Transport in Porous Media (2006) 62: 55–79 DOI 10.1007/s11242-005-4507-y
© Springer 2006
Effect of Thermal Modulation on the Onset of Convection in a Viscoelastic Fluid Saturated Porous Layer M. S. MALASHETTY1, , P. G. SIDDHESHWAR2 and MAHANTESH SWAMY1 1 2
Department of Mathematics, Gulbarga University, Gulbarga 585 106, India Department of Mathematics, Bangalore University, Bangalore 560 001, India
(Received: 18 October 2004; accepted in final form: 1 March 2005) Abstract. The effect of thermal modulation on the onset of convection in a horizontal, anisotropic porous layer saturated by a viscoelastic fluid is investigated by a linear stability analysis. Darcy’s law with viscoelastic correction is used to describe the fluid motion. The perturbation method is used to find the critical Rayleigh number and the corresponding wavenumber for small amplitude thermal modulation. The stability of the system characterized by a correction Rayleigh number is calculated as a function of the thermal and mechanical anisotropy parameters, the viscoelastic parameters and the frequency of modulation. It is found that the onset of convection can be delayed or advanced by the factors represented by these parameters. The results of the problem have possible implications in mantle convection. Key words: anisotropy, thermal modulation, stability, viscoelastic fluid, mantle convection.
Nomenclature specific heat. height of the porous layer. thermal diffusivity tensor Dx (ii + jj) + Dz kk. gravitational acceleration. permeability tensor Kx−1 (ii + jj) + Kz−1 kk. wavenumbers in x, y directions. pressure. basic state pressure. Prandtl number, ρ µDz . R velocity vector, (u, v, w). dKz Rayleigh number, βgT . νDz time.
c d D g K l, m p pH Pr q RD t
Author for correspondence: e-mail:
[email protected]
56 T TH TR (x, y, z)
M. S. MALASHETTY ET AL.
temperature. basic state temperature. reference temperature (unmodulated temperature of the upper and lower surfaces when T = 0). space co-ordinates.
Greek symbols α αc β γ T ε φ η λ¯ 1 λ¯ 2 µ ν ω ρ ρH ρR ξ ∇12
horizontal wavenumber. critical wavenumber. coefficient of thermal expansion. (ρc)f +(1−)(ρc)s . ratio of specific heats, (ρc)f unmodulated temperature difference between the walls. amplitude of modulation. phase angle. porosity of the media. thermal anisotropy parameter, Dx /Dz . stress relaxation coefficient. strain retardation coefficient. viscosity. kinematic viscosity, µ/ρR . nondimensional frequency, d 2 /Dz . frequency of modulation. density. basic state density. = ρ(TR ) reference density. mechanical anisotropy parameter, Kx /Kz . ∂2 ∂2 + ∂y 2. ∂x 2
∇2
∂ ∇12 + ∂z 2.
2
1. Introduction Thermal convection in a fluid saturated porous medium has attracted the interest of engineers and scientists for a long time due to its numerous applications in fields such as geothermal energy utilization, oil reservoir modeling, building thermal insulation, nuclear waste disposals and mantle convection, to mention a few. The problem has been investigated extensively by several researchers and the growing volume of work in this area is well documented by Ingham and Pop (1998), Nield and Bejan (1999) and Vafai (2000). Most of the previous studies have usually been concerned with homogeneous isotropic porous structures. However, during the last few years the effect of nonhomogeneity and anisotropy of porous medium has been studied. The geological and pedagogical processes rarely form isotropic medium as is usually assumed in transport studies. Processes such as sedimentation, compaction, frost action, and reorientation of the solid matrix are responsible for the creation of a naturally anisotropic porous medium. Anisotropy
EFFECT OF THERMAL MODULATION ON THE ONSET OF CONVECTION
57
can also be a characteristic of manufactured porous media like those made of irregularly shaped particles formed by extrusion or pelletting used in chemical engineering process or fiber materials used for insulating purposes. There are very few available studies on thermal convection in a fluidsaturated horizontal, anisotropic porous medium heated from below. Castinel and Combarnous (1974) have conducted an experimental and theoretical investigation on the Rayleigh–Benard convection in an anisotropic porous medium. Epherre (1975) extended the stability analysis to a porous medium with anisotropy in thermal diffusivity also. A theoretical analysis of non-linear 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. There are few investigations available in the literature concerning how a time-dependent boundary temperature affects the onset of Rayleigh–Benard convection. Most of the findings relevant to these problems have been reviewed by Davis (1976). In case of small amplitude temperature modulations, a linear stability analysis was performed by Venezian (1969). Later Rosenblat and Herbert (1970), Rosenblat and Tanaka (1971) and Roppo et al. (1984) have studied the effect of temperature modulation on the onset of thermal convection in a horizontal fluid layer. On the other hand the studies related to the effect of thermal modulation on the onset of convection in a fluid-saturated porous medium have received marginal attention. The effect of time dependent wall temperature on the onset of convection in a porous medium has been studied by Caltagirone (1976), Rudraiah and Malashetty (1990) and Malashetty et al. (1999). Quite recently non-Newtonian fluids housed in fluid-based systems, with and without porous matrix, have been extensively used in application situations and hence warrant the attention they have been duly getting. In the asthenosphere and the deeper mantle it is well known now that viscoelastic behavior is an important rheological process (see Lowrie, 1997). The other application areas of viscoelastic fluid saturated porous media are flow through composites, timber wood, snow systems and rheology of food transport. The present problem housed in a porous medium suggests an elastohydrodynamical model for geophysical applications and the likes of it (see O’Connell and Budiansky, 1977; Turner and Cambell, 1986; Griffiths, 1987; Brown et al., 1992; Buffet et al., 1996; Lowrie, 1997; Jackson, 1998; Rao, 2000; Siddheshwar and Srikrishna, 2001; Yoon et al., 2004). Regulation of convection in these application situations is important and the study of this is the motive for the paper.
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We investigate the combined effect of anisotropy of the porous medium and time dependent wall temperature on the onset of convection in a horizontal, anisotropic porous layer saturated with a viscoelastic fluid. The amplitude and frequency of the modulation are externally controlled parameters and hence the onset of convection can be delayed or advanced by a proper tuning of these parameters. The problem has potential application in achieving major enhancement of mass, momentum and heat transfer in the geothermal context and related areas. 2. Mathematical Formulation We consider a viscoelastic fluid-saturated anisotropic porous medium, confined between two infinite horizontal walls at z = 0 and z = d, heated from below and cooled from above. The porous medium is assumed to possess horizontal isotropy in mechanical and thermal properties. We ignore the deformation of the porous matrix to circumvent the use of the general theory of mixtures (Rajagopal and Tao, 1995). Before we embark on the linear stability analysis of the problem at hand, we present a brief explanation of the chosen mathematical model of the anisotropic porous medium saturated by a viscoelastic fluid describable by the Oldroyd model. The Jeffrey (Oldroyd) constitutive equation that is used for performing a linear stability analysis of Rayleigh–Benard convection in clear viscoelastic fluids, i.e., in viscoelastic fluids with no porous medium to impede their flow, is ∂ ∂ ¯ ¯ 1 + λ1 τ = 1 + λ2 eij , (2.1) ∂t ij ∂t where τij is the shear stress, eij = µ(qi,j + qj,i ), qi is the velocity component, λ¯ 1 and λ¯ 2 are, respectively, the stress relaxation and strain retardation coefficients. The conservation of linear momentum with the Boussinesq approximation gives us ρR
∂qi = τij,j − ρgδi3 , ∂t
(2.2)
where τij = −pδij + τij .
(2.3)
In arriving at the governing equations of porous media momentum transport based on the Jeffrey constitutive equation we will have to make use of the Dupuit’s equation, viz., qf i = qi ,
(2.4)
EFFECT OF THERMAL MODULATION ON THE ONSET OF CONVECTION
59
where is the porosity, qi is the actual velocity component in the absence of a porous medium and qf i is the filter velocity. Substituting Equation (2.3) into Equation (2.2), averaging the resulting equation in a fashion similar to that done in the derivation of the classical Darcy law for low-porosity media, and then using Equation (2.4) in the resulting equation, we get ρR ∂ ∂qi ∂ ∂ ¯ ¯ ¯ 1 + λ1 = −µ 1 + λ2 Kij qj + 1 + λ1 (−p,i − ρgδi3 ), ∂t ∂t ∂t ∂t (2.5) where µ is the viscosity and Kij is the permeability tensor. It is important to note here that the porosity is absorbed into the permeability tensor and hence does not appear explicitly in the first term on the right-hand side of Equation (2.5). We have dropped the subscript f with the velocity, for simplicity, in the above equation. In vector form Equation (2.5) may be written as ρR ∂ ∂ ∂ ∂q + µ 1 + λ¯ 2 K · q − 1 + λ¯ 1 (−∇p + ρg) = 0, 1 + λ¯ 1 ∂t ∂t ∂t ∂t (2.6) where g = (0, 0, −g) Assuming thermal equilibrium between the fluid and solid phases one can write a single-phase Fourier second law in the form: ∂T + q · ∇T = ∇ · (D · ∇T ). (2.7) ∂t In using the energy transport equation as above we are assuming that viscoelastic effects influence the transport implicitly through the velocity. The equations of continuity and state are γ
∇ · q = 0,
(2.8)
ρ = ρR [1 − β(T − TR )] ,
(2.9)
where β is the coefficient of volume expansion. It is clear from the above equation that the Oberbeck–Boussinesq approximation is used in the study. One can refer a recent study by Rajagopal et al. (1996) for the derivation of appropriate equations, giving a rigorous basis for the Oberbeck– Boussinesq approximation. The externally imposed wall temperatures are time dependent and are taken as T [1 + ε cos t] , T (0, t) = TR + 2 (2.10) T [1 − ε cos (t + φ)] , T (d, t) = TR + 2
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where ε represents amplitude, the frequency and φ the phase angle. We consider three types of thermal modulation, viz., Case (a): symmetric (in phase, φ = 0), Case (b): asymmetric (out of phase, φ = π) and Case (c): only lower wall temperature is modulated while the upper one is held at constant temperature (φ = −i∞). 2.1. basic state The basic state of the fluid is quiescent and is given by qH = 0,
T = TH (z, t),
p = pH (z, t),
ρ = ρH (z, t).
(2.11)
The temperature T = TH (z, t) is a solution of γ
∂ 2 TH ∂TH = Dz ∂t ∂z2
(2.12)
and pressure pH (z, t) balances the buoyancy force. The solution of Equation (2.12) subject to the boundary conditions (2.10) is 2z T 1− + ε Re a(λ)eλz/d + a(−λ)e−λz/d e−it , TH = TR + 2 d (2.13) where
γ d 2 λ = (1 − i) 2Dz
1/2
,
e−iφ − e−λ a(λ) = eλ − e−λ
and Re stands for the real part. We do not record the expressions of pH and ρH as these are not explicitly required in the remaining part of the paper. We now superimpose infinitesimal perturbations on the quiescent basic state and study the stability of the system. 2.2. linear stability analysis Let the basic state be disturbed by an infinitesimal perturbation. We now have q = qH + q ,
T = TH + T ,
p = pH + p ,
ρ = ρH + ρ ,
(2.14)
where the prime indicates that the quantities are infinitesimal perturbations.
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EFFECT OF THERMAL MODULATION ON THE ONSET OF CONVECTION
Substituting Equation (2.14) into Equations (2.6)–(2.9), and using the basic state solutions, we get the linearized equations governing the perturbations in the form ρR ∂ ∂q ∂ ∂ ¯ ¯ ¯ 1 + λ1 + µ 1 + λ2 K · q − 1 + λ1 (−∇p + ρ g) = 0, ∂t ∂t ∂t ∂t (2.15) 2 ∂T ∂ T ∂TH γ (2.16) = Dx ∇12 T + Dz 2 , + w ∂z ∂z ∂t ρ = −βρR T .
(2.17)
We eliminate ρ between Equations (2.15) and (2.17) and then eliminate ρ from the resulting equation by operating curl twice. We render the resulting equation and Equation (2.16) dimensionless by setting ∗
∗
∗
(x, y, z) = d(x , y , z ), (T , TH ) = T (T ∗ , TH∗ ), RD = λ1 =
ρR βg T dKz , µDz
λ¯ 1 Dz , d2
λ2 =
Pr =
d2 ∗ t= t , Dz Kx ξ= , KZ µ , ρR Dz
Da =
Dz w = w∗ , d
Kz , d2
ω=
d 2 , Dz
λ¯ 2 Dz d2
η=
Dx , Dz (2.18)
to obtain (after dropping the asterisks) Da ∂ ∂(∇ 2 w) ∂ 1 ∂2 2 1 + λ1 + 1 + λ2 ∇1 + w Pr ∂t ∂t ∂t ξ ∂z2 ∂ ∇12 T , = RD 1 + λ1 ∂t ∂2 ∂TH ∂ , γ − η∇12 − 2 T = −w ∂t ∂z ∂z
(2.19) (2.20)
where ∇12 = (∂ 2 /∂x 2 ) + (∂ 2 /∂y 2 ). We note here that Dz in the definition of Pr is the effective thermal diffusivity of the porous medium. We also note that µ is quite large and Kz is quite small for a viscoelastic fluidsaturated low-porosity medium. Since this implies (Da/Pr) 1, the first term in Equation (2.19) is omitted in the remaining part of the paper. The
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dimensionless groups that appear are RD , the Darcy–Rayleigh number, ξ , the mechanical anisotropy parameter, η, the thermal anisotropy parameter, ω, the frequency of the modulation, and λ1 and λ2 the stress relaxation and strain retardation parameters respectively. The important observation to make here is that RD is based on the vertical permeability, Kz . Equations (2.19) and (2.20) are to be solved subject to the boundary conditions, w = T = 0,
at z = 0, 1.
(2.21)
From the above equation it is clear that the boundaries are considered to be impermeable and perfect thermal conductors. We now combine Equations (2.19) and (2.20) to obtain a single differential equation for the vertical component of velocity w as ∂ ∂2 ∂ 1 ∂2 2 2 1 + λ2 γ − η∇1 − 2 + ∇1 + ∂z ∂t ξ ∂z2 ∂t
∂ ∂TH 2 +RD 1 + λ1 ∇ w = 0. (2.22) ∂t ∂z 1 The boundary conditions (2.21) together with Equation (2.20) give us the following conditions in terms of w as w=
∂ 2w = 0, ∂z2
at z = 0, 1.
(2.23)
Using Equation (2.13), the dimensionless temperature gradient appearing in Equation (2.22) may be written as ∂TH = −1 + εf, ∂z
(2.24)
where A(λ)eλz + A(−λ)e−λz e−iωt ,
γ ω 1/2 λ e−i − e−λ A(λ) = and λ = (1 − i) . 2 eλ − e−λ 2 f = Re
The vaue of γ is set equal to one in further analysis for simplicity. 3. Method of Solution We now seek the eigenfunctions w and eigenvalues RD for the basic temperature gradient given by Equation from the linear profile (∂TH /∂Z ) = −1 by quantities of eigenvalues of the present problem differ from that of
of Equation (2.22) (2.24) that departs order ε. Thus, the the Epherre (1975)
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EFFECT OF THERMAL MODULATION ON THE ONSET OF CONVECTION
problem by quantities of order ε. We therefore assume the solution of Equation (2.22) in the form: (w, RD ) = (w0 , RD0 ) + ε(w1 , RD1 ) + ε2 (w2 , RD2 ) + · · ·
(3.1)
Substituting Equation (3.1) into Equation (2.22) and equating the coefficients of various powers of ε on either side of the resulting equation, we obtain the following system of equations up to the order of ε 2 : Lw0 = 0,
(3.2)
Lw1 = RD1 (L4 )∇12 w0 − RD0 (L4 )f ∇12 w0 ,
(3.3)
Lw2 = RD2 (L4 )∇12 w0 −RD1 (L4 )f ∇12 w0 +RD1 (L4 )∇12 w1 −RD0 (L4 )f ∇12 w1 , (3.4) where L = (L1 )(L2 )(L3 ) − RD0 (L4 )∇12 with L1 = 1+λ2
∂ , ∂t
L2 = ∇12 +
1 ∂2 , ξ ∂z2
L3 =
∂ ∂2 −η∇12 − 2 , ∂t ∂z
L4 = 1+λ1
∂ ∂t
and w0 , w1 , w2 are required to satisfy the boundary conditions of Equation (2.23). Equation (3.2) is the one used in the study of convection in a viscoelastic fluid saturated anisotropic porous medium subject to a uniform basic temperature gradient. We now assume the marginally stable solutions for Equation (3.2) in the form w0 = W0 (z) exp[i(lx + my)], where W0 (z) = W0n (z) = sin(nπz), n = 1, 2, 3 . . . and l, m are the wavenumber in the xy(n) are plane such that l 2 + m2 = α 2 . The corresponding eigenvalues RD0 = RD0 given by
1 n2 π 2 (n) 2 2 2 2 RD0 = 2 (ηα + n π ) α + . (3.5) α ξ For a fixed value of the wavenumber α the least eigenvalue occurs for n = 1, and is given by
1 π2 2 2 2 RD0 = 2 (ηα + π ) α + . (3.6) α ξ RD0 assumes the minimum value 2
RD0c = π 2 1 + η/ξ ,
(3.7)
when α = α0 where α0 is given by
α02 = π 2 / ξ η.
(3.8)
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The above results are independent of viscoelastic parameters and concur with earlier results on Newtonian fluids occupying an anisotropic porous medium. These results were first obtained by Epherre (1975) (see Storesletten, 1998 for details). We note that the critical wavenumber α0 and the corresponding critical Rayleigh number RD0c depend on anisotropy parameters ξ and η. It is quite explicit from the expression (3.7) that an increase in the value of (η/ξ ) increases the critical value of the Rayleigh number and thus makes the system more stable. In the case of an isotropic porous medium ξ = η = 1 and Equations (3.7) and (3.8) give RD0c = 4π 2 and αC = π. These are the values reported by Lapwood (1948) for the isotropic problem. We now move on to obtain the correction Rayleigh number. Equation (3.3) is inhomogeneous and poses a problem because of the presence of the resonance terms. The solvability condition requires that the time independent part of the right-hand side must be orthogonal to w0 . The term independent of time on the right-hand side is RD1 ∇12 ω0 so that RD1 must be zero. It follows that all the odd coefficients, i.e., RD1 , RD3 , . . . in Equation (3.1) must vanish. The equation for w1 then takes the form Lw1 = RD0 α 2 Re{1 − iω λ1 }f sin(πz).
(3.9)
We solve Equation (3.9) for w1 by expanding the right-hand side in a Fourier series and inverting the operator L term by term. Thus, we obtain ∞ B (λ) n w1 = RD0 α 2 Re (1 − iω λ1 ) (3.10) e−iωt sin(n π z) . M(ω, n) n=1
The detail of the algebra is presented in the Appendix A through Equations (A1)–(A9). The equation for w2 becomes Lw2 = −RD2 α 2 w0 + RD0 α 2 (L4 )f w1 .
(3.11)
We do not require the solution of this equation but need to merely use it to determine RD2 , the first nonzero correction to RD0 . The solvability condition requires that the time-independent part of the right-hand side of Equation (3.11) be orthogonal to sin(πz) and this results in the following equation: ∞ 2 RD0 α2 RD2 = |Bn (λ)|2 Cn . 2
(3.12)
n=1
The detail is presented in the Appendix A through Equations (A10)–(A17). In the above equation the summation extends over even values of n for case (a), odd values of n for case (b) and all integer values of n for case (c).
EFFECT OF THERMAL MODULATION ON THE ONSET OF CONVECTION
65
The value of the Rayleigh number RD obtained by this procedure is the eigenvalue corresponding to the eigenfuction w which, though oscillating, remains bounded in time. Since RD is a function of the horizontal wave number α and the amplitude of the modulation ε, we use the following expansion: RD (α, ε) = RD0 (α) + ε2 RD2 (α) + · · · ,
(3.13)
α = α0 + ε2 α2 + · · ·
(3.14)
The critical value of the Rayleigh number RD is computed up to O(ε2 ) by evaluating RD0 and RD2 at α = α0 given by Equation (3.8). It is only when one wishes to evaluate RD4 that α2 must be taken into account. To the order of ε 2 , RD2 is obtained for the cases when the oscillating temperature field is (i) symmetric, (ii) asymmetric, and (iii) when only the lower wall temperature is oscillating while the upper wall is held at constant temperature. The infinite series in Equation (3.12) converges rapidly as the terms decrease with n−10 . The critical value of Rayleigh number RD is determined to order of ε 2 and, accordingly we have RDc = RD0c + ε2 RD2c ,
(3.15)
where RD0c is given by Equation (3.7) and RD2c can be obtained from Equation (3.12). If RD2c is positive, supercritical instability exists and RDc has minimum at ε = 0. When RD2c is negative, subcritical instability is possible. We have from Equation (3.12), ε 2 < (RD0c /RD2c ). Now we can calculate the maximum range of ε by assigning values to the physical parameters involved in the above condition. Thus, the range of the amplitude of modulation in which subcritical instabilities are possible in different physical situations can be explained. 4. Results and Discussion The effect of thermal modulation on the onset of convection in a horizontal viscoelastic fluid saturated porous layer is examined using a linear stability analysis. The expression for the critical correction Rayleigh number RD2c is computed as a function of the frequency of modulation for different parameter values and the results are depicted in Figures 1–8. The sign of RD2c characterizes the stabilizing or destabilizing effect of modulation. A positive RD2c means the modulation effect is stabilizing while a negative RD2c means the modulation effect is destabilizing, compared to the system in which the modulation is absent. The effect of a symmetric modulation of the wall temperature on the onset of convection in a horizontal, anisotropic porous layer saturated by
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a viscoelastic fluid is shown in Figures 1–4 for different parameter values. We find that for low frequency RD2c is negative indicating that the effect of symmetric modulation is destabilizing. On the other hand, for moderate and high frequency RD2c is positive indicating that the effect of symmetric modulation is to stabilize the system. There are two peak values of RD2c one positive and another negative. Let ω∗ represent the frequency at which RD2c changes its sign from negative to positive, then the modulated system may be classified as destabilized or stabilized, compared with the unmodulated system, according as ω < ω∗ or ω > ω∗ . First RD2c decreases to its maximum destabilizing value and then increases to its maximum stabilizing value to finally decrease to zero as the frequency increases from zero to infinity. The maximum stabilization or destabilization can be achieved at critical frequencies ω = ωp or ω = ωn depending on the value of the frequency. Further, at some particular value of the frequency ω = ω0 , the effect of modulation ceases, i.e., RD2c = 0. These critical frequencies depend on the parameters governing the system. We now discuss the results depicted in Figures 1–8. Figure 1 shows the variation of RD2c with ω and ξ for the case of symmetric modulation of the wall temperature for a fixed value of η, λ1 and λ2 each equal to 0.1. From the figure it is evident that an increase in the value of ξ increases the magnitude of the RD2c . At low modulation frequency, RD2c is negative indicating that the symmetric modulation destabilizes the system while at moderate and high frequency RD2c is positive and thereby indicating that it stabilizes the system. An increase in the value of ξ(= Kx /Kz ) can be interpreted as follows: Let us keep the vertical permeability Kz (or the horizontal permeability Kx ) fixed and vary the horizontal permeability Kx (or vertical permeability Kz ). Then an increased horizontal permeability increases the magnitude of RD2c . The peak negative or positive value of RD2c increases with ξ . The curves for ξ = 1.0, 0.5 and 0.1 are close to each other which means that the effect of ξ on stability is small for ξ < 1. The peak positive value of RD2c depends on ξ and this also depends on the frequency. The variation of RD2c with ω and η for a fixed value of ξ, λ1 and λ2 each equal to 0.1 is shown in Figure 2. We observe from the figure that increasing η increases the magnitude of RD2c . The peak value of |RD2c | increases with increasing value of η. The effect of thermal anisotropy parameter η on the stability of the system is similar to that of the mechanical anisotropy parameter ξ . The most important result of the problem can be extracted by comparing the results of Figures 1 and 2. We find that the nadir value of RD2c is lower in the case of the thermal anisotropy parameter variation compared to the mechanical anisotropy parameter. This clearly implies that amongst the anisotropy parameters the thermal
67
EFFECT OF THERMAL MODULATION ON THE ONSET OF CONVECTION 500 Symmetric temperature modulation ξ = 0.1 ξ = 0.5 ξ = 1.0 ξ = 10.0 η = 0.1, λ1= 0.1, λ2= 0.1 400
300
ω
200
100
0
-0.1
0.0
0.1
0.2
0.3
RD2c/RD0c
Figure 1. Variation of RD2c /RD0c with ω for different values of mechanical anisotropy parameter ξ .
anisotropy parameter is a vital parameter in advancing convection and mechanical anisotropy is a vital parameter in delaying convection. Figure 3 shows the variation of RD2c with ω and stress relaxation parameter λ1 for a fixed value of ξ, η and λ2 each equal to 0.1. We find from the figure that an increase in the value of λ1 is to increase the magnitude of RD2c . On the other hand the effect of modulation diminishes as
68
M. S. MALASHETTY ET AL. 500 Symmetric temperature modulation η = 0.1 η = 0.5 η = 1.0 η = 5.0 η = 10.0 ξ = 0.1, λ1= 0.1, λ2= 0.1 400
300
ω
200
100
0 -0.2
-0.1
0.0
0.1
0.2
0.3
0.4
RD2c/RD0c
Figure 2. Variation of RD2c /RD0c with ω for different values of thermal anisotropy parameter η.
the stress relaxation parameter λ1 become smaller and smaller. The peak negative or positive value of RD2c is found to increase with λ1 . The effect of the strain retardation parameter λ2 is found to be opposite of the stress relaxation time λ1 (Figure 4). On comparing Figures 3 and 4 we find that
69
EFFECT OF THERMAL MODULATION ON THE ONSET OF CONVECTION 500
Symmetric temperature modulation λ1 = 0.1 λ1 = 0.2 λ1 = 0.3 λ1 = 0.4 ξ = 0.1, η= 0.1, λ2 = 0.1
400
300
ω
200
100
0 -2
-1
0
1
2
3
4
5
6
RD2c/RD0c
Figure 3. Variation of RD2c /RD0c with ω for different values of stress relaxation parameter λ1 .
the effect of stress relaxation parameter is more pronounced in aiding onset of convection compared to the effect of the strain retardation parameter. The effect of asymmetric modulation of the wall temperatures on the onset of convection in a horizontal, anisotropic porous layer saturated by a viscoelastic fluid is, in general, to inhibit the onset of convection over
70
M. S. MALASHETTY ET AL. 500 Symmetric temperature modulation λ2 = 0.1 λ2 = 0.2 λ2 = 0.3 λ2 = 0.4 ξ = 0.1, η = 0.1, λ1 = 0.1 400
300
ω
200
100
0 -0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
RD2c/RD0c
Figure 4. Variation of RD2c /RD0c with ω for different values of strain retardation parameter λ2 .
the whole range of frequency (see Figures 5–8) except for low frequency in which case the asymmetric modulation is destabilizing for certain values of the strain retardation parameter (Figure 8). This clearly points to the fact that the strain retardation parameter is an important one in the study of convection involving asymmetric thermal modulation.
71
EFFECT OF THERMAL MODULATION ON THE ONSET OF CONVECTION 500
Asymmetric temperature modulation ξ = 0.1 ξ = 0.5 ξ = 1.0 ξ = 10.0 η = 0.1, λ1 = 0.1, λ2 = 0.1
400
300
ω
200
100
0 -0.25
0.00
0.25
0.50
0.75
1.00
1.25
RD2c/RD0c
Figure 5. Variation of RD2c /RD0c with ω for different values of mechanical anisotropy parameter ξ .
Figure 5 shows the variation of RD2c with ω and the anisotropy parameter ξ for fixed value of η, λ1 and λ2 each equal to 0.1. We observe that RD2c decreases with increase in the value of the anisotropy parameter ξ . It is important to note that this effect is quite opposite to that in case of symmetric modulation. The maximum stabilization can be achieved by
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M. S. MALASHETTY ET AL.
500 Asymmetric temperature modulation η = 0.1 η = 0.5 η = 1.0 η = 5.0 η = 10.0 ξ = 0.1, λ1 = 0.1, λ2 = 0.1
400
300
ω
200
100
0 -1
0
1
2
3 4 RD2c/RD0c
5
6
7
8
Figure 6. Variation of RD2c /RD0c with ω for different values of thermal anisotropy parameter η.
decreasing the value of ξ . However, for ξ = 10, there is range of ω for which the trend is opposite. The variation of RD2c with ω and the thermal anisotropy parameter η for fixed value of ξ, λ1 and λ2 each equal to 0.1 is shown in Figure 6. We observe that an increase in the value of the thermal anisotropy parameter η
73
EFFECT OF THERMAL MODULATION ON THE ONSET OF CONVECTION 500
Asymmetric temperature modulation λ1 = 0.1
λ1 = 0.2 λ1 = 0.3 λ1 = 0.4 ξ = 0.1, η = 0.1, λ2 = 0.1
400
300
ω
200
100
0
0
1
2
3
4
5
6
7
RD2c/RD0c
Figure 7. Variation of RD2c /RD0c with ω for different values stress relaxation parameter λ1 .
increases the correction Rayleigh number RD2c indicating that its effect is to stabilize the system. Figure 7 shows the variation of RD2c with ω and the stress relaxation parameter λ1 for fixed value of ξ, η and λ2 each equal to 0.1. We find from the figure that an increase in the value of λ1 increases the correction Rayleigh number. On the other hand the effect of modulation diminishes as the stress relaxation parameter λ1 become small. The effect of the strain retardation parameter λ2 is found to be opposite of the stress relaxation parameter λ1 . It is important to note that for λ2 > 0.1 there is
74
M. S. MALASHETTY ET AL. 500
Asymmetric temperaturemodulation λ2 = 0.1 λ2 = 0.2 λ2 = 0.3 λ2 = 0.4 ξ = 0.1, η = 0.1, λ1= 0.1
400
300
ω 200
100
0 -0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
RD2c/RD0c
Figure 8. Variation of RD2c /RD0c with ω for different values of strain retardation parameter λ2 .
a small range of values of the frequency for which RD2c becomes negative indicating that the effect is destabilizing. The effect of lower wall temperature modulation is found to be qualitatively similar to the case of asymmetric modulation and we therefore omit a graphical representation of the same. The effect of thermal modulation disappears for large frequency irrespective of the type of thermal modulation.
EFFECT OF THERMAL MODULATION ON THE ONSET OF CONVECTION
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5. Conclusion The effect of thermal modulation on the onset of convection in a horizontal, anisotropic porous layer saturated by a viscoelastic fluid is studied using a linear stability analysis and the following conclusions are drawn: (1) The low frequency symmetric thermal modulation is destabilizing while high frequency symmetric modulation is always stabilizing. The asymmetric modulation and lower wall temperature modulations are, in general, stabilizing for all frequencies. However, in these two cases there is a range of small ω for which the system becomes unstable when λ2 > 0.1. Thus, the thermal modulation can destabilize a mode that is stable in the unmodulated case, or stabilize an unstable mode, with the stability characteristics depending on the parameters governing the system. (2) The effect of mechanical anisotropy parameter ξ on symmetric modulation is to destabilize at low frequency and stabilize at moderate and high frequency. The effect is insignificant for values of ξ < 0.5. The effect of ξ on the other two types of modulations is opposite to that of symmetric modulation. The effect of thermal anisotropy parameter η is to stabilize the system in all cases except for a range of small frequency in the case of symmetric modulation. (3) The increase in stress relaxation parameter λ1 enhances the effect of modulation while increase in strain retardation parameter λ2 suppresses the effect of modulation. (4) The effect of thermal modulation disappears for large frequency in all the cases. Appendix A We solve Equation (3.9) for w1 by expanding the right-hand side in a Fourier series expansion and inverting the operator L term by term. So we take e sin(mπz) = λz
∞
gnm (λ) sin(nπz),
(A1)
n=1
where
1
gnm (λ) = 2
eλz sin(nπz) sin(mπz)dz 0
=−
4nmπ 2 λ[1 + (−1)n+m+1 eλ ] . [(n − m)2 π 2 + λ2 ][(n + m)2 π 2 + λ2 ]
(A2)
We now define M(ω, n) = B1 + iωB2 ,
(A3)
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M. S. MALASHETTY ET AL.
where
B1 = ω λ2 2
π4 η n2 π 2 2 2 2 + (1 − n4 ), + α π (1 − n ) 1 + α + ξ ξ ξ 2
n2 π 2 n2 π 2 2 B2 = α + + λ2 α + (ηα 2 + n2 π 2 ) − ξ ξ π2 2 −λ1 α + (ηα 2 + π 2 ). ξ
(A4)
2
(A5)
It is easily seen that L[sin(nπz)e−iωt ] = M(ω, n) sin(nπz)e−iωt . From Equation (3.9), we obtain Lw1 = RD0 α 2 Re (1 − iωλ2 )
∞
(A6)
[A(λ)gn1 (λ)+
n=1
+A(−λ)gn1 (−λ)] sin(nπz)e so that
−iωt
(A7)
,
∞ Bn (λ) sin(nπz)e−iωt w1 = RD0 α 2 Re (1 − iωλ1 ) M(ω, n)
(A8)
n=1
where Bn (λ) = A(λ)gn1 (λ) + A(−λ)gn1 (−λ).
(A9)
The equation for w2 can then be written as Lw2 = −RD2 α 2 w0 + RD0 α 2 (L4 )f w1 .
(A10)
To simplify Equation (A10) for w2 , we need (L4 )f w1 = Re{1 − 2iωλ1 }f w1 .
(A11)
Equation (A10) then reads as Lw2 = −RD2 α 2 w0 + RD0 α 2 Re{1 − 2iωλ1 }f w1 .
(A12)
We do not require the solution of Equation (A12) but shall use it to determine the correction Rayleigh number RD2 . The solvability condition requires that the time-independent part of right-hand side of Equation (A12) be orthogonal to w0 = sin(πz). To that
EFFECT OF THERMAL MODULATION ON THE ONSET OF CONVECTION
77
end we multiply the right-hand side of Equation (A12) by sin(πz) and integrate between the limits 0 and 1 to obtain 1 RD2 = 2RD0 Re{1 − 2iωλ1 } f w1 sin(πz)dz, (A13) 0
where the over bar indicates time average. Now from Equation (3.9) we obtain, ∞ |Bn (λ)|2 RD0 α 2 2 sin (nπz) Re (1 + iωλ1 ) f w1 sin(πz) = M(ω, n) 2 n=1
so that 1 0
∞ |Bn (λ)|2 RD0 α 2 Re (1 + iωλ1 ) . f w1 sin(πz)dz = 4 M(ω, n)
(A14)
n=1
Substituting Equation (A14) into Equation (A13), we get ∞ 2 RD0 α 2 |Bn (λ)|2 RD2 = Re (1 + iωλ1 )(1 − 2iωλ1 )M ∗ (ω, n) . (A15) 2 2 |M(ω, n)| n=1
Now, Re{(1 + iωλ1 )(1 − 2iωλ1 )M ∗ (ω, n)} = B1 B3 − ωB2 B4 , and |M(ω, n)|2 = B12 + (ωB2 )2 , where B3 = 1 + 2ω2 λ21 and B4 = ωλ1 . Thus, we may write RD2 =
∞ 2 RD0 α2 |Bn (λ)|2 Cn , 2
(A16)
n=1
where |Bn (λ)|2 =
16π 4 n2 ω2 , ω2 + (n + 1)4 π 4 ω2 + (n − 1)4 π 4
Cn =
B1 B3 − ωB2 B4 . B12 + (ωB2 )2 (A17)
Acknowledgement The work was supported by the University Grants Commission, New Delhi, under the Special Assistance Programme of DRS and CAS respectively at the Universities of Gulbarga and Bangalore. The authors thank the reviewers for their useful comments.
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