Origin of ejecta in the water impact problem

Abstract

This work presents the analysis of the flow resulting from the flat plate impact on the surface of an incompressible viscous liquid at zero deadrise angle—of particular interest is the flow structure near the plate edge, r → 0, evolving at early times, t → 0. The deduced mathematical formulation proves to be of a singular perturbation type with the underlying governing equations having a linear structure. The key goals here are to elucidate the effects of viscosity and surface tension, which turn out to contribute to the solution at the leading order, and to resolve both t → 0 and r → 0 limit singularities in the classical pressure-impulse theory. In the course of construction of the solution, first the standard assumptions behind the existence of the inviscid approximation are revisited, which leads to correcting the previously known interpretation of the self-similarity of the classical inviscid solution near the plate edge. Second, new scalings of the solution structure near the plate edge are determined, with which the viscous solution near the edge is constructed analytically and matched to the inviscid one. Finally, the analysis of both the Stokes and inviscid limits of this uniformly valid solution allows us to uncover the scalings for the early time-evolution of ejecta—a jet forming during the impact—as well as to clarify the applicability of the Kutta-Joukowsky condition used in previous studies.