Fluid Mechanics and Waves Seminar - Fall 2019

The prediction of interactions between nonlinear laser beams is a longstanding open problem. A traditional assumption is that these interactions are deterministic. We have shown, however, that in the nonlinear Schrodinger equation (NLS) model of laser propagation, beams lose their initial phase information in the presence of input noise. Thus, the interactions between beams become unpredictable as well. Not all is lost, however. The statistics of many interactions are predictable by a universal model.

Computationally, the universal model is efficiently solved using a novel spline-based stochastic computational method. Our algorithm efficiently estimates probability density functions (PDF) that result from differential equations with random input. This is a new and general problem in numerical uncertainty-quantification (UQ), which leads to surprising results and analysis at the intersection of probability and approximation theory.

Coarse-grained continuum models of active fluids capture important aspects of self-organizing behavior seen in experiments while providing a computationally tractable basis for their simulation. Producing accurate solutions to these models is nevertheless challenging, and numerical schemes for their solution have been largely limited to simple, stationary geometries or low-accuracy methods. In this talk, I will describe a spectrally accurate method for the simulation of active fluids in complex geometries, and show how to use this method to simulate deformable droplets of active fluids. When multiple drops are immersed in a Stokesian fluid, their interactions are captured through a robust and scalable boundary integral method, allowing for the simulation of thousands of such particles which may come into near-contact with each other. We explore some of the emergent phenomenon that occurs with such large aggregates of active droplets.