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

We study the stability of stratified gas-liquid flow in a horizontal rectan gular channel using viscous potential flow. The analysis leads to an explic it dispersion relation in which the effects of surface tension and viscosit y on the normal stress are not neglected but the effect of shear stresses i s. Formulas for the growth rates, wave speeds and neutral stability curve a re given in general and applied to experiments in air-water flows. The effe cts of surface tension are always important and determine the stability lim its 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 visc osity ratio is equal to the density ratio; surprisingly the neutral curve f or 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 that for inviscid fluid. For air at 20 degreesC and liquids with density rho = I g cm(-3) the liquid viscosity for the critical conditions is 15 cP : the critical velocity for liquids with viscosities larger than 15 cP is o nly slightly smaller but the critical velocity for liquids with viscosities smaller than 15 cP, like water, can be much lower. The viscosity of the li quid has a strong effect on the growth rate. The viscous potential flow the ory fits the experimental data for air and water well when the gas fraction is greater than about 70%.