The Rayleigh–Taylor instability in viscous fluids: The finite thickness effect revisited

A quasi-irrotational approximation is found for the problem of the Rayleigh–Taylor instability at the interface between a viscous fluid layer attached to a rigid wall and a semi-infinite ideal fluid. It takes into account the first order effects of the vorticity, and this turns out to be sufficient to produce a very accurate dispersion relation in a close and explicit analytical form. The resulting asymptotic growth rate reduces to the one obtained in the limit of pure irrotational flow for very thick layers, which is known to be accurate within a 10 %. For arbitrary thickness, the accuracy progressively improves for thinner layers retrieving the well-known exact results for very thin layers. For the more general case of two superposed layers of viscous fluids bounded, respectively, at the top and at the bottom by rigid walls, an interpolation formula is proposed for the dispersion relation on the basis of the problem symmetry, which, although it introduces a further approximation, still improves considerably the existing results.