Control and optimization problems in fluid mechanics

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The proposed research will address a range of topics that bridge the theoretical and computational fluid mechanics with the modern theory of control and optimization. Our goal is to develop and validate a suite of advanced computational methods for a number of fundamental problems in this field. The theoretical foundations established by this project will provide the basis for more applied investigations to be pursued in the future. In one part of the proposed effort we will revisit vortex-based models for inviscid and incompressible flows characterized by continuous vorticity distributions, such as vortex patches and vortex sheets. These models are relevant for understanding the mechanisms of hydrodynamic propulsion important, for instance, for fish-like locomotion. Our research will add new value to these classical models by providing a suite of tools for solution of control and optimization problems formulated based on such models. In another part of the proposed effort we will extend the methods of control and optimization to flow phenomena involving multiphysics effects and featuring interfaces and contact lines. Such flows are ubiquitous as models in a number of important industrial applications. Mathematically, all of these phenomena are described as 'free-boundary problems' in which the shape of the domain must be determined as a part of the solution. This fact has far-reaching consequences for the mathematical formulation of inverse problems for such systems. As a result, one is required to develop special approaches combining classical methods of applied mathematics with a number of novel techniques. In addition, in this investigation we will also seek to make contributions to some outstanding problems in classical hydrodynamics concerning flows past obstacles. The funding provided by this grant will allow Dr. Protas to train several graduate students in interdisciplinary research rooted in applied mathematics and involving also elements of scientific computing, physics and engineering science.