A New Upper Bound for the Largest Growth Rate of Linear Rayleigh--Taylor Instability

We investigate the effect of surface tension on the linear Rayleigh--Taylor (RT) instability in stratified incompressible viscous fluids with or without (interface) surface tension. The existence of linear RT instability solutions with largest growth rate $\Lambda$ is proved under the instability condition (i.e., the surface tension coefficient $\vartheta$ is less than a threshold $\vartheta_{\rm c}$) by modified variational method of PDEs. Moreover we find a new upper bound for $\Lambda$. In particular, we observe from the upper bound that $\Lambda$ decreasingly converges to zero, as $\vartheta$ goes from zero to the threshold $\vartheta_{\rm c}$. The convergence behavior of $\Lambda$ mathematically verifies the classical RT instability experiment that the instability growth is limited by surface tension during the linear stage.