Simulation of viscoelastic free-surface flows

Free-surface flows are ubiquitous in our lives. Almost all flows we encounter as human beings are of this type. However, when we think of flows we expect mostly water flows. The latter is an example of a Newtonian flow, which only show viscous behavior. Subject of this thesis, however, are the viscoelastic flows, which - in addition to the viscous behavior - behave elastically to a certain degree. In many cases, this elasticity is due to a polymeric component in the fluid. This thesis will take the reader from the microscopic world of polymer chains to the macroscopic description of a polymer solution or polymer melt. Moreover, the thesis highlights important properties that carry over from the microscopic to the macroscopic world. One of these properties is that the conformation tensor should stay positive definite. The latter then subsequently motivates the derivation of a numerical scheme, based on the work of Fattal and Kupferman, that by replacing the conformation tensor with the log-conformation tensor in the numerical model ensures this positivity by design. The novelty of our approach is that it allows for the easier application of many analytical tools. E.g., Newton's method can be easily applied to the non-linear equation system, but also the numerical analysis of schemes becomes easier. The central application, in which this thesis culminates, is the simulation of a rising air bubble in a viscoelastic fluid. The latter, so simple it may seem, shows phenomena that are to date not fully understood. It has long been observed experimentally that the bubble velocity is not always a continuous function of the bubble volume. This thesis tries to shed some new light on the mechanism behind this discontinuity by showing the emergence of vortices on the bubble surface