On the nonlinear stability of a fluid layer of a mixture heated and salted from below

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

“…In Section 4 we give a nonlinear exponential stability condition in the asymptotic case ( 2 → ∞). We improve the results of [9] and obtain the same asymptotic behavior as in the linear case. In Section 5 we study the case when the domain is a layer bounded by two rigid planes or by one rigid plane and one stressfree plane.…”

“…This effect has been proved also in the nonlinear context in [6,9]. Precisely, in [6], the coincidence of the critical linear and nonlinear Rayleigh numbers was obtained only for 2 = 2 1 (the stability result is not exponential), while |in [9] exponential stability results are obtained without coincidence. Here we prove the coincidence of the critical linear and nonlinear limits for 2 small and stress-free boundary conditions.…”

“…He introduced the coupling parameters method and obtained critical nonlinear stability values near to the linear ones for small 2 [6] are not exponential. In [9], nonlinear exponential results have been obtained both in the cases p < 1 and p > 1. In the last case, in [9], although analytical expression for the critical nonlinear Rayleigh numbers has been given, nevertheless the coincidence of the critical linear and nonlinear limits has not been obtained, and, in the asymptotic case 2 → ∞ , the nonlinear critical Rayleigh numbers go to infinity as (and not 2 as in the linear case).…”

“…In [9], nonlinear exponential results have been obtained both in the cases p < 1 and p > 1. In the last case, in [9], although analytical expression for the critical nonlinear Rayleigh numbers has been given, nevertheless the coincidence of the critical linear and nonlinear limits has not been obtained, and, in the asymptotic case 2 → ∞ , the nonlinear critical Rayleigh numbers go to infinity as (and not 2 as in the linear case). The aim of the present paper is to study, for p > 1, global nonlinear exponential stability to (i) reach critical nonlinear Rayleigh numbers near to the linear ones for 2 → ∞, and…”

“…In [9][10][11], the study of nonlinear exponential stability of motions of a binary fluid mixture heated and salted below has been given. In particular, in the case of a fluid layer bounded by two horizontal parallel planes, necessary and sufficient conditions of global nonlinear exponential stability of the conduction-diffusion solution (the motionless state with linear temperature and concentration profiles) have been obtained when the Schmidt number P C is less than or equal to the Prandtl number, P T , for any 2 ( [10,11]).…”