STABILITY ANALYSIS OF REGULARIZED VISCOUS VORTEX SHEETS

Abstract

Abstract. A vortex sheet is susceptible to the Kelvin-Helmhotz insta-bility, which leads to a singularity at finite time. The vortex blob modelprovided a regularization for the motion of vortex sheets in an inviscidfluid. In this paper, we consider the blob model for viscous vortex sheetsand present a linear stability analysis for regularized sheets. We showthat the diffusing viscous vortex sheet is unstable to small perturbations,regardless of the regularization, but the viscous sheet in the sharp limitbecomes stable, when the regularization is applied. Both the regulariza-tion parameter and viscosity damp the growth rate of the sharp viscousvortex sheet for large wavenumbers, but the regularization parametergives more significant effects than viscosity. 1. IntroductionA vortex sheet is an interface in an incompressible fluid across which thetangential velocity is discontinuous [3]. It serves as a simple model for a shearlayer at a high Reynolds number. In a free shear flow, strong roll-ups evolveon the vortex sheet, which results in a small-scale structure and mixing of thefluid [10, 12, 18]. A variety of flows are described by a vortex sheet; for example,Rayleigh-Taylor instability, water waves and Hele-Shaw flows [4, 7].The motion of vortex sheets suffers from the Kelvin-Helmholtz instability [3].The small perturbation of a flat sheet proportional to exp(λt + ikΓ), in aninviscid fluid, has the dispersion relation(1) λ(k) =k2,where k represents the wavenumber of the solution, t is time and Γ is thecirculation parameter. This relation indicates that short-wave disturbancesgrow spuriously and cause instability in the evolution of the sheet. As a result,