The linear stability of a viscoelastic liquid saturated horizontal anisotropic porous layer heated from below and cooled from above is investigated by...

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The onset of convection in a viscoelastic liquid saturated anisotropic porous layer M. S. Malashetty · Mahantesh Swamy

Received: 14 June 2005 / Accepted: 11 April 2006 / Published online: 28 September 2006 © Springer Science+Business Media B.V. 2006

Abstract The linear stability of a viscoelastic liquid saturated horizontal anisotropic porous layer heated from below and cooled from above is investigated by considering the Oldroyd type liquid. A generalized Darcy model, which takes into account the viscoelastic properties, the mechanical and thermal anisotropy is employed as momentum equation. The critical Rayleigh number, wavenumber, for stationary and oscillatory states and frequency of oscillation are determined analytically. It is shown that oscillatory instabilities can set in before stationary modes are exhibited. The effect of the viscoelastic parameter, the mechanical and thermal anisotropy parameters and specific heat ratio on the linear stability of the system is analyzed and presented graphically. Keywords Stationary/oscillatory convection · Mechanical/thermal anisotropy · Viscoelastic liquid Nomenclature √ a wavenumber, l2 + m2 d height of the fluid layer D thermal diffusivity tensor, Dx ii + jj + Dz (kk) g gravitational acceleration, (0, 0, −g) K permeability tensor, Kx−1 ii + jj + Kz−1 (kk) p pressure q velocity vector (u, v, w) RD Darcy–Rayleigh number, β g T dKz ν Dz

Communicated by M. S. Malashetty. M. S. Malashetty (B)· M. Swamy Department of Mathematics, Gulbarga University, Jnana Ganga, Gulbarga- 585 106, India e-mail: [email protected]

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T T t x, y, z

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temperature temperature difference between the walls time space coordinates

Greek symbols β thermal expansion coefficient γ ratio of specific heats (ρc)m (ρc)f ε porosity η thermal anisotropy parameter, Dx Dz λ1 stress relaxation time λ1 dimensionless stress relaxation time, Dz λ1 d2 λ2 strain retardation time dimensionless strain retardation time, Dz λ2 d2 λ2 µ dynamic viscosity ν kinematic viscosity, µ ρ0 dimensionless amplitude of temperature perturbation ρ density σ growth rate ξ mechanical anisotropy parameter, Kx Kz ψ stream function

dimensionless amplitude of stream function ω frequency Subscripts b basic state c critical 0 reference value Superscripts * dimensionless quantity / perturbed quantity Osc oscillatory S stationary 1 Introduction Thermal convection in fluid saturated porous media is of considerable interest due to its numerous applications in different fields such as geothermal energy utilization, oil reservoir modeling, building thermal insulation, and nuclear waste disposals to mention a few. The problem of convective instability of a horizontal fluid saturated porous layer has been extensively investigated and the growing volume of work devoted to this area is well documented by Ingham and Pop (1998), Nield and Bejan (1999), and Vafai (2000). The substantial part of theoretical and experimental works on convective flow in porous media has dealt with isotropic materials. However, in many practical situations the porous materials are anisotropic in their mechanical and thermal properties. Anisotropy is generally a consequence of preferential orientation or asymmetric geometry of porous matrix or fibers and is in fact encountered in numerous systems in industry and in nature. Anisotropy can also be a characteristic of artificial porous

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materials like pelletting used in chemical engineering process and fiber material used in insulating purpose. Despite the practical importance of the topic, in contexts varying from fibrous insulating material to sedimentary rocks, only few studies have been reported on convection in an anisotropic porous medium uniformly heated from below. The review of research on convective flow through anisotropic porous media has been well documented by McKibbin (1985,1992) and Storesletten (1998, 2004). The flow of non-Newtonian fluids in a porous medium is of considerable importance in various areas in science, engineering, and technology, such as material processing, petroleum, chemical and nuclear industries, geophysics, bioengineering, and reservoir engineering. The performance of a reservoir depends to a large extent upon the physical nature of crude oil present in the reservoir. The light crude is essentially Newtonian and is studied extensively using the Darcy equation. On the other hand, the heavy crude is non-Newtonian and a study of such fluids is based on a generalized Darcy equation, which takes into account the non-Newtonian effects. Such an equation is useful in the study of mobility control in oil displacement mechanism, which improves the efficiency of the oil recovery. Furthermore, some oil sand contains waxy crude at shallow depths of the reservoirs, which are considered to be viscoelastic fluids. In such situations, a viscoelastic model of a fluid will be more realistic than the Newtonian model. Besides viscoelastic models are interesting because they fit quite well the data found in experiments of many polymeric fluids. Although the problem of Rayleigh–Benard convection 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 there in). Flow instability and turbulence are far less widespread in viscoelastic fluids than in Newtonian fluids because of the high viscosity of the polymeric fluids. Since elastic behavior is inherent in non-Newtonian fluids, it may be expected that oscillatory convection will sets up in such fluids. The study of Rayleigh–Benard convection 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 a certain viscoelastic fluids. Other than the importance of this problem from the rheological point of view, viscoelastic fluids exhibit unique patterns of instabilities such as the overstability that is not predicted or observed in Newtonian flow. The Rayleigh–Benard convection in the case of a porous medium has also not received much attention and it is in much-to-be desired state. Rudraiah et al. (1989) have studied the linear stability of a viscoelastic fluid in a densely packed porous layer using an Oldroyd model. It is found that the effect of elasticity of the fluid is to destabilize the system and that of porosity is to stabilize the same. A theoretical analysis of thermal instability in a horizontal porous layer saturated with viscoelastic liquid is conducted by Kim et al. (2003). It is found that the overstability is a preferred mode for a certain parameter range. They further reported on the basis of non-linear theory that the convection has the form of a supercritical and stable bifurcation independent of the values of the elastic parameters. Yoon et al. (2004) have studied the onset of thermal convection in a horizontal porous layer saturated with viscoelastic liquid using linear theory. A simple constitutive model was employed to examine the effects of relaxation times. It is shown that the oscillatory instabilities can set in before stationary mode. Recently Malashetty et al. (2006) have studied the linear stability of viscoelastic fluid saturated porous layer using a thermal non-equilibrium model by considering the Oldroyd-B type fluid. They reported that due to the competition

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between the processes of viscous relaxation and thermal diffusion, the first convective instability to be oscillatory rather than stationary. The effect of viscoelastic parameters and the thermal non-equilibrium on the stability of the system is analyzed. The interest in the study of convective instability of viscoelastic fluids saturating a porous layer is of two fold. On the one hand, the influence of the viscoelastic properties however, small can lead to important changes in the convective pattern. On the other hand, the study can be a useful tool to determine some rheological properties of a given fluid, because experiments can be performed with a greater accuracy in convection cells and in turn they could provide a test for some viscoelastic properties. In this paper, we study the onset of convection in a viscoelastic fluid saturated porous layer heated from below with emphasis on how the condition for the onset of convection is modified by elastic effects. The Oldroyd model, which allows to fit the data of many polymeric solutions, is employed to include the viscoelastic properties, as this model is more general in nature compared to Maxwell and Jeffrey’s models.

2 Mathematical formulation 2.1 Basic equations For an Oldroyd-B fluid, the extra-stress tensor T is given by the constitutive equation (Tan and Masuoka 2005) DS DA ¯ ¯ T = −pI + S, S + λ1 = µ A + λ2 , (2.1) Dt Dt where p is the hydrostatic pressure, I the identity tensor, µ the viscosity of the fluid, and S the extra stress tensor, λ¯ 1 and λ¯ 2 are constant relaxation and retardation times, respectively. A = ∇q + ∇qT is the strain-rate tensor, q the velocity vector, ∇ the gradient operator, and T DS ∂ = + q · ∇ S − S ∇q − ∇q S, (2.2) Dt ∂t DA = Dt

T ∂ + q · ∇ A − A ∇q − ∇q A. ∂t

(2.3)

It should be noted that this model includes the classical viscous Newtonian fluid as a special case for λ¯ 1 = λ¯ 2 = 0, and to be the Maxwell fluid when λ¯ 2 = 0. It is well known that in flow of viscous Newtonian fluid at a low speed through a porous medium the pressure drop caused by the frictional drag is directly proportional to velocity, which is the Darcy’s law. By analogy with Oldroyd-B constitutive relationships, the following phenomenological model, which relates pressure drop and velocity for a viscoelastic fluid in an anisotropic porous medium is been given by ∂ ∂ 1 + λ¯ 1 (2.4) ∇p = −µ 1 + λ¯ 2 K · qD ∂t ∂t (see, e.g., Khuzhayorov et al. 2000), where K is the modified permeability tensor, qD is Darcian velocity, which is related to the usual (i.e. volume averaged over a volume element consisting of fluid only in the pores) velocity vector q by qD = ε q, ε is porosity of the porous medium. We note that when λ¯ 1 = λ¯ 2 = 0, Eq. 2.4 simplified to Darcy’s

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law for flow of viscous Newtonian fluid through an anisotropic porous medium. Thus Eq. 2.4 can be regarded as an approximate form of an empirical momentum equation for flow of Oldroyd-B fluid through an anisotropic porous medium. Under consideration of the balance of forces acting on a volume element of fluid, the local volume average balance of linear momentum is given by ρ

dq = −∇p + ρg + ∇ · S + r, dt

(2.5)

where d is the material time derivative, r is Darcy resistance for an Oldroyd-B fluid dt in the porous medium. Since the pressure gradient in Eq. 2.4 can also be interpreted as a measure of the resistance to flow in the bulk of the porous medium, and r is a measure of the flow resistance offered by the solid matrix, thus r can be inferred from Eq. 2.4 to satisfy the following equation ∂ ∂ 1 + λ¯ 1 r = −µε 1 + λ¯ 2 K · q. (2.6) ∂t ∂t Substituting Eq. 2.6 into Eq. 2.5, we obtain dq ∂ ∂ ρ + ∇p − ρg − ∇ · S = −µε 1 + λ¯ 2 K · q. 1 + λ¯ 1 ∂t dt ∂t

(2.7)

For Darcy model, ignoring the material derivative term dq and the viscous term ∇ · S, dt Eq. 2.7 can be simplified to (after dropping the suffix D on q for simplicity) ∂ ∂ ρg − ∇p = µ 1 + λ¯ 2 K · q. (2.8) 1 + λ¯ 1 ∂t ∂t The equations of energy, state, and continuity, for a homogeneous anisotropic (mechanical and thermal) porous medium are given by γ

∂T + q · ∇ T = ∇ · ( D · ∇T), ∂t

(2.9)

ρ = ρ0 1 − β (T − T0 ) ,

(2.10)

∇ · q = 0,

(2.11)

where D is the thermal diffusivity tensor of the fluid saturated porous medium. We consider a horizontal anisotropic porous layer of depth d, saturated with a viscoelastic liquid, which is heated from below and cooled from above. A constant temperature gradient T is maintained between the two walls. We assume that the porous layer possesses horizontal isotropy in both mechanical and thermal properties and that the Oberbeck–Boussinesq approximation is valid, taking into account the fact that in the case of densely packed porous medium the non-linear terms in the velocity and viscous dissipation are negligible. 2.2 Basic state The basic state is assumed to be quiescent and is given by q = 0,

T = Tb (z),

ρ = ρb (z),

p = pb (z).

(2.12)

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The basic state is governed by ∂pb = −ρb g, ∂z

(2.13)

d2 Tb =0 dz2

(2.14)

and ρb = ρ0 [1 − β(Tb − T0 )]

(2.15)

2.3 Perturbed state To study the stability of the system we superimpose infinitesimal perturbations on the basic state, which are of the form, q = 0 + q ,

T = Tb + T ,

p = pb + p ,

ρ = ρb + ρ ,

(2.16)

where the prime indicates that the quantities are infinitesimal perturbations. Using Eq. 2.16 in Eqs. 2.8–2.11 and using the basic state solutions, we obtain the linearized equations governing the infinitesimal perturbations in the form, µ ∂ ∂ 1 K · q = − 1 + λ1 βT g + ∇p , 1 + λ2 (2.17) ρ0 ∂t ∂t ρ0 γ

∂T + w ∂t

dTb dz

= ∇ · ( D · ∇T ),

(2.18)

ρ = −ρ0 βT .

(2.19)

We consider the two-dimensional perturbations by introducing the stream function ∂ψ ψ such that u = ∂ψ ∂z and w = − ∂x . We eliminate p from Eq. 2.17 by operating curl twice on it and then render the resulting equation and Eq. 2.18 dimensionless using the following transformations (x, z) = d (x∗ , z∗ ),

t=

d2 ∗ t , Dz

ψ = Dz ψ ∗ ,

and T = (T)T ∗ (2.20)

to obtain linearized non-dimensional equations as (on dropping the asterisks for simplicity) 2 ∂ ∂ ∂ ∂T 1 ∂2 ψ = −R 1 + λ + 1 + λ2 , (2.21) D 1 ∂t ξ ∂z2 ∂t ∂x ∂x2 2 ∂2 ∂ ∂ψ ∂ , T=− γ − η 2 + 2 ∂t ∂x ∂x ∂z where x ξ= K Kz , the mechanical anisotropy parameter,

Dx Dz , the thermal anisotropy parameter, ¯ λ1 = Ddz2λ1 , the non-dimensional stress relaxation time, ¯ λ2 = Ddz2λ2 , the non-dimensional strain retardation time,

η=

(2.22)

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dKz , the Darcy–Rayleigh number. RD = βgT νDz γ = ratio of specific heats, (ρc)m (ρc)f . Here (ρc)m = ε(ρc)f + (1 − ε)(ρc)s is the effective heat capacity of the porous medium and (ρc)f and (ρc)s are the heat capacities of the fluid and solid, respectively. The boundaries are considered to be impermeable and isothermal so that the appropriate boundary conditions are ψ = T = 0,

at z = 0, 1.

(2.23)

It is important to observe that the Maxwell limit is recovered by setting λ2 = 0 in Eqs. 2.21 and 2.22 and the Newtonian flow is recovered in the limit λ1 = λ2 = 0. Most of the studies on convection in porous media assume the specific heats of the fluid and that of the fluid saturated porous medium equal. However, Kladias and Prasad (1990) in their study relaxed this assumption and have shown that the specific heat ratio has significant effect on the oscillatory instability in a fluid saturated porous medium. Accordingly, in this study we have analyzed the effect of this parameter on the stability of the system and we too found that the specific heat ratio indeed has significant effect on the oscillatory instability and the stationary motions are independent of this parameter in agreement with the results of Kladias and Prasad (1990).

3 Linear stability analysis In this section, we study the onset of convection in a horizontal anisotropic porous layer saturated with viscoelastic fluid using linear theory and predict the thresholds of both marginal and oscillatory motions. The eigenvalue problem (2.21)–(2.23) is solved using the time-dependent periodic disturbances in a horizontal plane, upon assuming that amplitudes are small enough and can be expressed as ψ = sin(nπz) sin(ax) eσ t ,

(3.1)

T = sin(nπz) cos(ax) eσ t ,

(3.2)

where a = (l2 + m2 )1/ 2 , is the resultant horizontal wavenumber and σ , the temporal growth rate. Infinitesimal perturbations of the rest state may either damp or grow depending on the value of the parameter σ . Substituting Eqs. 3.1 and 3.2 into the Eqs. 2.21 and 2.22 we obtain a matrix equation

2 2 (1 + λ2 σ ) a2 + n ξπ (1 + λ1 σ ) a RD

0 = . (3.3) 2 2 2 0 a γ σ + ηa + n π For the non-trivial solutions of and the above Eq. 3.3 simplifies to the following dispersion relationship Aσ 2 + Bσ + C = 0,

(3.4)

where

2 2 2 B = γ + λ2 ηa2 + n2 π 2 a + A = γ λ2 a2 + n ξπ , 2 n2 π 2 2 2 2 2 a + ξ − a RD . and C = ηa + n π

n2 π 2 ξ

− λ1 a2 RD (3.5)

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Equation 3.4 may admit essentially two solutions, depending on whether the instability is stationary or oscillatory. The growth rate σ is in general complex such that σ = σr + iσi . The system with σr < 0 is always stable, while for σr > 0 it will become unstable. When σ = 0, that is σr = 0 and σi = 0, the system is marginally stable. With σr = 0 and σi = 0, the overstable motion may occur. 3.1 Stationary Convection For stationary convection σ in Eqs. 3.1 and 3.2 is real and for marginal stability σ = 0, therefore, the corresponding characteristic values of the Rayleigh number associated with stationary convection are obtained by substituting σ = 0 in Eq. 3.4 and are given by 1 2 n2 π 2 S 2 2 2 RD = 2 ηa + n π a + . (3.6) ξ a Since we are interested in the most dangerous mode, our consideration is confined to the lowest-order mode. Therefore, we set n = 1 for further study. Minimizing RSD with respect to wavenumber ‘a’ yields the critical Rayleigh number for stationary convection 1/ 2 2 η S 2 (3.7) RD,c = π 1 + ξ and the corresponding critical wavenumber aSc = π (ηξ )−1/ 4 .

(3.8)

We observe that the stationary mode is independent of the viscoelastic parameters and specific heat ratio and therefore, it is identical with that obtained by Storesletten (1998), for the Newtonian fluid. This result is in agreement with the results of Rosenblat (1986). The stationary critical Rayleigh number and critical wavenumber are independent of viscoelastic parameters because of the absence of base flow in the present case. This is in contrast to viscoelastic Taylor–Couette flow, where the base flow depends on viscoelastic parameters, thus leading to critical conditions that are influenced by elastic effects. It is obvious, on mathematical ground also, that for neutral stationary mode, the solution is identical to that for an ordinary viscous fluid through porous media. Furthermore for isotropic porous medium, that is when ξ = η = 1 Eqs. 3.7 and 3.8 give the classical results of Lapwood (1948), namely RSD,c = 4π 2 and aSc = π. 3.2 Oscillatory convection The oscillatory instability will set in for the disturbances whose growth rate is such that σr = 0 and σi = 0. Therefore, the condition for marginally stable oscillatory mode is AC > 0 The first conditions gives ROsc D <

1 a2

a2 +

and B = 0. π2 ξ

ηa2 + π 2

(3.9)

(3.10)

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and the second yields ROsc D =

1 π2 2 2 2 γ + λ ηa + π + . a 2 ξ a2 λ1

(3.11)

The critical Rayleigh number that marks the oscillatory convection is obtained as η 1 η γ 1 Osc 2 +2 π + π2 + (3.12) RD,c = γ + λ2 π π 1 + λ1 ξ ξ λ2 λ2 and the corresponding critical wavenumber is 1/ 4 1 γ Osc 1/ 2 2 π + . ac = π ηξ λ2

(3.13)

depends on the parameters It is important to note that the critical wavenumber aOsc c namely, ξ , η, γ , and λ2 and is independent of λ1 . We observe from Eq. 3.12 that the oscillatory Rayleigh number increases with increase in the value of λ2 where as it decreases with increase in the value of λ1 . We also observe from Eq. 3.13 that the critical wavenumber becomes independent of λ2 for large value of λ2 . Then the condition for the existence of overstable motions, using Eqs. 3.10 and 3.11 reduces to λ1 > λ2 +

γ π π + ηξ π 2 +

γ λ2

.

(3.14)

This indicates that for the oscillatory mode the retardation time λ2 must be smaller than the relaxation time λ1 and as soon as the retardation time exceeds this limiting value the convection will attain the steady state, which is indicated in Fig. 1. The dimensionless frequency for neutrally oscillatory mode at the critical state can be obtained in the from 4 2 2 π 4 η aOsc aOsc + π 1 + ηξ − ROsc + ξ c c D,c 2 2 σ i = ωc = . (3.15) 2 2 γ λ2 aOsc + πξ c As λ2 → 0, the following interesting relation on the critical frequency is obtained η 1/ 4 π 1/ 2 ωc = 3 4 . (3.16) γξ λ2/ This indicates that for the small retardation time λ2 , the critical frequency ωc for the oscillatory mode becomes independent of the relaxation time λ1 . This fact can be verified from Fig. 4. Therefore, the retardation time affects the critical frequency much significantly than the relaxation time. In case of isotropic porous medium, that is when ξ = η = 1, Eqs. 3.15 and 3.16 give Osc 4 2 2 aOsc + 2π − ROsc + π4 ac c D,c 2 2 σ i = ωc = (3.17) 2 γ λ2 aOsc + π2 c 1/ 3 and ωc = π 1/ 2 γ λ32 . (3.18)

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Fig. 1 Variation of critical Rayleigh number RD,c with strain retardation time λ2 for different values of stress relaxation time λ1

It is important to note that, in the above analysis, ξ =η does not represent the isotropic porous medium case, even though some of the results coincide with those of isotropic case.

4 Results and discussion The expressions for stationary and oscillatory critical Rayleigh numbers, 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 specific heat ratio and therefore concurs with that obtained by Storesletten (1998) for the case of anisotropic porous layer saturated with the Newtonian fluid. This result is in agreement with the results of Rosenblat (1986). The stationary critical Rayleigh number and critical wavenumber are independent of viscoelastic parameters because of the absence of base flow in the present case. This is in contrast to viscoelastic Taylor– Couette flow, where the base flow depends on viscoelastic parameters, thus leading to critical conditions that are influenced by elastic effects. The critical Rayleigh number for the most dangerous mode (i.e. the oscillatory mode) is derived as a function of both viscoelastic and anisotropy parameters and specific heat ratio. It is observed

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clearly that the overstable motions are possible prior to the steady motions. Further the interesting constraint on the viscoelastic parameters to maintain the overstable motions has also been obtained. The effect of strain retardation time λ2 on the critical Rayleigh number for the fixed values of the anisotropy parameters is shown in Fig. 1 for different values of the stress relaxation time λ1 . It is observed that for each value of λ1 initially the instability sets in, in the form overstable motions and the corresponding critical Rayleigh number increases with λ2 and as soon as the inequality in Eq. 3.14 fails the convection ceases to be oscillatory and bifurcates into steady state. Then the Rayleigh number takes the value given by Eq. 3.7, which characterizes the stationary convection and consequently becomes independent of λ2 . Figure 2 indicates the influence of the mechanical anisotropy parameter ξ on the stability of the system for the fixed values of λ1 = 0.25 and η = 0.2. The behavior of the system is found to be similar to the one shown by Fig. 1. The critical Rayleigh number RD,c decreases with an increase in the value of ξ , for both oscillatory and stationary modes. This result is similar to the one with Newtonian fluid. Thus, the effect of increasing mechanical anisotropy parameter ξ is to destabilize the system.

Fig. 2 Variation of critical Rayleigh number RD,c with strain retardation time λ2 for different values of mechanical anisotropy parameter ξ . Note: For each value of ξ the region above the horizontal line represents the stationary region while the one between horizontal line and the corresponding slanted line represents the oscillatory region

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Fig. 3 Variation of critical Rayleigh number RD,c with strain retardation time λ2 for different values of thermal anisotropy parameter η. Note: For each value of η the region above the horizontal line represents the stationary region while the one between horizontal line and the corresponding slanted line represents the oscillatory region

The effect of thermal anisotropy parameter η on RD,c for fixed values of λ1 = 0.25 and ξ = 0.2 is shown in Fig. 3. The nature of the graph is similar to that shown in Fig. 2. It is observed that with the increasing values of η, the value of RD,c for both oscillatory and stationary modes increases. This indicates that the effect of increasing thermal anisotropy parameter η is to stabilize the system in both oscillatory and stationary regions. Again this result is similar to the one with Newtonian fluid. In Fig. 4, the effect of specific heat ratio γ on the stability of the system is depicted for the fixed values of ξ = 0.1, η = 0.2, and λ1 = 0.75. This figure shows that the oscillatory motions are possible up to a certain range of values of λ2 for which the inequality in Eq. 3.14 is valid. Beyond this range the stationary onset is attained, which is independent of specific heat ratio γ . The critical Rayleigh number ROsc D,c for oscillatory instability is found to increase with γ indicating that the effect of increasing γ is to inhibit the onset of oscillatory instability. Further it is clear from the figure that the effect of specific heat ratio on the stability of the system is significant for γ >1. The variation of critical frequency ωc for the oscillatory mode with the stress relaxation time λ1 for different values of strain retardation time λ2 is shown in Fig. 5 for the values of anisotropy parameters ξ and η fixed to 0.1 and 0.2, respectively. It is

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Fig. 4 Variation of critical Rayleigh number RD,c with strain retardation time λ2 for different values of specific heat ratio γ

observed that for each value of λ2 the critical frequency ωc increases sharply over a small range of values of λ1 and thereafter it remains almost independent of λ1 . Further it is observed that the critical frequency ωc increases with the decrease in the value of λ2 and as λ2 → 0, the critical frequency ωc will become independent of the stress relaxation time λ1 . This fact can be easily verified by Eq. 3.16. Fig. 6 indicates the variation of the critical frequency ωc with anisotropy parameter ξ for different values of the stress relaxation time λ1 . We observe from this figure that the critical frequency ωc increases exponentially when the mechanical anisotropy parameter ξ is very small (ξ < 1) and λ1 ≤ 1, attains a maximum value and then it decreases slowly as ξ takes the larger value. Further for the values of λ1 > 1, the critical frequency decreases with increasing ξ . The variation of critical frequencyωc with the thermal anisotropy parameter η for different values of the stress relaxation time λ1 is shown in Fig 7. We find that the critical frequency ωc increases with increase in the value of the thermal anisotropy parameter η. In Fig. 8 the variation of critical frequency ωc for the oscillatory mode with the specific heat ratio γ for different values of stress relaxation time λ1 is reported. From this figure, it is clear that the critical frequency ωc reduces greatly with γ , this is because

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w w

Fig. 5 Variation of critical frequency ωc with stress relaxation time λ1 for different values of strain retardation time λ2

w

=

Fig. 6 Variation of critical frequency ωc with mechanical anisotropy parameter ξ for different values of stress relaxation time λ1

of the fact that the specific heat ratio stabilizes the system in oscillatory mode (see Fig. 4). Further this figure indicates that the increase in the value of relaxation time λ1 increases the critical frequency ωc .

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w

Fig. 7 Variation of critical frequency ωc with thermal anisotropy parameter η for different values of stress relaxation time λ1

h

w

Fig. 8 Variation of critical fequency ωc with specific heat ratio γ for different values of strain retardation time λ1

5 Conclusion The onset of thermal convection in a horizontal anisotropic porous layer saturated with viscoelastic liquid heated from below is studied using linear theory. The critical Rayleigh number and critical wavenumber in case of stationary mode are independent of viscoelastic parameters because of the absence of base flow. Further the critical

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Rayleigh number and critical wavenumber for stationary mode are independent of the specific heat ratio. The oscillatory instabilities can set in before stationary modes are exhibited. The interesting constraint on the viscoelastic parameters to maintain the overstable motions is given by Eq. 3.14. The elasticity of saturated liquid is a destabilizing factor and it leads to oscillatory motion. The effect of increasing the mechanical anisotropy parameter ξ is to destabilize the system while that of thermal anisotropy parameter η is to stabilize the system. The oscillatory Rayleigh number depends on the specific heat ratio and the specific heat ratio strongly influences the oscillatory instability. It is found that the specific heat ratio inhibits the onset of oscillatory instability. Acknowledgements This work is supported by UGC New Delhi, under the Special Assistance Programme DRS. The authors thank the reviewers for their useful comments and suggestions.

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