Rionero s critical perturbations method via weighted energy for stability of convective motions in anisotropic porous media

Transcription

1 1 Rionero s critical perturbations method via weighted energy for stability of convective motions in anisotropic porous media F. Capone,a, M. Gentile,a, A. A. Hill,b a Department of Mathematics and Applications R. Caccioppoli, University of Naples Federico II, Via Cinthia, 8016 Naples, Italy. b Department of Mathematical Sciences, Durham University, South Road, DH1 3LE, UK Abstract The linear and nonlinear stability analysis of a vertical throughflow in an anisotropic porous medium, is performed. In particular the effect of a non homogeneous porosity and a constant anisotropic thermal diffusivity have been taken into account. 1 Introduction Thermal convection in anisotropic porous media has received considerable interest in the last 30 years due to its great relevance in engineering applications such as insulating techniques [4], [6], [9], [10], [14], [15], [18]. In the present paper, in the framework of thermal convection in a horizontal anisotropic porous layer filled by an homogeneous newtonian fluid, we study the linear and nonlinear stability of a basic (stationary) vertical throughflow existing in the layer, via the Rionero s way of using the critical perturbations. As it is well known, the perturbations destabilizing the basic motion, in the linear stability, are called critical since them usually destabilize the basic motion also in the nonlinear theory. If, in the space H of the kinematically admissible perturbations one can introduce a scalar product with respect to which the principle of exchange of stabilities holds true, then choosing as a measure of the perturbations the associated norm, according to the way suggested by Rionero in [13] a perturbation is destabilizing if and only if it is critical. Hence, the first aim is to look for a scalar product with respect to which the principle of exchange of stabilities holds true, then it is suficient to confine the attention to the destabilizing perturbations which are the critical ones. The plan of the paper is the following. In fcapone@unina.it m.gentile@unina.it a.a.hill@durham.ac.uk

2 Section, we consider the Darcy-Boussinesq model: p, i = µ k(z) v i + ρ 0 αgt k i, v i,i = 0, (1) T, t +v i T, i = K ij T, ij, where k(z) = k 0 ϕ(z), K = diag(k 1, K 1, K 3 ), () ϕ(z) C([0, d]) C 1 (0, d) and k = (0, 0, 1), determine a steady vertical throughflow (convective motion) and the perturbation equations in dimensionless form. On introducing a suitable weighted norm, we perform nonlinear stability analysis of m 1 and find that the conditions ensuring the (local) nonlinear stability of m 1 ensures also (linear) instability (Section 3). Finally (Section 4), on solving numerically the associated variational problem, we obtain the stability threshold of the convective motion. Weighted Nonlinear Stability Analysis of the Convective Motion. Adding to (1) the boundary conditions T = T L, on z = 0, T = T U, on z = d v = (0, 0, T f ), on z = 0, d (3) with T f constant and T L > T U, (1)-(3) admit the steady state m 1 = (ṽ, T ) such that: ṽ = (0, 0, T f ), T (z) = T (4) L T U e cd 1 (ecz 1) + T L, c = T f /k 3 which is a (stationary) convective motion [], [8]. In order to study the stability of m 1, we introduce the perturbation (u, θ, π) and use the dimensionless variables x i = dx i, t = T t, u i = Uu i, θ = T θ, π = P π,, P = µud k 0 (5) βk 3 µ ρ 0 αgk 0 βd T =, R =, ρ 0 αgk 0 µk 3 T = d, K 3 U = K 3 d, and hence, dropping all primes, the dimensionless perturbations equations are: π, i = 1 f(z) u i + Rθk i, u i,i = 0, θ, t +u i θ, i = RF (z)w Qθ, 3 +ζ 1 θ + θ, 33, (6)

3 where u = (u, v, w), F (z) = (Qe Qz )/(e Q 1) and Q = T f d/k 3, f(z) = ϕ(dz) is the porosity of the medium in the dimensionless variable z and K = diag(ζ, ζ, 1), with ζ = K 1 /K 3. To system (6) we append the following boundary conditions: w = θ = 0 on z = 0, 1. (7) From now on, we will assume that the perturbation fields (u, θ, π), defined on IR [0, 1], are periodic in the x and y direction and we shall denote by = [0, π/a x ] [0, π/a y ] [0, 1] the periodicity cell. Moreover, on denoting by, and the L ()-scalar product and L ()-norm, respectively, to ensure the uniqueness we will assume that u d = v d = 0. (8) 3 It is easily verified that the linear version of (6), i.e. π, i = 1 f(z) u i + Rθk i, u i,i = 0, θ, t = RF (z)w Qθ, 3 +ζ 1 θ + θ, 33, (9) is symmetric with respect to the weighted L ()-scalar product g(z),, with g(z) = 1. Hence since the strong version of the Principle of Exchange of Stabilities holds F (z) true, choosing as a measure of the perturbations the associated weighted norm E(t) = 1 g(z)θ d, (10) a perturbation is destabilizing if and only if is critical [13]. Along the solutions of (6) we find: dt = RI D + 1 [Qg (z) + g (z)] θ d + 1 g (z)θ wd, (11) where Since it follows that: I = [1 + g(z)f (z)]θwd, [ ] 1 D = f(z) u + g(z)(ζ 1 θ + θ,3 ) d. (1) g(z) = 1 F (z) = Q 1 (e Q 1)e Qz, (13) Qg (z) + g (z) 0, Q IR, z [0, 1] (14)

4 4 and hence from (11) one obtains: dt = RI D + 1 eq Following the standard energy method, on defining: where e Qz θ w d. (15) 1 I = max R E H D, (16) H = {u, θ : u = 0; u, θ periodic in x and y of periods π/a x, π/a y respectively, satisfying (7)- (8) and such that D <.} (17) is the class of kinematically admissible perturbations, it turns out that: ) (1 dt D RRE + 1 eq e Qz θ w d, (18) Moreover the use of Schwarz and Sobolev inequalities leads to: e Qz θ w d C u θ, (19) where C = C() is a positive constant. On the other hand, from (6) 1 it follows that: u µr θ, µ 1 = min f(z) > 0. (0) 0 z 1 By virtue of (0), from (19) one obtains: 1 e Q e Qz θ w d γe 1/ D, (1) with γ = CµR Qe Q (e Q 1) min(ζ, 1) > 0, () and hence, from (18), we get: ) (1 dt D RRE + γe 1/ D. (3) From the last inequality, applying recursive arguments, the following theorem holds true. Theorem - If provided R < R E, (4) E 1/ (0) < γ 1 ( 1 R R E ), (5) then the convective motion m 1 is nonlinearly stable with respect to the weighted norm E.

5 3 Numerical results The Euler-Lagrange equations of (16) are 1 f(z) u i + R (1) E θk i = ω, i, u i,i = 0, 5 (6) R (1) E F (z)w Qθ, 3 +ζ 1 θ + θ, 33 = 0, with ω a Lagrange multiplier, and they coincide at the criticality with the linear version of (6). Equations (6) have been solved numerically, through the compound matrix method, on choosing f(z) = 1 + λz, λ = const. > 0. The critical Rayleigh numbers R E for the weighted (local) stability and linear instability of m 1 are plotted in Figures 1-3. Figure 1: Linear instability and weighted local nonlinear stability thresholds of m 1, plotted against Q, with λ = 0.5 and ζ = 1. Acknowledgements This work has been performed under the auspices of the GNFM of INDAM and MIUR (PRIN 005): Nonlinear Propagation and Stability in Thermodynamics Processes of Continuous Media and the Research Project Grant of the Leverhulme Trust - Grant Number F/0018/AK. References [1] G.I. Barenblatt, V.M. Entov, V.M. Ryzhik (1990), Theory of fluid flows through natural rocks. Kluwer Academic Publishers.

6 6 Figure : Linear instability and weighted local nonlinear stability thresholds of m 1, plotted against λ, with Q = 1, Q = 5 and ζ = 1. Figure 3: Linear instability and weighted local nonlinear stability thresholds of m 1, plotted against ζ, with Q = 1, Q = 5 and λ = 0.5.

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