A Study of Singularity Formation in the Kelvin–Helmholtz Instability with Surface Tension

Abstract

The possibility of singularity formation in vortex sheets acted on by surface tension is considered. In the evolution of a nearly flat vortex sheet with viscosity and surface tension neglected, Kelvin-Helmholtz instability has been shown to lead to the development of curvature singularities on the sheet. This is not unexpected, as the linearized equations of motion for this flow exhibit short-wavelength instability: infinitesimal perturbations grow at an infinite rate. Since the inclusion of surface tension stabilizes (on linear theory) the high-wavenumber modes, it has been presumed that it also suppresses the formation of singularities. The possibility of singularity formation is examined by generalizing Moore's approximation for vortex sheets to include the presence of surface tension. For the approximate theory, traveling wave solutions that develop singularities on the interface in finite time are found. Computations of the traveling wave solution are provided to reveal other features, and comparisons between these solutions and previously computed solutions of the full equations are made.