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 bifurcations at the onset of convection). For any boundary condition (rigid or stress-free), the coincidence of the critical linear 2L and nonlinear 2ERayleigh 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