Viscous effects on perturbed spherical flows

Abstract

The problem of two viscous, incompressible fluids separated by a nearly spherical free surface is considered in general terms as an initial-value problem to first order in the perturbation of the spherical symmetry. As an example of the applications of the theory, the free oscillations of a viscous liquid drop and of a bubble in a viscous liquid are studied in some detail. It is shown that the oscillations are initially describable in terms of an irrotational approximation, and that the normal-mode results are recovered as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">t \to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In between these asymptotic regimes, however, the motion is significantly different from either approximation.