The Changing Role of Applied Mathematics in Fluid Research Dynamics

Fluid dynamics and applied mathematics have a long shared history, with fluid dynamics providing some of the most challenging problems for applied mathematicians to tackle, and applied mathematics supplying the needed tools to further research in fluid dynamics. In some cases, in fact, fluid mechanics has provided the impetuous for developments in various branches of applied mathematics. This talk describes current developments in the application of applied mathematics to fluid dynamics research. In particular, it is argued that dynamical systems theory and optimal control theory are "disruptive innovations" that are revolutionizing fluid dynamics research. Being an infinite dimensional, nonlinear dynamical system, fluid dynamics is a perfect testbed for modern developments in dynamical systems theory, including chaos theory, reduced-order modeling, Lagrangian coherent structures, etcetera. On the other hand, but for the same reason, fluid dynamics represents a very challenging application of optimal control theory. Current developments in the application of optimal control theory in fluid dynamics are discussed with illustrations from our own research.