The investigators develop and apply efficient boundary integral methods for the motion of interfaces in 3D flow. The methods address a significant difficulty in the numerical computation of fluid interfaces with surface tension or elastic forces in 3D flow. Such forces introduce high order (i.e., high derivative) terms into the evolution equations, which lead to severe stability constraints or `stiffness' for explicit time-integration methods. Furthermore, the high order terms appear in nonlinear and nonlocal operators, making the efficient application of stable implicit methods difficult. The investigators' method relies on using the first and second fundamental coefficients of the surface as dynamical variables, and employs a special parameterization of the interface combined with an analysis of the governing equations at small scales. This enables the efficient application of implicit time-integration methods for 3D flow. The investigators implement the method in canonical interface problems for inviscid fluids, including the Kelvin-Helmholtz, Rayleigh-Taylor, and water wave problems, and study the dynamics of inextensible elastic sheets in inviscid flow and vesicles in 3D viscous flow. Most importantly, they develop a version of the numerical method which uses domain decomposition or overlapping coordinate patches to describe the interface. This has the added benefit of providing a framework for the implementation of spectrally accurate and spatially adaptive methods.
Moving boundary problems occur in many diverse areas in, for example, fluid dynamics, materials science, and biology. Specific examples include traveling ocean waves, growing cancer tumors, beating hearts and moving cells and organisms. The investigators develop accurate and efficient `boundary integral' numerical methods for the simulation of moving boundaries in applications. Boundary integral methods are among the most accurate numerical methods for the simulation of moving interfaces, but are often inefficient when the interface is acted on by surface tension or elastic forces. The development of fast and accurate boundary integral methods for 3D interfacial flow with surface tension or elastic forces will be of great benefit in understanding existing applications and developing technology further.
