The convective instability of Maxwell fluid-saturated porous layer using a thermal non-equilibrium model

J. Non-Newtonian Fluid Mech. 162 (2009) 29–37

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The convective instability of Maxwell fluid-saturated porous layer using a thermal non-equilibrium model M.S. Malashetty ∗ , Sridhar Kulkarni Department of Mathematics, Gulbarga University, Jnana Ganga Campus, Gulbarga 585 106, India

a r t i c l e

i n f o

Article history: Received 24 August 2008 Received in revised form 26 April 2009 Accepted 7 May 2009

Keywords: Maxwell fluid Overstability Thermal non-equilibrium Porous media

a b s t r a c t A linear stability analysis is carried out to study viscoelastic fluid convection in a horizontal porous layer heated from below and cooled from above when the solid and fluid phases are not in a local thermal equilibrium. The modified Darcy–Brinkman–Maxwell model is used for the momentum equation and two-field model is used for the energy equation each representing the solid and fluid phases separately. The conditions for the onset of stationary and oscillatory convection are obtained analytically. Linear stability analysis suggests that, there is a competition between the processes of viscoelasticity and thermal diffusion that causes the first convective instability to be oscillatory rather than stationary. Elasticity is found to destabilize the system. Besides, the effects of Darcy number, thermal non-equilibrium and the Darcy–Prandtl number on the stability of the system are analyzed in detail. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Thermal convection in fluid-saturated porous media is of considerable interest due to its numerous applications in different fields such as thermal insulation engineering, nuclear waste repository, grain storage, mantle convection, geothermal energy utilization, and oil reservoir modeling to mention a few. The problem of convective instability of a horizontal fluid-saturated porous layer heated from below has been extensively investigated and the growing volume of work devoted to this area is well documented by Ingham and Pop [1,2], Vafai [3,4] and Nield and Bejan [5]. Most of the works on convection in porous media have mainly been investigated under the assumption that the fluid and the porous medium are everywhere in local thermodynamic equilibrium (LTE). This is a common practice for most of the studies where the temperature gradient at any location between the two phases is assumed to be negligible. For many practical applications, involving high-speed flows or large temperature differences between the fluid and solid phases, the assumption of local thermal equilibrium is inadequate and it is important to take account of the local thermal non-equilibrium (LTNE) effects. Due to applications of porous media theory in drying, freezing of foods and other mundane materials and applications in everyday technology such as microwave heating, rapid heat transfer from computer chips via use of porous metal foams and their use in heat pipes, it is believed that local thermal non-equilibrium theory will play a major role in future

∗ Corresponding author. Tel.: +91 08472 263296/250086; fax: +91 08472 263206. E-mail address: [email protected] (M.S. Malashetty). 0377-0257/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2009.05.003

developments. Typical situations which require the use of the LTNE model include hyper-porous materials, and media in which there is a significant difference in conductivities between the phases. Recently, attention has been given to the local thermal nonequilibrium model in the study of convection heat transfer in porous media. Much of this work has been reviewed by Nield and Bejan [5]. Rees and co-workers [6–9] in a series of studies have investigated thermal non-equilibrium (LTNE) effect on free convective flows in a porous medium. Free convection in a square porous cavity using a thermal non-equilibrium model is studied by Baytas and Pop [10]. A review of thermal non-equilibrium free convection in a cavity filled with non-Darcy porous medium is also given by Baytas [11]. The problem of two-dimensional steady mixed convection in a vertical porous layer using thermal non-equilibrium model is investigated numerically by Saeid [12]. The effect of thermal non-equilibrium on the onset of convection in a porous layer using the Lapwood–Brinkman model and also including anisotropy in permeability and thermal diffusivity in a densely packed porous layer have been investigated by Malashetty et al. [13,14]. Recently, Straughan [15] has considered a problem of thermal convection in a fluid-saturated porous layer using a global nonlinear stability analysis with a thermal non-equilibrium model. An analysis of viscoelastic effects on linear stability analysis of a densely packed porous layer saturated with Oldroyd fluid by considering a LTNE model has been carried out by Malashetty et al. [16]. This problem has been extended to include the Brinkman effect by Shivakumara et al. [17]. More recent works on thermal non-equilibrium effects on convection includes those by Sheu [18] and Malashetty et al. [19]. It has long been a common belief that oscillatory Rayleigh– Benard convection is hard to realize/cannot be observed in vis-

30

M.S. Malashetty, S. Kulkarni / J. Non-Newtonian Fluid Mech. 162 (2009) 29–37

coelastic fluids in realistic experimental conditions (see e.g. Larson [20]). However, in a series of experiments, Perkins et al. [21,22] have shown that dilute suspensions of long DNA molecules have low enough viscosities and high enough Deborah numbers to present an oscillatory instability as first convective instability. This possibility has been recently confirmed by Kolodner [23], who observed oscillatory convection in DNA suspensions. He convincingly proved that viscoelasticity is the cause of the oscillations obtained in his experiments. These experiments triggered a renewed interest for thermal convection in viscoelastic fluids, especially from the theoretical point of view. 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. [24] and references therein). The corresponding problem in the case of a porous medium has also not received much attention until recently. Viscoelastic fluid flows through porous media is of interest for many engineering fields such as enhanced oil recovery, paper and textile coating, and composite manufacturing processes. The performance of an oil reservoir depends to a great extent upon the physical nature of the crude oil present in the reservoir. Light crude oil is essentially Newtonian while heavy crude oil is non-Newtonian and a study of such fluid 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 the oil displacement mechanism, which improves the efficiency of oil recovery. Besides, at shallow depth in the reservoirs, some oil sand contains waxy crude oil, which is considered to be a viscoelastic fluid. In such situations, a viscoelastic model of a fluid will be more realistic than an inelastic non-Newtonian fluid model. The published work on thermal convection of viscoelastic fluids in porous media is fairly limited. Rudraiah and co-workers [25], in a series of investigations, have studied the stability of viscoelastic fluids. A theoretical analysis of thermal instability in a porous layer saturated with viscoelastic fluid is carried out by Kim et al. [26]. They found that the overstability is a preferred mode for a certain range of parameters and the onset of convection has the form of a supercritical and stable bifurcation independent of the values of the elastic parameters. Yoon et al. [27] have analyzed the onset of thermal convection in a horizontal porous layer saturated with viscoelastic liquid using a linear theory. A simple constitutive model is used to examine the effects of relaxation times and it is shown that oscillatory instabilities can set in before stationary modes. Sheu et al. [28] investigated the chaotic thermal convection in an Oldroyd fluid-saturated porous medium using a thermal non-equilibrium model. Sheu et al. [29] have derived, a novel, unified system with six-order dynamics of chaotic convection to investigate thermal convection in both pure Newtonian/viscoelastic fluid layers and in Newtonian/viscoelastic fluid-saturated porous media. Recently, Tan and Masuoka [30] have studied the stability of Maxwell fluid in a porous medium using modified Darcy–Brinkman–Maxwell model. Most of the works mentioned are based on the LTE condition and have used the Darcy model. It is well known that Darcy’s law is not valid for the non-Newtonian fluid flows in porous media. The modified Darcy law is independent of shear rate due to the neglect of viscous shear effect. Thus it cannot satisfy the full set boundary conditions and cannot predict the boundary layer region near the boundaries of the porous layer. The modified Dacy–Brinkman–Maxwell model has been developed on the basis of the local volume averaging technique and the balance of forces acting on a volume element of viscoelastic fluids in porous media [30]. The Maxwell constitutive relation is deemed sufficient to reveal the basic effects of viscoelasticity on thermal instability, particularly in view of the extremely low shear rates involved and the linearization process used in the analysis. Therefore in this paper we study the onset of convection in a viscoelastic fluid-saturated

porous layer using modified Darcy–Brinkman–Maxwell model with emphasis on how the condition for the onset of convection is altered by elastic effects when the solid and fluid phases are not in local thermal equilibrium. The critical Rayleigh number, wavenumber and frequency for the oscillatory mode are determined. 2. Mathematical formulation We consider an infinite horizontal porous layer of depth d, which is confined between two planes z = 0 and z = d, respectively. A Cartesian frame of reference is chosen with the origin in the lower boundary and the z-axis vertically upwards. The lower surface is held at a temperature Tl , while the upper surface is at Tu (Tl > Tu ) with a fixed temperature difference T = (Tl − Tu ). The porous medium with porosity ε, permeability K is saturated with a Maxwell fluid with constant relaxation time . 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. We use the modified-Darcy–Brinkman–Maxwell model to describe the flow in the porous media that take care of boundary effect also [30]



1+

∂ ∂t





0 ∂q + ∇ p − g ε ∂t

+

 q − ∇ 2 q = 0, K

(2.1)

where q = (u, v, w) is the volume averaged velocity obtained by the local volume averaging technique, p is the pressure, g is the acceleration due to gravity,  is the density,  is the fluid viscosity. In modeling energy equation for a fluid-saturated porous system, two kinds of theoretical approaches have been used. In the first model, the fluid and solid structures are assumed to be in local thermal equilibrium. This assumption is satisfactory for small-pore media such as geothermal reservoirs and fibrous insulations and small temperature differences between the phases. In the second kind of approach, the fluid and solid structures are assumed to be in thermal non-equilibrium. For many applications involving highspeed flows or large temperature difference between the fluid and solid phases, it is important to take account of the thermal nonequilibrium effects. If the temperatures difference between phases is a very important parameter, the thermal non-equilibrium model in the porous media is an indispensable model. The local thermal non-equilibrium, which account for the transfer of heat between the fluid and solid phases is considered. A two-field model that represents the fluid and solid phase temperature fields separately, is employed for the energy equation [5] ε(0 c)f

∂Tf ∂t

+ (0 c)f (q · ∇ )Tf = εkf ∇ 2 Tf + h(Ts − Tf ),

(1 − ε)(0 c)s

∂Ts = (1 − ε)ks ∇ 2 Ts − h(Ts − Tf ), ∂t

(2.2) (2.3)

where c is the specific heat, k is the thermal conductivity with the subscripts f and s denoting fluid and solid phase respectively and h is the inter-phase heat transfer coefficient. In two-field model the energy equations are coupled by means of the terms, which accounts for the heat lost to or gained from the other phase. The inter-phase heat transfer coefficient h depends on the nature of the porous matrix and the saturating fluid and the value of this coefficient has been the subject of intense experimental interest. Large values of h correspond to a rapid transfer of heat between the phases (LTE limit) and small values of h gives rise to relatively strong thermal non-equilibrium effects. In Eqs. (2.2)–(2.3) Tf and Ts are intrinsic average of the temperature fields and this allows one to set Tf = Ts = Tb whenever the boundary of the porous medium is maintained at the temperature Tb . The continuity and state equations are

∇ · q = 0,

(2.4)

M.S. Malashetty, S. Kulkarni / J. Non-Newtonian Fluid Mech. 162 (2009) 29–37

 = 0 [1 − ˇ(Tf − Tl )],

(2.5)

where ˇ is the thermal expansion coefficient. 2.1. Basic state The basic state is assumed to be quiescent and is given by u = v = w = 0,

Tf = Tfb (z),

Ts = Tsb (z),

h = 0.

(2.6)

The basic state temperatures of fluid phase and solid phase satisfy the equations d2 Tfb dz 2

d2 Tsb = 0, dz 2

= 0,

(2.7)

with boundary conditions Tfb = Tsb = Tl

at z = 0,

Tfb = Tsb =

Tu

(2.8)

at z =

d,

(2.9)

so that the conduction state solutions are given by Tfb = Tsb = −

T z + Tl . d

(2.10)

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

Tf = Tfb + Tf ,

p = pb + p ,

Ts = Tsb + Ts ,

 = b +  .

(2.11)

where the prime indicates that the quantities are infinitesimal perturbations. Using Eq. (2.11) in Eqs. (2.1)–(2.5) and using the basic state solutions, we obtain the linearized equations governing the infinitesimal perturbations in the form,



1+

ε(0 c)f

∂ ∂t



∂Tf ∂t

 + (0 c)f w

(1 − ε)(0 c)s 

 =

0 ∂q +  gk + ∇ p ε ∂t dTfb



+

 q − ∇ 2 q = 0, K

(2.12)

 = εkf ∇ 2 Tf + h(Ts − Tf ),

dz

(2.13)

∂Ts = (1 − ε)ks ∇ 2 Ts − h(Ts − Tf ), ∂t

(2.15)

where k is the unit vector in the positive z-direction. We eliminate  between the Eqs. (2.12) and (2.15) and p from Eq. (2.12) by using the curl operator twice on it and then render the resulting equation dimensionless using the following transformations: (x∗ , y∗ , z ∗ ) = Tf∗ =

(x, y, z) , d Tf T

,

t∗ =

Ts∗ =

Ts T

kf (0 c)f d2

t,

w∗ =

(0 c)f d εkf

w ,

 

∂ 1+ ∂t

∂ − ∇2 ∂t



,

1 ∂ (∇ 2 w) + Ra∇ 21 Tf Pr D ∂t



∂ ˛ − ∇2 ∂t

RaLTE =

(2.16)

Tf = w + H(Ts − Tf ),



1+

Ra =

ˇgdKT (0 c)f [εkf + (1 − ε)ks ]

,

(2.20)

∂2 w =0 ∂z 2

Tf = Ts = 0

at z = 0 and 1,

at z = 0 and 1.

(2.21) (2.22)

3. Linear stability analysis + ∇ w − Da ∇ w = 0, 2

4

(2.17)

We assume the normal mode solutions for the eigenvalue problem defined by Eqs. (2.17)–(2.19) subject to the boundary conditions (2.21) and (2.22) in the form

(2.18)

⎛

 Ts = H(Tf − Ts ),

 

is the Rayleigh number based on the mean properties of the porous medium and it is this latter definition, which is used in the thermal equilibrium model. Eqs. (2.17)–(2.19) are solved for impermeable isothermal boundaries. Hence the boundary conditions are w=

to obtain non-dimensional equations as (on dropping the asterisks for simplicity),



where  = kf /(0 c)f d2 = f /d2 , the stress relaxation parameter, PrD = ε(0 c)f d2 /0 kf K = εd2 / f K, the Darcy–Prandtl number, Ra = ˇgdKT/ε f , the Darcy–Rayleigh number, Da = K/d2 , the Darcy number, H = hd2 /εkf , the inter-phase heat transfer coefficient, ˛ = (0 c)s kf /(0 c)f ks = f / s , the ratio of diffusivities, = εkf /(1 − ε)ks , the porosity modified conductivity ratio. It should be noted that the Darcy–Prandtl number PrD = εPr/Da, where Pr = / f is the Prandtl number. Further,  = /0 , is the kinematic viscosity, f = kf /(0 c)f and s = ks /(0 c)s are the thermal diffusivities of the fluid and solid phases respectively. The buoyancy due to thermal gradient is characterized by Rayleigh number, the ratio between viscous and thermal diffusivities is given by the Prandtl number, the viscoelastic character of the fluid appears in the relaxation parameter  . The smaller the viscoelastic parameter, the more fluid the material appears. The viscoelastic parameter  that relates to the relaxation time to the thermal diffusion time is of order one for most viascoelastic fluids. The scaled interface heat transfer coefficient H depends on the nature of the porous matrix. For large values of H the temperature fields of the two phases are almost identical which represent LTE limit. On the other hand, when H is small, there is no transfer of heat from fluid phase to solid and the porous media act in a way which is independent of the solid phase and hence one obtain an LTE limit. Between these two extremes H gives rise to strong thermal non-equilibrium (LTNE) effects. The other parameters namely, PrD depend on the properties of the fluid and Darcy number, Da on nature of porous matrix. It is suggested and verified by Katto and Masuoka [31] that the porous medium is said to be sparsely packed (Brinkman porous medium) when the Darcy number Da > 10−3 , and densely packed (Darcy porous medium) when Da is much smaller than 10−3 . This model bridges the gap between non-porous case in which Da → ∞ and Darcy porous medium in which Da → 0. The value for viscoelastic parameter  for dilute polymeric solution is most likely in the range [0.1, 2]. For a sparse porous media, Da ∈ [10−2 , 1], ε = 0.5 and typical value for Prandtl number for viscoelastic fluid is Pr = 10. Since PrD is magnified by a factor εDa−1 the range for PrD will be [5, 5 × 102 ]. At present there are no experimental data available for comparison and therefore we have considered a wide range of values for the parameters. It is worth mentioning that the Rayleigh number Ra defined above is based on the properties of the fluid while

(2.14)

−0 ˇTf ,

31

(2.19)

w

⎞

⎛

W (z)

⎞

⎜ ⎟ ⎜ ⎟ ⎝ Tf ⎠ = ⎝ (z) ⎠ exp{i(lx + my) + ωt}, Ts

˚(z)

(3.1)

32

M.S. Malashetty, S. Kulkarni / J. Non-Newtonian Fluid Mech. 162 (2009) 29–37

where l and m are the horizontal wave numbers and ω is the growth rate. Substituting Eq. (3.1) into Eqs. (2.17)–(2.19) we obtain (1 + ω)

 ω



PrD

(D2 − a2 )W + Ra a2

3.1. Stationary convection 2

+ (D2 − a2 )W − Da(D2 − a2 ) W = 0,

(3.2)

2

2

(3.3)

2

2

(3.4)

[(D − a ) − ω] + W + H(˚ − ) = 0,

where D = d/dz and a2 = l2 + m2 . The boundary conditions now become, W = D2 W = = ˚ = 0,

at z = 0 and 1.

(3.5)

We assume the solution to W, and ˚ in the form,



W ˚



=

W0 0 ˚0



(n = 1, 2, 3, . . .)

sin nz,

(3.6)

which satisfy the boundary conditions (3.5). Substituting Eq. (3.6) into Eqs. (3.2)–(3.4) we obtain the following matrix equation



⎛ ⎜ ⎝

ı2n

ω 1 + Da ı2n + PrD 1 + ω −1 0



⎞ −Ra a2 ω

0 0 0

=

0

+ ı2n

+H − H

 

−H ˛ω + ı2n + H

⎟ ⎠



W0 0 ˚0

,





(˛ω + ı2 )H (ω + ı ) + ˛ω + ı2 + H

1 =

a2



1 + Da ı2 1 +  2 ωi2

 − ωi2

1 − PrD

ı2 +

{˛2 ωi2 + ı2 (ı2 + H)}H ˛2 ωi2 + (ı2 + H)

(1 + Da ı2 )



1+

1 +  2 ωi2

Ra

,

(3.8)



 +

1 + Da ı2 1 +  2 ωi2



2



˛ H 2

1+

1 (1 + Da ı2 ) − PrD 1 +  2 ωi2

˛ H 2



˛2 ωi2 + ı2 + H





2

ı +



1 + Da ı2



1 +  2 ωi2

 − ωi2

2

ı +

{˛2 ωi2 + ı2 (ı2 + H)}H

1 (1 + Da ı2 ) − Pr 1 +  2 ωi2

˛2 ωi2 + (ı2 + H)



1+



2



˛ H 2 ˛2 ωi2 + (ı2 + H)

2

, (3.13)

and the frequency ωi is given by a1 (ωi2 ) + a2 (ωi2 ) + a3 = 0, where a1

= ˛2  2 (ı2

(3.14)

+ H),

2

a2 = ˛ PrD (1 + Da ı2 ) + ˛2 (ı2 + H)[1 − PrD  (1 + Da ı2 )] +  2 ı2 (ı2 + H)[ı2 + H(1 + )],

2

a3 = PrD (1 + Da ı2 )[(ı2 + H) + ˛ H 2 ] + ı2 (ı2 + H)[ı2 + H(1 + )][1 − Pr D  (1 + Da ı2 )].

> ,

2

 2 

˛2 ωi2 + ı2 ı2 + H



(3.12)

A careful examination of the expressions for ai s reveals that the necessary condition for the occurrence of the oscillatory convection is

(3.10)

ı2 2 = 2 a

ı2 = 2 a

2



˛2 ωi2 + (ı2 + H)

.

3.2. Oscillatory convection

(3.9)

where



We observe that the expression for RaSt is independent of viscoelastic parameter. This result is consistent with the one reported by Rosenblat [32] for a viscoelastic fluid without porous medium. Thus, as far as the steady onset is concerned, there is no distinction between the viscous fluid and viscoelastic fluid. The stationary onset of convection of viscous Newtonian fluid in a porous medium with LTNE model has studied in detail by Malashetty et al. [13]. The interested reader may refer to that paper. We discuss below the results about the oscillatory motions.

osc



2

Ra = 1 + iωi 2 ,



Ra

(3.7)

1 + Da ı2 ω + PrD 1 + ω



(2 + a2 ) H = [1 + Da(2 + a2 )] 1 + 2 a2  + a2 + H

For oscillatory onset 2 = 0 (ωi = / 0), Eq. (3.9) with 2 = 0 gives

where consideration has been confined to the lowest-order mode, n = 1 so that ı2 = 2 + a2 . The growth ω is in general a complex quantity such that ω = ωr + ωi . The system with ωr < 0 is always stable, while for ωr > 0 it will become unstable. For neutral stability state ωr = 0. Therefore, we set ω = iωi in Eq. (3.8) and clear the complex quantities from the denominator, to obtain

ı2

St



where ı2n = n2 2 + a2 is the total wavenumber. For the nontrivial solution of the above matrix equation (3.7), we require ı2 Ra = 2 a

The steady convection occurs when ωi = 0 and in that case Eq. (3.9) gives 2

[(D − a ) − ˛ω]˚ + H( − ˚) = 0,



Since Ra is a physical quantity, it must be real. Hence, from Eq. / 0, (3.9) it follows that either ωi = 0 (steady onset) or 2 = 0 (ωi = oscillatory onset).

˛2 ωi2 + ı2 + H

 

2

H

.

(3.11)

1 . (1 + Da ı2 )PrD

(3.15)

From the above inequality it is evident that the oscillatory convection is possible only if the elasticity parameter  exceeds a threshold value that in tern depends on Prandtl number and the Darcy number. The critical Rayleigh number Raosc c , at which the oscillatory convection sets in as preferred mode of instability is determined numerically as a function of wavenumber a for any fixed value of physical parameters as follows: First, Eq. (3.14) is solved to obtain positive solutions for ωi2 . If there are none, then no oscillatory convection is possible and the critical Rayleigh number is given by Eq. (3.12). Otherwise, the minimum over a of Eq. (3.13) with ωi2 (> 0) given by Eq. (3.14) gives the required critical Rayleigh number Raosc c , corresponding to the critical wavenumber 2 . This procedure ac and the critical frequency of the oscillation ωi,c

M.S. Malashetty, S. Kulkarni / J. Non-Newtonian Fluid Mech. 162 (2009) 29–37

33

Fig. 1. Variation of critical Rayleigh number Rac with H for different values of (a) porosity modified conductivity ratio , (b) viscoelastic parameter  , (c) thermal diffusivity ratio ˛ and (d) Darcy number Da.

is repeated for various values of the physical parameters involved and the results are presented graphically. Figs. 1–4 illustrate the influence of different physical parameters in the range ∈ [0.01, 10],  ∈ [0.1, 5], ˛ ∈ [0.01, 2], Da ∈ [0.01, 0.5] and PrD ∈ [3, 50] on the critical Rayleigh number, wavenumber and frequency of oscillation as a function of the interface heat transfer coefficient H. Excluding the varying parameter considered in a particular figure, the results shown are for  = 0.2, = 0.5, ˛ = 0.25, Da = 0.05 and PrD = 10. The observations made from these figures are discussed in Section 4. 4. Results and discussion The analytical expression for the oscillatory Rayleigh number given by Eq. (3.13) is minimized with respect to the wavenumber numerically, after substituting for ωi2 (> 0) from Eq. (3.14), as described in the last section, for various values of physical parameters in order to know their effects on the onset of oscillatory convection. The behavior of the critical values of Rayleigh number, wavenumber and frequency of the oscillatory mode as the functions of inter-phase heat transfer coefficient H is depicted through Figs. 1–4. In general, it is observed that for very small and large values of H the stability criterion is found to be independent of H. However, the effect of H on the stability of the system is significant only for intermediate values of H. The physical reason for this is that when H → 0 there is almost no transfer of heat between the fluid and solid phases and the properties of solid phase have no significant influence on the onset criterion. When H → ∞ the fluid and solid phase have almost equal temperatures and therefore may be treated as single phase. Between

these two extremes H gives rise to a strong non-equilibrium effect. The variation of critical Rayleigh number with inter-phase heat transfer coefficient H for different parameter values is shown in Fig. 1(a)–(d). These figures indicate that the critical Rayleigh number increases from the LTE value when H is small to a LTNE value when H is large. Thus, the inter-phase heat transfer coefficient makes the system more stable for its intermediate values. We find that the oscillatory curves in all these figures lies below the stationary curve (broken curves), indicating that the convection sets in through oscillatory mode prior to the stationary mode. The variation of the critical Rayleigh number with H for different values of porosity modified conductivity ratio is depicted in Fig. 1(a). We observe from this figure that for H → 0, the critical Rayleigh is independent of and is close to that of the LTE case, since for very small values of H, there is no significant transfer of heat between the phases and the onset criterion is not affected by the properties of the solid phase. On the other hand, for large values of H, though the stability criterion is independent of H, the condition for the onset of convection is based on the mean properties of the medium and therefore, the critical Rayleigh number is function of . This figure also indicates that for moderate and large values of H, oscillatory critical Rayleigh number decreases with the increasing values of . The effect of porosity modified conductivity ratio therefore, is to reduce the stabilizing effect of inter-phase heat transfer coefficient. It is important to note that for sufficiently large values of (≥10), the critical Rayleigh number of oscillatory convection becomes independent of H. This figure also shows the critical Rayleigh number for stationary mode (the broken curves) and we

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M.S. Malashetty, S. Kulkarni / J. Non-Newtonian Fluid Mech. 162 (2009) 29–37

Fig. 2. Variation of critical wavenumber ac with H for different values of (a) porosity modified conductivity ratio , (b) viscoelastic parameter  , (c) thermal diffusivity ratio ˛ and (d) Darcy number Da.

find that it too has similar behavior. It is important to note that the oscillatory curves are far below the stationary curves for the same values of , indicating that the oscillatory instabilities occur well ahead of the stationary instabilities. Fig. 1(b) shows the effect of viscoelastic parameter  on the oscillatory convection. For a fixed value of the other governing parameters, the critical Rayleigh number for the onset of oscillatory convection decreases with an increase in the value of  indicating that the effect of increasing viscoelastic parameter is to advance the onset of oscillatory convection. Thus it is confirmed that the elastic behavior of the non-Newtonian fluids leads to the oscillatory motions. Further, for large values of elastic parameter  , the critical Rayleigh number for oscillatory convection becomes independent of the interface heat transfer coefficient H. In Fig. 1(c) the variation of RaOsc with H for different values of c diffusivity ratio ˛ is indicated. We observe that for small values of H the diffusivity ratio does not affect the stability criterion, while for large values of H, the diffusivity ratio ˛ has significant only for large ˛ ≥ 0.5. This figure indicates that for moderate and large values of H the oscillatory critical Rayleigh number increases with increasing ˛. As ˛ increases, the contribution of heat conduction from the solid phase becomes negligible, and therefore the critical Rayleigh number for oscillatory mode increases towards a constant value. The diffusivity ratio therefore reinforces the stabilizing effect of inter-phase heat transfer coefficient in case of the overstable mode. Fig. 1(d) shows the effect of Darcy number on the oscillatory Rayleigh number. We observe that the increase in the Darcy number increases the critical Rayleigh number. Thus the Darcy number stabilizes the system. The effect of Darcy number is more pro-

nounced on stationary mode compared to its effect on oscillatory mode. Examining Fig. 1(d), it is clear that the oscillatory curve lies far below that of the corresponding stationary curve for the same Da. This implies that oscillatory instability can set in much ahead of the stationary instability. There is a general trend towards the value of the critical Rayleigh number becoming smaller as Da decreases. The magnitude of the Da is related to the importance of viscous effects at the boundaries, and reduction in Da decreases this effect, which allows the fluid to move more easily, thereby decreasing the critical Rayleigh number. Fig. 2(a)–(d) shows the variation of critical wavenumber for overstable mode with inter-phase heat transfer coefficient H for a range of values of the parameters ,  , ˛, and Da for a fixed value of PrD . We observe that the critical wavenumber remains constant for very small and very large values of the inter-phase heat transfer coefficient H, for intermediate values, it increases with H, attains a maximum value and then decreases back to another constant value with further increase of H. This is because the solid phase ceases to affect the thermal field of the fluid when H → 0 and on the other hand when H → ∞; the solid and fluid phases have almost equal temperatures. It is found that the critical wavenumber under these two limiting values of H are not same in case of oscillatory mode. However, in case of stationary mode, the critical wavenumber approaches to a common LTE value in the two limiting values of H (see Fig. 2(a)). For small H, the porosity modified conductivity ratio and diffusivity ratio ˛ have no influence on the critical wavenumber (see Fig. 2(a) and (c)). Further the oscillatory critical wavenumber at moderate values of H is found to decrease with increasing values of ,  and increases with an increase in the value

M.S. Malashetty, S. Kulkarni / J. Non-Newtonian Fluid Mech. 162 (2009) 29–37

35

Fig. 3. Variation of critical frequency ωc2 with H for different values of (a) porosity modified conductivity ratio , (b) viscoelastic parameter  , (c) thermal diffusivity ratio ˛ and (d) Darcy number Da.

of ˛ and Da. However, in case of the stationary mode, the effect of Darcy number Da is reversing (Fig. 2(d)). In Fig. 3(a)–(d) we display the effect of inter-phase heat transfer coefficient H on the critical frequency of the overstable motions for different values of ,  , ˛ and Da for a fixed value of PrD . It is clear from these figures that the critical frequency remains constant for small and large values H and attains a maximum for the intermediate values of H. This behavior is similar to that of critical wavenumber. It is interesting to note that the frequency decreases with increasing values of ,  while it becomes independent of H for the values of  > 1 and Da ≤ 0.001. The frequency decreases with decreasing values of Darcy number Da and conductivity ratio ˛. Fig. 4(a)–(c) shows the effect of Prandtl number on the critical Rayleigh number, the critical wavenumber and the critical frequency. From Fig. 4(a) we observe that the effect of Prandtl number is to decrease the critical Rayleigh number indicating that the effect of increasing Prandtl number is to destabilize the system. It is interesting to note that the oscillatory motions are not possible for values of the Darcy–Prandtl number less than 3 for the parameter values chosen for this figure. Further, it is found that for Darcy–Prandtl number PrD = 3, the oscillatory motions are not possible for values of H < 0.5. When H > 10, the oscillatory instability occurs prior to the stationary instability. Fig. 4(b) and (c) indicates that an increasing Prandtl number increases both critical wavenumber and the critical frequency. Fig. 4(b) reveals that the critical wavenumber for oscillatory mode is always larger than that of the stationary mode. In Table 1, we listed the critical values of the interface heat transfer coefficient H at which the critical wavenumber and critical frequency attains peak values for different parameter values. We find that these peak values strongly depend on and H. It is

Table 1 Critical values of the interface heat transfer coefficient H that produces the peak values for critical wavenumber and critical frequency.



˛

Da

PrD

(ac )max

(ωc2 )max

(H)critical

0.01 0.1 0.2 0.5 1.0 10.0

0.2

0.25

0.05

10

6.47 4.85 4.41 4.08 3.98 3.89

151.0 102.0 89.0 78.2 72.6 65.8

309.0300 91.2011 61.6595 23.9883 10.4713 2.05116

0.5

0.1 0.2 0.3 0.5 1.0 5.0

0.25

0.05

10

4.20 4.08 3.97 3.86 3.77 3.74

90.4 78.2 59.1 38.9 20.8 14.3

20.8930 23.9883 24.5471 25.7040 25.7040 25.7040

0.5

0.2

0.01 0.10 0.25 0.50 1.00 2.00

0.05

10

4.03 4.05 4.08 4.09 4.35 4.86

77.1 77.3 78.2 79.7 81.5 88.9

17.7828 19.4984 23.9883 50.1180 76.3936 99.9899

0.5

0.2

0.25

0.001 0.01 0.05 0.07 0.1

10

4.05 4.06 4.08 4.10 4.13

21.5 31.2 78.2 102.0 138.0

22.3872 22.3872 23.9883 24.5471 25.1189

0.5

0.2

0.25

0.05

3 5 10 15 25 50

3.98 4.01 4.08 4.17 4.24 4.30

6.18 26.60 78.2 132.00 188.00 244.00

25.7040 24.5471 23.9883 23.4423 22.3872 20.4174

36

M.S. Malashetty, S. Kulkarni / J. Non-Newtonian Fluid Mech. 162 (2009) 29–37

diffusion that causes the first convective instability to be oscillatory rather than stationary. The critical Rayleigh number is independent of for small H, while for intermediate values of H, it decreases with increasing . The critical Rayleigh number is independent of the inter-phase heat transfer coefficient H for the value of ≥ 10. An increase in the thermal diffusivity ratio ˛ increases the critical Rayleigh number for the overstable mode and thus its effect is to delay the onset of convection. For a fixed value of H, the effect of increasing viscoelastic parameter  is to decrease the critical Rayleigh number indicating its destabilizing effect. The oscillatory critical wavenumber at moderate values of H is found to decrease with increasing values of ,  and increases with an increase in the value of ˛ and Da. However, in case of the stationary mode, the Darcy number has reverse effect. The critical wavenumber and critical frequency remains constant for small and large values of H and for the intermediate values they attain the maximum values. The effect of increasing Prandtl number is to destabilize the system. We find that the local thermal equilibrium results are recovered in the large-H limit. Of particular interest is the fact that under some circumstances the critical wavenumber (which corresponds to all thin convection cell) and critical frequency attain large values. Numerical analysis of these situation reveals the fact that this phenomenon occurs when H is large, is small, but with H = O(1); it would be of very considerable interest for this unusual result to be confirmed by direct experiment.

Acknowledgements This work is supported by UGC New Delhi, under the Special Assistance Programme DRS Phase II. The authors are grateful to the Reviewers for their insightful and useful comments which have enhanced the quality of the paper.

References

Fig. 4. Variation of (a) critical Rayleigh number Rac , (b) critical wavenumber ac and (c) critical frequency ωc2 with interface heat transfer coefficient H for different values of Darcy–Prandtl number PrD .

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