Equation of motion for a drop or bubble in viscous compressible flows

Abstract

The problem of unsteady motion of a spherical bubble or drop is analyzed in the limit of vanishing Mach and Reynolds numbers. Linearized viscous compressible Navier-Stokes equations are solved inside and outside of the spherical bubble/drop and an expression of the transient force is first obtained in the Laplace domain and then transformed to the time domain. The total force is separated into the quasi-steady, the inviscid unsteady, and the viscous unsteady contributions. The new force expression reduces to known results in the limits of a drop in an incompressible flow or a rigid particle in a compressible flow. We observe that in all compressible flow cases, the viscous unsteady kernel shows a \documentclass[12pt]{minimal}\begin{document}$1/\sqrt{t}$\end{document}1/t decay at sufficiently short times. This is in contrast to the behavior in an incompressible flow where the viscous unsteady kernel on the bubble reaches a constant value at short times.