Global nonlinear exponential stability of the conduction-diffusion solution for Schmidt numbers greater than Prandtl numbers

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 bifurcati ons at the onset of convection). For any boundary condition (rigid or stres s-free), the coincidence of the critical linear R-L(2) and nonlinear R-E(2) Rayleigh numbers is shown when the Rayleigh numbers for the concentration V are small. This result is obtained using a Lyapunov function equivalent t o the classical energy and choosing in an optimal way the Lyapunov paramete rs. Critical nonlinear Rayleigh numbers close to the linear ones arc also o btained for lame Rayleigh numbers for the solute concentration. (C) 2001 Ac ademic Press.