Solitons of geometric flows and their applications

In this thesis we construct solitons of geometric flows with applications in three different settings. The first setting is related to nonuniqueness for geometric heat flows. We show that certain double cones in Euclidean space have several self-expanding evolutions under mean curvature flow. The construction of the associated self-expanding solitons leads to an application in fluid dynamics. We present a new model for the behaviour of oppositely charged droplets of fluid, based on the mean curvature flow of double cones. If two oppositely charged droplets of fluid are close to each other, they start attracting each other and touch eventually. Surprisingly, experiments have shown, that if the strength of the charges is high enough, then the droplets are repelled from each other, after making short contact. The constructed self-expanders can be used to correctly predict the experimental results, using our theoretical model. Secondly we employ space-time solitons of the mean curvature flow to give a geometric proof of Hamilton's Harnack estimate for the mean curvature flow. This proof is based on the observation that the associated Harnack quantity is the second fundamental form of a space-time self-expander. Moreover the self-expander is asymptotic to a cone over the convex initial hypersurface. Hence the self-expander can be seen as the mean curvature evolution of a convex cone, which we exploit to show that preservation of convexity directly implies the Harnack estimate. In the last chapter we study solutions of the mean curvature flow in a Ricci flow backgound. We show that the space-time track of such a solution can be seen as a soliton. Moreover the second fundamental form of this soliton matches the evolution of a functional, which is the analogue of G. Perelman's F-functional for the Ricci flow on a manifold with boundary and which also has relations to quantum gravity. Furthermore our construction provides a link between the Harnack estimate for the mean curvature flow and the Harnack estimate for the Ricci flow

Bouncing, bursting, and stretching: the effects of geometry on the dynamics of drops and bubbles

In this thesis, we develop a physical understanding of the effects of viscosity and geometry on the dynamics of interfacial flows in drops and bubbles. We first consider the coalescence of pairs of conical water droplets surrounded by air. Droplet pairs can form cones under the influence of an electric field and have been observed to coalesce or recoil depending on the angle of this cone. With high resolution numerical simulations we show the coalescence and non-coalescence of these drop pairs is negligibly affected by the electric field and can be understood through a purely hydrodynamic process. The coalescence and recoil dynamics are shown to be self similar, demonstrating that for these conical droplet pairs viscosity has a negligible effect on the observed behavior. We generalize this result to the coalescence and recoil of droplets with different cone angles, and focus on droplets coalescing with a liquid bath and flat substrate. From the simulations of these droplets with different cone angles, an equivalent angle is found that describes the coalescence and recoil behavior for all water cones of any cone angle. While viscosity is found to negligibly affect the coalescence of conical water drops, it plays a key role in regulating the coalescence process of bursting gas bubbles. When these gas bubbles burst, a narrow liquid jet is formed that can break up into tiny liquid jet drops. Through consideration of the effects of viscosity, we show that these jet drops can be over an order of magnitude smaller than previously thought. Here, viscosity plays a key role in balancing surface tension and inertial forces and determining the size of the jet drops. Finally, we investigate the drainage of surfactant free, ultra-viscous bubbles where surface tension serves only to set the initial shape of the bubble. We use interferometry to find the thickness profiles of draining bubble films up to the point the of rupture. A theoretical film drainage model considering the balance of viscous and gravitational stresses is developed and numerically computed. The numerical results are found to be consistent with the experimentally obtained thickness profiles. In this work we provide insight into the role of viscosity in the outlined interfacial flows. The results of this thesis will advance the understanding of drop production in clouds, the marine climate, and the degassing of glass melts