Transience to instability in a liquid sheet

Abstract

Series solutions are found which describe the evolution to absolute and convective instability in an inviscid liquid sheet flowing in a quiescent ambient gas and subject to a localized perturbation. These solutions are used to validate asymptotic stability predictions for sinuous and varicose disturbances. We show how recent disagreements in growth predictions stem from assumptions made when arriving at the Fourier integral response. Certain initial conditions eliminate or reduce the order of singularities in the Fourier integral. If a Gaussian perturbation is applied to both the position and velocity of a sheet when the Weber number is less than one, we observe absolutely unstable sinuous waves which grow liket1/3. If only the position is perturbed, we find that the sheet is stable and decays liket−2/3at the origin. Furthermore, if both the position and velocity of a sheet are perturbed in theabsenceof ambient gas, we observe a new phenomenon in which sinuous waves neither grow nor decay and varicose waves grow liket1/2with a convective instability.