Incompressible flows with interfaces occur in a wide variety of physical phenomena as well as technological processes. Mathematically, the motion is governed by the incompressible Navier-Stokes equations together with interfacial conditions.
In this thesis, we present a numerical approach to simulate the two-dimensional viscous, incompressible flows with interfaces. First we introduce some new coordinates so that the interface is mapped into a coordinate line which enables us to work on a rectangular domain instead of a deformed geometry. Then an iterative approach combined with an implicit time marching method is applied to update the motion in time. At each iterate, the Fourier transform and the pseudo-spectral technique are applied in the horizontal direction, X, under the assumption that the solutions are periodic in X. Then we write the semi-discretized equations as a 1st-order ODE system with respect to the vertical coordinate, Z, and an efficient ODE solver is developed to construct the solutions.
As an application of our numerical approach, we study the problem of steady progressive interfacial waves (Stokes waves). In contrast to all the previous work which was concerned with inviscid fluids, we study Stokes waves in the presence of viscosity. Our numerical results show that the effect of viscosity is somehow equivalent to the decay of the expansion parameter in the series expansion of the inviscid Stokes waves. Our work suggests a new xpansion form for Stokes waves in viscous fluids. In addition, we perform a similar study for the viscous effects on standing waves.
Finally, some analysis is applied for the linearized motion. In particular, the asymptotic solution to the linearized interfacial flow is derived.