WEAKLY NONLINEAR-ANALYSIS OF THE VIBRATIONAL-CONVECTIVE INSTABILITY IN A FLUID LAYER

A weakly nonlinear analysis of the onset of the vibrational-convective instability of a fluid layer with thermally nearly insulating boundar ies that performs high-frequency vibration is presented. It is based o n a set of incompressible fluid and heat equations averaged with respe ct to the ''fast time'' related to the vibrational frequency. A nonlin ear amplitude equation describing the spatiotemporal evolution of the temperature and velocity fields is derived and studied analytically an d numerically. It is found that if the boundaries are nearly insulatin g the primary bifurcation is supercritical for small deviations of the vibration axis from the longitudinal direction and subcritical for la rge ones. The primary bifurcation is subcritical if the boundaries are insulating. Numerical experiments show that the pattern formation is strongly affected by the departure of the vibration axis from its ''sy mmetric'' longitudinal direction. The evolution equation derived here is found to be common for a class of convective problems with a broken left-right symmetry.