rayleigh number RaTc decreases with the increase of ξ for small values of Taylor number T a, and for the moderate and large values of T a, the critical Rayleigh number increases with increasing ξ . However, contrary to its usual influence on the onset of convection in the absence of rotation, the mechanical anisotropy parameter ξ show contrasting effect on the stationary and oscillatory onset at moderate and high rotation rates. Thus the mechanical anisotropy parameter ξ has dual effect on oscillatory and stationary convection in the presence of rotation. Figure 10 indicates the variation of RaTc with T a for different values of thermal anisotropy parameter η. It is observed that when η > 1, the critical Rayleigh number RaTc increases with the increase of η indicating that the effect of increasing thermal anisotropy parameter is to delay the onset of stationary and oscillatory convection as
123
The Effect of Rotation on the Onset Double Diffusive Convection
335
900 η = 0.3, ξ = 0.5, RaS = 100 PrD = 10, Ta = 100, Da = 0.1 λ = 0.7
800
RaT
700
600
Le = 5
500
10
300
2
4
Stationary Oscillatory
100 150
50
400
6
8
10
12
7
8
a Fig. 6 Neutral stability curves for different values of Lewis number Le
540
520
PrD = 1
RaT
500
480
10
η = 0.3, ξ = 0.5, RaS = 100, Ta = 100, Le = 10, Da = 0.1 λ = 0.7
460
2
50
5
440 3
4
5
6
a Fig. 7 Neutral stability curves for different values of Darcy–Prandtl number PrD
compared to the isotropic case. On the other hand, when η < 1, the critical Rayleigh number RaTc decreases with the decrease of η indicating that the effect of decreasing thermal anisotropy parameter is to advances the onset of stationary and oscillatory convection as compared to the isotropic case. Figure 11 presents the variation of RaTc /Da with T a for different values of Darcy number Da. We find that the critical Rayleigh number RaTc /Da decreases with the increase of Da. It is important to note that the convection mode switches from oscillatory to stationary
123
336
M. S. Malashetty, I. Begum 900
750
λ = 0.3
RaT
0.4 600
0.5 0.7
450 η = 0.3, ξ = 0.5, RaS = 100
0.9
PrD = 10, Le = 10, Da = 0.1 Ta = 100
1
Stationary Oscillatory
300 0
2
4
6
8
10
a Fig. 8 Neutral stability curves for different values of normalized porosity λ 1300
Stationary Oscillatory
1200
1000
RaS = 100, η = 0.3, PrD = 10 Le = 10, Da = 0.1 λ = 0.7
RaT,c
800
600
ξ = 0.1, 0.2, 0.5, 2, 10 400
St Isotropic Case Osc Isotropic Case
ξ = 0.1, 0.2, 0.5, 2, 10 200
1
10
100
1000
Ta Fig. 9 Variation of critical Rayleigh number for different values of mechanical anisotropy parameter ξ
when the Taylor number exceeds certain critical value. The variation of RaTc with T a for different values of solute Rayleigh number RaS and Lewis number Le on the onset criteria is shown in Figs. 12 and 13, respectively. We observe from these figures that the effect of RaS is to delay the onset of convection while the effect of Le is to advance the onset of double diffusive convection. That is the solute Rayleigh number makes the system more stable while the Lewis number is responsible for the advancement of oscillatory convection.
123
The Effect of Rotation on the Onset Double Diffusive Convection
337
2000
η =2
1.5 1500
RaT,c
η =2
0.5
1.5
1000
0.3
0.3
500
RaS = 100, ξ = 0.5, PrD = 10, Le = 10, Da = 0.1, λ = 0.7
0.1 St Isotropic Case Osc Isotropic Case
Stationary Oscillatory
0 1
10
100
1000
Ta Fig. 10 Variation of critical Rayleigh number for different values of thermal anisotropy parameter η
5
2x10
Stationary Oscillatory
5
10
RaTc/Da
Da = 0.01
0.05 4
10
0.1
RaS = 100, η = 0.3 , PrD = 10 Le = 10, ξ = 0.5, λ = 0.7 3
10
0
10
1
2
10
10
3
10
Ta Fig. 11 Variation of critical Rayleigh number for different values of Darcy number Da
The variation of RaTc with T a for different values of Darcy–Prandtl number PrD is presented in Fig. 14. From this figure it is evident that for small and moderate values of PrD the critical value of oscillatory Rayleigh number decreases with the increase of PrD ; however, this trend is reversed for large values of PrD . In Fig. 15 the variation of RaTc with T a for different values of normalized porosity λ is shown for the fixed values of other parameters.
123
338
M. S. Malashetty, I. Begum 1500 η = 0.3, ξ = 0.5
PrD = 30, Le = 7, Da = 0.1, λ = 0.7
RaS = 200 150
RaT,c
1000
100 50 500
20 10 Stationary Oscillatory
0 1
10
100
1000
Ta Fig. 12 Variation of critical Rayleigh number for different values of solute Rayleigh number Ras 1000 η = 0.3, ξ = 0.5 PrD = 30, RaS = 100, λ = 0.7, Da = 0.1
800
RaT,c
10
600
Le = 5 50 100 150
400
Stationary Oscillatory
200 1
10
100
1000
Ta Fig. 13 Variation of critical Rayleigh number for different values of Lewis number Le
Osc decreases with the increase of λ. Therefore, the effect of It is important to note that RaTc normalized porosity is to advance the onset of oscillatory convection. The variation of the critical Rayleigh number for stationary, oscillatory and finite amplitude modes with Taylor number Ta for different values of mechanical anisotropy parameter ξ , thermal anisotropy parameter η and Darcy number Da and for fixed values of other parameters are shown in Fig. 16, 17, and 18. It is observed that critical Rayleigh number for finite amplitude, stationary and oscillatory mode increases with increasing Taylor number. We find that for small anisotropy parameters (both mechanical and thermal) the convection sets in
123
The Effect of Rotation on the Onset Double Diffusive Convection
339
400
RaT,c
350
PrD = 1
50
300
2
RaS = 100, ξ = 0.5, η = 0.3, Le = 10, Da = 0.1, λ = 0.7
250
200
5
10
1
10
100
Ta Fig. 14 Variation of critical Rayleigh number for different values of Darcy–Prandtl number PrD 1050
RaS = 100, η = 0.3, PrD = 10, Le = 10, Da = 0.1, ξ = 0.5
900
RaT,c
750
600
λ = 0.3
0.4
450
0.5 0.7
300
0.9 1
Stationary Oscillatory
150 1
10
100
1000
Ta Fig. 15 Variation of critical Rayleigh number for different values of normalized porosity λ
through finite amplitude mode while for large anisotropy the first bifurcation will be through oscillatory mode (Figs. 16, 17). It is also important to note that the finite amplitude convection sets in prior to oscillatory and stationary convection when the Darcy number is small and as Da is increased the transition is through the oscillatory mode (Fig. 18). In Figs. 19, 20, 21, and 22, the effect of various parameters on heat and mass transfer is displayed. Figure 19 shows that the Nusselt number and Sherwood number decrease with increasing mechanical anisotropy parameter. The Nusselt number decreases with increasing
123
340
M. S. Malashetty, I. Begum 1200 Stationary Oscillatory Finite amplitude
1000
ξ = 2, 0.5, 0.1 800
RaT,c
2 600
ξ = 2, 0.5, 0.1
400
0.5 200
RaS = 100, Da = 0.01, PrD = 10, Le = 10, η = 0.3, λ = 0.7
ξ = 0.1 0 1
10
100
1000
Ta Fig. 16 Variation of critical Rayleigh with Taylor number Ta for different values of mechanical anisotropy parameter ξ
2000
η=2 Stationary Oscillatory Finite amplitude
RaT,c
1500
RaS = 100, Da = 0.01, PrD = 10, Le = 10, ξ = 0.5, λ = 0.7
1000
η = 0.1, 0.3, 2
η=2
0.3 0.3
500
0.1 0.1 0 1
10
100
1000
Ta Fig. 17 Variation of critical Rayleigh with Taylor number Ta for different values of thermal anisotropy parameter η
thermal anisotropy parameter while Sherwood number increases (Fig. 20). The Taylor number reduces both heat and mass transfer and its effect is more significant on heat transfer (Fig. 21). Both the Nusselt number and Sherwood number decrease with increasing Darcy number (Fig. 22).
123
The Effect of Rotation on the Onset Double Diffusive Convection
341
5
2x10
RaS = 100, η = 0.3 , PrD = 10 Le = 10, ξ = 0.5, λ = 0.7
5
10
RaTc/Da
Da = 0.01
0.05
Da = 0.01 4
10
0.1
0.1
0.05
Stationary Oscillatory Finite amplitude 3
10
0
10
1
2
10
3
10
10
Ta Fig. 18 Variation of critical Rayleigh with Taylor number Ta for different values of Darcy number Da 3.0
Nu/Sh
2.5
2.0
ξ = 0.1, 0.5, 1, 2 1.5
RaS = 100, η = 0.3, Ta = 100 Le = 10, Da = 0.01 1.0 1.0
1.5
2.0
Nu Sh
2.5
F
RaT /RaTc Fig. 19 Variation of Nusselt number and Sherwood number with critical Rayleigh number for different values of mechanical anisotropy parameter ξ
6 Conclusions The effect of rotation on the onset of double diffusive convection in a sparsely packed anisotropic porous layer, which is heated and salted from below, is investigated analytically using the linear and nonlinear theories. The usual normal mode technique is used to solve the linear problem. The truncated Fourier series method is used to make the finite amplitude
123
342
M. S. Malashetty, I. Begum 3.0
η = 2, 0.5, 0.3 , 0.1
Nu/Sh
2.5
2.0
Nu Sh
1.5
RaS = 100,
ξ = 0.5, Da = 0.01
Le = 10, Ta = 100 1.0 1.0
1.5
2.0
2.5
F
RaT /RaTc Fig. 20 Variation of Nusselt number and Sherwood number with critical Rayleigh number for different values of thermal anisotropy parameter η 3.0
Nu/Sh
2.5
2.0
Ta = 0,10,100,500 Nu Sh
1.5
RaS = 100,
η = 0.3, Da = 0.01 Le = 10, ξ = 0.5
1.0 1.0
1.5
2.0
2.5
F
RaT /RaT,c
Fig. 21 Variation of Nusselt number and Sherwood number with critical Rayleigh number for different values of Taylor number Ta
analysis. The following conclusions are drawn: Contrary to its usual influence on the onset of convection in the absence of rotation, the mechanical anisotropy parameter ξ shows contrasting effect on the stationary and oscillatory onset at moderate and high rotation rates. The mechanical anisotropy parameter ξ has dual effect on stationary, oscillatory and finite amplitude convection. When the horizontal component of thermal diffusivity is large (η > 1), the system becomes more stable while the vertical component of thermal diffusivity is large
123
The Effect of Rotation on the Onset Double Diffusive Convection
343
3.0
Nu/Sh
2.5
2.0
Da = 0.001, 0.01, 0.1 1.5
1.0 1.0
RaS = 100, η = 0.3, Ta = 100 Le = 10, ξ = 0.5
Nu Sh
1.5
2.0
2.5
F
RaT /RaTc Fig. 22 Variation of Nusselt number and Sherwood number with critical Rayleigh number for different values of Darcy number Da
(η < 1), the system becomes more unstable. The Taylor number T a has a stabilizing effect on the double diffusive convection in sparsely packed anisotropic porous medium. The striking effects of rotation found in this study is that it reverses the effect of mechanical anisotropy parameter and the convection mode switches from oscillatory to stationary mode when its value exceeds certain threshold value. The effect of solute Rayleigh number is to delay, both stationary and oscillatory convection. The effect of Lewis number is to delay the stationary convection and it advances the oscillatory convection. The Darcy–Prandtl PrD has a dual effect on the oscillatory mode. The effect of normalized porosity is to advance the onset of oscillatory convection. The first bifurcation is through the finite amplitude mode when the anisotropy parameters and Darcy number are small. The increasing thermal anisotropy parameter η suppresses the heat transport while the mass transport is reinforced. The effect of increasing mechanical anisotropy parameter ξ , Taylor number Ta, and Darcy number Da is to suppress the heat and mass transport. Acknowledgments This study is supported by UGC New Delhi, under the Special Assistance Programme DRS Phase-II. The authors thank the reviewers for their useful suggestions.
References Amahmid, A., Hasnaoui, M., Mamou, M., Vasseur, P.: Double-diffusive parallel flow induced in a horizontal Brinkman porous layer subjected to constant heat and mass fluxes: analytical and numerical studies. Heat Mass Transf. 35, 409–421 (1999) Bahloul, A., Boutana, N., Vasseur, P.: Double-diffusive and soret-induced convection in a shallow horizontal porous layer. J. Fluid Mech. 491, 325–352 (2003) Bennacer, R., Mohammad, A.A., Ganaoui, M.El.: Analytical and numerical investigation of double diffusion in thermally anisotropy multi layer porous medium. Heat Mass Transf. 41, 298–305 (2005) Brand, H., Steinberg, V.: Nonlinear effects in the convection instability of a binary mixture in a porous medium near threshold. Phys. Lett. 93A, 333–336 (1983)
123
344
M. S. Malashetty, I. Begum
Castinel, G., Combarnous, M.: Critere d’apparition de la convection naturelle dans une couche poreuse anisotrope horizontal. C.R. Acad. Sci. B 278, 701–704 (1974) Chakrabarti, A., Gupta, A.S.: Nonlinear thermohaline convection in a rotating porous medium. Mech. Res. Commun. 8, 9–22 (1981) Chen, C.F., Chen, F.: Double-diffusive fingering convection in a porous medium. Int. J. Heat Mass Transf. 36, 793–807 (1993) Epherre, J.F.: Criterion for the appearance of natural convection in an anisotropic porous layer. Int. Chem. Eng. 17, 615–616 (1977) Givler, R.C., Altobelli, S.A.: A determination of the effective viscosity for the Brinkman–Forchheimer flow model. J. Fluid Mech. 258, 355–370 (1994) Govender, S.: On the effect of anisotropy on the stability of convection in a rotating porous media. Transp. Porous Media 64, 413–422 (2006) Govender, S.: Coriolis effect on the stability of centrifugally driven convection in a rotating anisotropic porous layer subjected to gravity. Transp. Porous Media 67, 219–227 (2007) Govender, S., Vadasz, P.: The effect of mechanical and thermal anisotropy on the stability of gravity driven convection in rotating porous media in the presence of thermal non-equilibrium. Transp. Porous Media 69, 55–66 (2007) Guo, J., Kaloni, P.N.: Nonlinear stability problem of a rotating doubly diffusive porous layer. J. Math. Anal. Appl. 190, 373–390 (1995) Hill, A.A.: Double-diffusive convection in a porous medium with a concentration based internal heat source. Proc. R. Soc. A 461, 561–574 (2005) Ingham, D.B., Pop, I. (eds.): Transport Phenomena in Porous Media. Pergamon Press, Oxford (1998) Ingham, D.B., Pop, I. (eds.): Transport Phenomena in Porous Media, vol. III. Elsevier, Oxford (2005) Knobloch, E.: Oscillatory convection in binary mixtures. Phys. Rev. A34, 1538–1549 (1986) Kvernvold, O., Tyvand, P.A.: Nonlinear thermal convection in anisotropic porous media. J. Fluid Mech. 90, 609–624 (1979) Lombardo, S., Mulone, G.: Necessary and sufficient conditions of global nonlinear stability for rotating double diffusive convection in a porous medium. Contin. Mech. Thermodyn. 14(16), 527–540 (2002) Malashetty, M.S.: Anisotropic thermo convective effects on the onset of double diffusive convection in a porous medium. Int. J. Heat Mass Transf. 36, 2397–2401 (1993) Malashetty, M.S., Swamy, M.: The effect of rotation on the onset of convection in a horizontal anisotropic porous layer. Int. J. Therm. Sci. 46, 1023–1032 (2007) Malashetty, M.S., Swamy, M.: The onset of convection in a binary fluid saturated anisotropic porous layer Int. J. Therm. Sci. 49, 861–878 (2010) Malashetty, M.S., Heera, R.: The effect of rotation on the onset of double diffusive convection in a horizontal anisotropic porous layer. Transp. Porous Media 74, 105–127 (2008) Malashetty, M.S., Pop, I., Heera, R.: Linear and nonlinear double diffusive convection in a rotating sparsely packed porous layer using a thermal non-equilibrium model. Contin. Mech. Thermodyn. 21, 317–339 (2009) Mamou, M., Vasseur, P.: Thermosolutal bifurcation phenomena in porous enclosures subject to vertical temperature and concentration gradients. J. Fluid Mech. 395, 61–87 (1999) Mansour, A., Amahmid, A., Hasnaoui, M., Bourich, M.: Multiplicity of solutions induced by thermosolutal convection in a square porous cavity heated from below and submitted to horizontal concentration gradient in the presence of Soret effect. Numer. Heat Transf. A 49, 69–94 (2006) McKibbin, R.: Thermal convection in layered and anisotropic porous media: a review. In: Wooding, R.A., White, I. (eds.) Convective Flows in Porous Media, pp. 113–127. Department of Scientific and Industrial Research, Wellington (1985) McKibbin, R. : Convection and heat transfer in layered and anisotropic porous media. In: Quintard, M., Todorovic, M. (eds.) Heat and Mass Transfer in Porous Media, pp. 327–336. Elsevier, Amsterdam (1992) Murray, B.T., Chen, C.F.: Double diffusive convection in a porous medium. J. Fluid Mech. 201, 147–166 (1989) Nield, D.A.: Onset of thermohaline convection in a porous medium. Water Resour. Res. 4, 553–560 (1968) Nield, D.A., Bejan, A.: Convection in Porous Media, 3rd edn. Springer, New York (2006) Nilsen, T., Storesletten, L.: An analytical study on natural convection in isotropic and anisotropic porous channels. Trans. ASME J. Heat Transf. 112, 396–401 (1990) Poulikakos, D.: Double diffusive convection in a horizontal sparsely packed porous layer. Int. Commun. Heat Mass Transf. 13, 5897–5898 (1986) Rosenberg, N.D., Spera, F.J.: Thermohaline convection in a porous medium heated from below. Int. J. Heat Mass Transf. 35, 1261–1273 (1992) Rudraiah, N., Malashetty, M.S.: The influence of coupled molecular diffusion on the double diffusive convection in a porous medium. ASME J. Heat Transf. 108, 872–876 (1986)
123
The Effect of Rotation on the Onset Double Diffusive Convection
345
Rudraiah, N., Shivakumara, I.S., Friedrich, R.: The effect of rotation on linear and nonlinear double diffusive convection in a sparsely packed porous medium. Int. J. Heat Mass Transf. 29, 1301–1317 (1986) Rudraiah, N., Shrimani, P.K., Friedrich, R.: Finite amplitude convection in a two component fluid saturated porous layer. Int. J. Heat Mass Transf. 25, 715–722 (1982) Straughan, B., Hutter, K.: A priori bounds and structural stability for double diffusive convection incorporating the Soret effect. Proc. R. Soc. Lond. A 455, 767–777 (1999) Storesletten, L.: Effects of anisotropy on convective flow through porous media. In: Ingham, D.B., Pop, I. (eds.) Transport Phenomena in Porous Media, pp. 261–283. Pergamon Press, Oxford (1998) Storesletten, L.: Effects of anisotropy on convection in horizontal and inclined porous layers. In: Ingham, D.B. et al. (eds.) Emerging Technologies and Techniques in Porous Media, pp. 285–306. Kluwer, Dordrecht (2004) Vadasz, P. (ed.): Emerging Topics in Heat and Mass Transfer in Porous Media. Springer, New York (2008) Vafai, K. (ed.): Handbook of Porous Media. Marcel Dekker, New York (2000) Vafai, K. (ed.): Handbook of Porous Media. Taylor & Francis/CRC Press, London/Boca Raton, FL (2005)
123
