Mathematical analysis of bubble rings in ideal fluids

Basic data for this project

Description

In this project, the evolution of toroidal bubble vortices is to be investigated. Bubble vortices are special vortices that occur in two-phase fluids. A typical and fascinating example is an air bubble ring in water created by dolphins or beluga whales. The underlying mathematical model is given by the two-phase Euler equations with surface tension. One major goal is a thorough mathematical construction of steady rings that move without changing their shape, and of perturbations of these. Such traveling waves are known for the classical Euler equations, but their existence is unknown for surface tension dependent models. Of particular interest is the role of the surface tension for the shape of the ring, which will be investigated. A second objective of this project is to understand how the effect of surface tension can be exploited to rigorously justify certain nonlinear motion laws of one or more interacting bubble rings. The understanding of such motion laws for the classical Euler equations is poor, and it is expected that the regularising effect of surface tension helps to mathematically tame the problem.