On stabilizing effect of elasticity in the Rayleigh-Taylor problem of stratified viscoelastic fluids

We investigate the stabilizing effect of elasticity in the Rayleigh-Taylor (RT) problem of stratified immiscible viscoelastic fluids, separated by a free interface and in the presence of a uniform gravitational field, in a horizontally periodic domain where the velocities of the fluids are non-slip on both upper and lower fixed flat boundaries, while the internal surface tension is omitted. We establish a discriminant Cr for the stability of the stratified viscoelastic RT problem. More precisely, if Cr<1, then the stratified viscoelastic RT equilibrium state is exponentially stable. This means that a sufficiently large elasticity coefficient has stabilizing effect so that it can inhibit viscoelastic RT instability. On the other hand, if Cr>1, then we show that the RT equilibrium state is linearly unstable in the Hadamard sense. Moreover, for the case of a nonhomogeneous incompressible viscoelastic fluid, the condition Cr>1 will lead to the nonlinear instability of the RT equilibrium state; and this shows that the RT instability still occurs in viscoelastic fluids when the elasticity coefficient is small.