The Two-Phase Periodic Stokes Flow in Two Dimensions: Well-Posedness and Stability

In this thesis, we study the two-phase horizontally periodic quasistationary Stokes flow in two dimensions. This system models the evolution of a sharp interface, given by the graph of a periodic function, separating two immiscible Newtonian fluids with possibly different densities and viscosities. The interface dynamics are driven by surface tension effects and we may incorporate a gravitational force as well. Such two-phase flows are of great significance in many industrial processes and life sciences. The first part of this thesis focuses on the analysis of stationary two-phase Stokes systems with a fixed interface. We derive the horizontally periodic Stokeslet and use potential theory to define and analyze the hydrodynamic single- and double-layer potentials. Moreover, the invertibility of the hydrodynamic double-layer potential operator is extensively studied by means of a new class of (singular) integral operators. Using these layer potentials, we are able to construct the velocity field and the pressure of the fluids for any sufficiently regular interface. In the second part, we use the formula obtained for the velocity field together with the kinematic boundary condition to reformulate the two-phase Stokes system as a fully nonlinear and nonlocal evolution equation for the function parametrizing the interface. We show that this evolution equation is of parabolic type and employ abstract parabolic theory to obtain local well-posedness in subcritical Sobolev spaces whose exponent is arbitrarily close to a certain critical value. Additionally, we establish, by means of a classical parameter trick, a parabolic smoothing property. Furthermore, we present a full overview of equilibrium solutions to the two-phase Stokes system. We then study the stability properties of these equilibria and show that flat interfaces are exponentially stable in the Rayleigh--Taylor stable regime. Moreover, we show that finger-shaped equilibria occur when the denser fluid lies on top of the less dense fluid and prove that these equilibrium solutions are unstable due to the onset of a Rayleigh--Taylor instability pattern.