Singularities in the classical Rayleigh-Taylor flow: formation and subsequent motion

Abstract

This paper is concerned with the creation and subsequent motion of singularities of solution to classical Rayleigh-Taylor flow (two-dimensional inviscid, incompressible fluid over a vacuum). For a specific set of initial conditions, we give analytical evidence to suggest the instantaneous formation of one or more singularity(ies) at specific point(s) in the unphysical plane, whose locations depend sensitively to small changes in initial conditions in the physical domain. One-half power singularities are created in accordance with an earlier conjecture; however, depending on initial conditions, other forms of singularities are also possible. For a specific initial condition, we follow a numerical procedure in the unphysical plane to compute the motion of a one-half singularity. This computation confirms our previous conjecture that the approach of a one-half singularity towards the physical domain corresponds to the development of a spike at the physical interface. Under some assumptions that appear to be consistent with numerical calculations, we present analytical evidence to suggest that a singularity of the one-half type cannot impinge the physical domain in finite time.