STABILIZING EFFECT OF CAPILLARITY IN THE RAYLEIGH-TAYLOR PROBLEM TO THE VISCOUS INCOMPRESSIBLE CAPILLARY FLUIDS

e investigate the stabilizing effect of capillarity in the Rayleigh-Taylor problem to the viscous incompressible capillary fluids driven by gravity in a smooth bounded domain. It is shown that if the steady density p<overline> is heavier with increasing height, i.e., p<overline>'(x(3))>0 , then there exists a finite critical capillary coefficient kappa(c) , such that when the capillary coefficient kappa<kappa(c), the steady-state is linear unstable, while for the case kappa>kappa(c), the steady-state is linear stable. On the other hand, if kappa<kappa(c) and is suitably small, we can also prove that the steady-state is nonlinearly unstable in the sense of Hadamard.