Boundary integral equations for contact problems of plane quasi-steady viscous flows

Abstract

Plane, quasi-steady, free-boundary flows of an incompressible viscous fluid with surface tension in a container are considered. The mathematical problem is decomposed into an auxiliary elliptic problem for the Stokes system in a fixed flow domain, whose solution leads to the Cauchy problem for the free boundary with the so-called ‘normal velocity’ operator. By introducing the complex stress-stream function and applying time-dependent conformal mapping, the auxiliary problem is reduced to a boundary integral equation via consideration of two Hilbert problems for analytic functions in a unit disc. As an application, plane capillary flow with moving contact points is investigated asymptotically for small capillary numbers. We prove that in the case when a dynamic contact angle is equal to π, this problem is well-posed for a filling regime, and ill-posed for a drying one.