Quasi-static motion of a capillary drop, II: the three-dimensional case

Abstract

A theory is presented to analyze the nonlinear stability of a drop of incompressible viscous fluid with negligible inertia. The theory is developed here on the three-dimensional version of the relevant free-boundary model for Stokes equations. Within this context we show that if the free-boundary initiates close to a sphere r=1+ελ 0(ω), |ε| small, ω=(θ,ϕ), then there exists a global-in-time solution with free boundary r=1+λ(ω,t,ε)=1+ ∑ n⩾1 λ n(ω,t)ε n, which approaches a sphere exponentially fast as t→∞. Moreover, we prove that if λ 0( ω) is analytic (resp. C ∞) in ω, then the velocity u → (x,t,ε) , the pressure p( x, t, ε) and the free-boundary λ are all jointly analytic (resp. C ∞) in ( x, ε). In an earlier paper, we considered the analogous problem for a two-dimensional drop. Although the three-dimensional problem proceeds along similar lines, the analysis is more complicated due to the fact that we work here with spherical harmonics and vector spherical harmonics.