Rayleigh-Taylor Instability of Viscoelastic Drops at High Weber Numbers

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 to 10 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 λ1. The most unstable wave is a sensitive function of the retardation time λ2 which fits experiments when λ2/λ1= O(10). 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.