Modeling of fluids and waves with analytics and numerics

Abstract

Capillary instability (Plateau-Rayleigh instability) has been playing an important role in experimental work such as multimaterial fiber drawing and multilayer particle fabrication. Motivated by complex multi-fluid geometries currently being explored in these applications, we theoretically and computationally studied capillary instabilities in concentric cylindrical flows of N fluids with arbitrary viscosities, thicknesses, densities, and surface tensions in both the Stokes regime and for the full Navier-Stokes problem. The resulting mathematical model, based on linear-stability analysis, can quickly predict the breakup lengthscale and timescale of concentric cylindrical fluids, and provides useful guidance for material selections and design parameters in fiber-drawing experiments. A three-fluid system with competing breakup processes at very different length scales is demonstrated with a full Stokes flow simulation. In the second half of this thesis, we study large-scale PDE-constrained microcavity topology optimization. Applications such as lasers and nonlinear devices require optical microcavities with long lifetimes Q and small modal volumes V. While most microcavities are designed mostly by hand using some understanding of the physical principles of the confinement, we let the computer discover its own structures. We formulate and solve a full 3d optimization scheme over all possible 2d-lithography patterns in a thin dielectric film. The key to our formulation is a frequency-averaged local density of states (LDOS), where the frequency averaging corresponds to the desired bandwidth, evaluated by a novel technique: solving a single electromagnetic wave scattering problem at a complex frequency.