Optimal Control Design for Fluid Mixing: from Open-Loop to Closed-Loop

Date: Thu. May 19, 2022
Organized by: Durham University, Seminars in Mathematical Sciences
Title: Optimal Control Design for Fluid Mixing: from Open-Loop to Closed-Loop

Speaker: Visitor scientist Prof. Dr. Weiwei Hu
Affiliation: University of Georgia (USA)

Abstract. The question of what velocity fields effectively enhance or prevent transport and mixing, or steer a scalar field to a desired distribution, is of great interest and fundamental importance to the fluid mechanics community. In this work, we mainly discuss the problem of optimal mixing of an inhomogeneous distribution of a scalar field via active control of the flow velocity, governed by the Stokes or the Navier-Stokes equations. Specifically, we consider that the velocity field is steered by a control input which acts tangentially on the boundary of the domain through the Navier slip boundary conditions. This is motivated by mixing within a cavity or vessel by rotating or moving walls. Our main objective is to design a Navier slip boundary control for achieving optimal mixing. Non-dissipative scalars governed by the transport equation will be our main focus. In the absence of molecular diffusion, mixing is purely determined by the flow advection. This essentially leads to a nonlinear control and optimization problem. A rigorous proof of the existence of an optimal control and the first-order necessary conditions for optimality will be addressed. Moreover, feedback laws will be constructed based on the idea of instantaneous control as well as a direct approximation of the optimality system. Finally, numerical experiments will be presented to demonstrate our ideas and control designs.