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 is. 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 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 that for inviscid fluid. For air at 20°C and liquids with density ρ = 1 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 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 effect 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%.

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Movies of the breakup of viscous and viscoelastic drops in the high speed airstream behind a shock wave in a shock tube have been reported by Joseph, Belanger and Beavers [1999]. A Rayleigh-Taylor stability analysis for the initial breakup of a drop of Newtonian liquid was presented in that paper. The movies, which may be viewed at http://www.aem.umn.edu/ research/Aerodynamic_Breakup, show that for the conditions under which the experiments were carried out the drops were subjected to initial accelerations of orders 10 4 to 10 5 times the acceleration of gravity. In the Newtonian analysis of Joseph, Belanger and Beavers the most unstable Rayleigh-Taylor wave fits nearly perfectly with waves measured on enhanced images of drops from the movies, but the effects of viscosity cannot be neglected. Here we construct a Rayleigh-Taylor stability analysis for an Oldroyd B fluid using measured data for acceleration, density, viscosity and relaxation time λ 1 . The most unstable wave is a sensitive function of the retardation time λ 2 which fits experiments when λ 2 /λ 1 = O(10 -3 ). The growth rates for the most unstable wave are much larger than for the comparable viscous drop, which agrees with the surprising fact that the breakup times for viscoelastic drops are shorter. We construct an approximate analysis of Rayleigh-Taylor instability based on viscoelastic potential flow which gives rise to nearly the same dispersion relation as the unapproximated analysis.

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The problem of radial fingering in two phase gas/liquid flow in a Hele-Shaw cell under injection of gas is studied here. The fingers arise as an instability of a time-dependent flow. The instability is analyzed as a viscous potential flow, in which potential flow analysis of Paterson ͓L. Paterson, J. Fluid Mech. 113, 513 ͑1981͔͒ and others is augmented to account for the effects of viscosity on the normal stress at the gas/liquid interface. The addition of these new effects brings our theory into a much better agreement with experiments of Maxworthy ͓T. Maxworthy, Phys. Rev. A 39, 5863 ͑1989͔͒ than other theories.