Quasistatic Motion of a Capillary Drop I. The Two-Dimensional Case

AbstractA theory is presented for analyzing the nonlinear stability of a drop of incompressible viscous fluid with negligible inertia. The theory is developed here on the two-dimensional version of the relevant free-boundary model for Stokes equations. As we show, the two-dimensional problem presents most of the difficulties expected from a projected three-dimensional study while allowing for simpler manipulation of the spherical harmonics. Within this context we show that if the free-boundary initiates close to a circler=1+ελ0(θ),|ε|small, then there exists a global-in-time solution with free boundaryr=1+λ(θ, t, ε)=1+∑n⩾1λn(θ, t)εn, which approaches a circle exponentially fast as t→∞. Moreover, we prove that if λ0(θ) is analytic (resp. C∞) in θ, 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)