AEM Home page> People > Faculty > Daniel D. Joseph> Archive on Irrotational Motions of Viscous and Viscoelastic Fluids

Viscous potential flow analysis of Kelvin-Helmholtz instability in a channel

By T. FUNADA1 AND D. D. JOSEPH2

1Department of Digital Engineering, Numazu College of Technology, Ooka 3600, Numazu, Shizuoka, Japan
1Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA

(Received ?? and in revised form ??)

Abstract

We study the stability of stratified gas-liquid flow in a horizontal rectangular channel using viscous potential flow. The analysis leads to an explicit dispersion relation in which the effects of surface tension and viscosity on the normal stress are not neglected but the effect of shear stresses are neglected. Formulas for the growth rates, wave speeds and neutral stability curve are given in general and applied to experiments in air-water flows. The effects of surface tension are always important and actually determine the stability limits for the cases in which the volume fraction of gas is not too small. The stability criterion for viscous potential flow is expressed by a critical value of the relative velocity. The maximum critical value is when the viscosity ratio is equal to the density ratio; surprisingly the neutral curve for this viscous fluid is the same as the neutral curve for inviscid fluids. The maximum critical value of the velocity of all viscous fluids is given by inviscid fluids. For air at 20C and liquids with density g/cm3 the liquid viscosity for the critical conditions is 15 cp; the critical velocity for liquids with viscosities larger than 15 cp are only slightly smaller but the critical velocity for liquids with viscosities smaller than 15 cp, like water, can be much lower. The viscosity of the liquid has a strong affect on the growth rate. The viscous potential flow theory fits the experimental data for air and water well when the gas fraction is greater than about 70%.