Global Nonlinear Exponential Stability of the Conduction-Diffusion Solution for Schmidt Numbers Greater than Prandtl Numbers

Abstract

The nonlinear exponential stability of the conduction-diffusion solution of a binary fluid mixture heated and salted from below is studied in the case of a horizontal layer when the Schmidt numbers are bigger than the Prandtl numbers (i.e., when the linear theory does not exclude Hopf-type bifurcations at the onset of convection). For any boundary condition (rigid or stress-free), the coincidence of the critical linear 2 L and nonlinear 2 E Rayleigh numbers is shown when the Rayleigh numbers for the concentration 2 are small. This result is obtained using a Lyapunov function equivalent to the classical energy and choosing in an optimal way the Lyapunov parameters. Critical nonlinear Rayleigh numbers close to the linear ones are also obtained for large Rayleigh numbers for the solute concentration.

Non‐linear stability in the Bénard problem for a double‐diffusive mixture in a porous medium

“…The coincidence between the non-linear critical Rayleigh number R E with the linear one R c , still holds for Le¿1 and C¡ 4 2 Le(Le 2 −1) . To prove this we follow the technique described by Lombardo et al [26].…”

“…For Le¿1 and large C (asymptotic case) we again follow the method of Reference [26]. By proceeding as in [26] we choose =Ĉ − Le R and = 1= . Then for R¡Ĉ=Le we have = 0; Á = 0 and…”

Global stability in the B�nard problem for a mixture with superimposed plane parallel shear flows

“…(but this problem is much more complicated and will be object of a forthcoming paper implies m(1 and, hence, global non-linear exponential stability (see [11], where the asymptotic case has been examined for the motionless state).…”

“…The non-linear stability of a mixture heated and salted from below is also studied for any boundary condition (rigid or stress-free) in [15,11], where global non-linear exponential stability theorems for the motionless state are given for any value of the ratio p"P ! /P 2 between the Schmidt and the Prandtl numbers (in [15] for p)1, and in [11] for p'1).…”

A new approach to nonlinear L2-stability of double diffusive convection in porous media: Necessary and sufficient conditions for global stability via a linearization principle

“…The stability of (6) has been considered by several authors (also when rotation about the vertical axis is incorporated) [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Precisely, denoting by (i) are periodic in the x and y directions respectively of periods 2π a x and 2π a y , (ii) on the periodicity cell Ω = [0, 2π a x ] × [0, 2π a y ] × [0, 1] (in order to guarantee uniqueness) u and v have zero mean value, (iii) belong to L 2 (Ω), ∀t ∈ R + , the results on nonlinear energy stability [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], as far as we know, can be summarized as follows: there exists a bounded positive number R * ∞ such that…”