Quasi-steady deformation of a two-dimensional bubble placed within a potential viscous flow (Q1332905)

The shape evolution of a two-dimensional bubble, bounded by a simple closed curve, which is initially placed within a potential viscous flow, is analysed. Reformulating the problem for Stokes equations with relevant boundary conditions at the free surface in terms of the bianalytic stress-stream function, and using the time-dependent conformal mapping \(z(\zeta,t)\) of a unit disk onto an unbounded flow domain sought, an infinite system of ordinary differential equations for the Laurent coefficients of \(z(\zeta, t)\) is derived. A class of exact solutions is found for the case when the principal part of the complex velocity of the dominant flow at infinity is a polynomial, and the problem of formation of a pointed bubble is discussed.