PhD Comprehensive Exam | Hanzhe (Tom) Chen, Two and Three Dimensional Floating Objects with Surface Tension

M3 4206

Candidate

Hanzhe(Tom) Chen | Applied Math, University of Waterloo

Title

Two and Three Dimensional Floating Objects with Surface Tension

Abstract

The study of floating objects can be traced back to Archimedes’ time. With the consideration of surface tension, the ability and stability of flotation gets complicated but more interesting. We propose two basic problems: the infinity long horizontal floating cylinder and the floating ball on an unbounded reservoir. For the floating cylinder problem (two-dimensional), circular and strictly convex cross-sections are both considered. The circular floating cylinder has a symmetric configuration. As the solution of one-dimensional capillary equation can be translated horizontally, there is a possibility that the fluid interfaces intersect. We apply both the force and the energy approaches, at most two equilibrium configurations can be obtained, one is stable, the other is not, in consideration of the possible of intersection of interfaces. Unlike the circular cylinder, the configuration of strictly convex body usually admits asymmetric configurations. Conditions are imposed to guarantee a unique configuration for a fixed position. An example of an ellipse with rotation will be given. For the floating ball problem (three-dimensional), we assume the floating configuration is radially symmetric. The fluid interface has to be solved numerically. An example with two force balanced points will be given, we will show both of them are stable. More examples and applications are anticipated.