Purely irrotational theories for the viscous effects on the oscillations of drops and bubbles

J.C. Padrino^a, T. Funada^b and D.D. Joseph^a'*

a^ Department of Aerospace Engineering and Mechanics, University of Minnesota, 110 Union Street SE, Minneapolis, MN 55455, USA

b^ Department of Digital Engineering, Numazu College of Technology, Ooka 3600, Numazu, Shizuoka, 410-8501, Japan

Abstract

Two purely irrotational theories of the motion of viscous fluids, viscous potential flow (VPF) and a viscous correction of viscous potential flow (VCVPF), have recently been applied to study capillary instability and to obtain the rate of decay of free gravity waves. The results from these analyses are, in general, in good to excellent agreement with exact solutions. Furthermore, the application of the dissipation method to these problems gives the same results as VCVPF. In this paper, we apply the VPF and VCVPF approximations as well as the dissipation method to the problem of the decay of waves on spheres. We treat the problem of the decay of small disturbances on a viscous drop surrounded by gas of negligible density and viscosity and a bubble immersed in a viscous liquid. The instantaneous velocity field in the viscous liquid is assumed to be irrotational. The linearized governing equations are solved by normal modes. The dissipation method applied to this problem gives rise to the same dispersion relation as VCVPF. Detailed comparisons of the eigenvalues from VPF and VCVPF and the exact solution are presented. Good agreement with the exact solution for the two purely irrotational solutions is demonstrated.

Key words: drops, bubbles, potential flow, two-phase flow, gas-liquid flow, viscous potential flow.

* Corresponding author. Tel: +1 612 625 0309; fax: +1 612 626 1558. Email address: joseph@aem.umn.edu (D.D. Joseph).