THE 2 1 STEADY/HOPF MODE INTERACTION IN THE 2-LAYER BENARD-PROBLEM/

The interaction of Hopf and steady modes with wavenumber ratio 2:1 is investigated for a critical situation. Two different immiscible liquid s lie in layers between horizontal walls and are heated from below. A situation with a pair of complex conjugate eigenvalues at wavenumber r r and a real eigenvalue at wavenumber 2 alpha is at criticality. Weakl y nonlinear amplitude equations are derived for the interaction of the se oscillatory and steady modes. The two modes generate a two-paramete r bifurcation. The coefficients involved in the equations are determin ed numerically, based on the physical parameters of the system at crit icality. Three obvious equilibrium solutions of the amplitude equation s are the steady solution, the traveling waves and the mixed standing waves, The eigenvalues governing the stability of these solutions are found explicitly. Numerical results and bifurcation diagrams are given for the critical situation. The steady solution and the traveling wav e solution are unstable, There is a region of stability for the standi ng wave solution, A new equilibrium solution, the asymmetric mixed mod e, is found to be stable in a parameter range. Bifurcations from the s tanding wave solution and the asymmetric mixed mode are described.