## Abstract

We carry out an analysis of the stability of a liquid jet into a gas or another liquid using viscous potential flow. The instability may be driven by Kelvin-Helmholtz KH instability due to a velocity difference and a neckdown due to capillary instability. Viscous potential flow is the potential flow solution of Navier-Stokes equations; the viscosity enters at the interface.

KH instability is induced by a discontinuity of velocity at a gas-liquid interface. Such discontinuities cannot occur in the flow of viscous fluids. However, the effects of viscous extensional stresses can be obtained from a mathematically consistent analysis of the irrotational motion of a viscous fluid carried out here. An explicit dispersion relation is derived and analyzed for temporal and convective/absolute (C/A) instability. We find that for all values of the relevant parameters, there are wavenumbers for which the liquid jet is temporally unstable. The cut-off wavenumber and wavenumber of maximum growth are most important; the variation of these quantities with the density and viscosity ratios, the Weber number and Reynolds is computed and displayed as graphs and asymptotic formulas. The instabilities of a liquid jet are due to capillary and KH instabilities. We show that KH instability cannot occur in a vacuum but capillary instability can occur in vacuum.We present comprehensive results, based on viscous potential flow, of the effects of the ambient.

Temporally unstable liquid jet flows can be analyzed for spatial instabilities by C/A theory; they are either convectively unstable or absolutely unstable depending on the sign of the temporal growth rate at a singularity of the dispersion relation. The study of such singularities is greatly simplified by the analysis here which leads to an explicit dispersion relation; an algebraic function of a complex frequency and complex wavenumber. Analysis of this function gives rise to an accurate Weber-Reynolds criterion for the border between absolute and convective instabilities. Some problems of the applicability to physics of C/A analysis of stability of spatially uniform and nearly uniform flows are discussed.

Keywords: Viscous potential flow; Kelvin-Helmholtz instability; Capillary instability; Temporal instability; Absolute and conveetive instability

^{*}Corresponding author. Tel.: +1 612 626 8000; fax: +1 612 626 1558. E-mail address: joseph@aem.umn.edu (D.D. Joseph).