Quasi-irrotational growth rate for the Rayleigh-Taylor instability in accelerated elastic and viscous layers

A quasi-irrotational approximation is used to calculate the growth rate of the Rayleigh-Taylor instability in uniformly accelerated elastic or viscous layers, which, in addition to the unstable interface, also have a free surface. The approximation yields simple and analytical expressions for the growth rate which are still very accurate. In the limiting case of semi-infinite media, for which the well-known results of the purely irrotational approximation are retrieved, the maximum error is about 11-15%, and the accuracy progressively improves for finite thickness layers becoming exact in the very thin limit. This excellent agreement with the exact growth rate obtained by normal modes would indicate that vorticity, while introducing a considerable complexity in the mathematical problem, does not play a very significant role in the development of the instability.