Cellular flow control design for mixing based on the least action principle

TL;DR

This paper introduces a novel fluid mixing control strategy based on the Least Action Principle, using cellular flows to optimize energy efficiency and mixing effectiveness within a finite time.

Contribution

It formulates a new optimal control problem for fluid mixing using cellular flows and proves the existence of a global solution with derived optimality conditions.

Findings

Optimal control of cellular flows enhances mixing efficiency.

Numerical experiments confirm the effectiveness of the proposed control strategy.

Abstract

We consider a novel approach for the enhancement of fluid mixing via pure stirring strategies building upon the Least Action Principle (LAP) for incompressible flows. The LAP is formally analogous to the Benamou--Brenier formulation of optimal transport, but imposes an incompressibility constraint. Our objective is to find a velocity field, generated by Hamiltonian flows, that minimizes the kinetic energy while ensuring that the initial scalar distribution reaches a prescribed degree of mixedness by a finite time. This formulation leads to a ``point to set" type of optimization problem which relaxes the requirement on controllability of the system compared to the classic LAP framework. In particular, we assume that the velocity field is induced by a finite set of cellular flows that can be controlled in time. We justify the feasibility of this constraint set and leverage Benamou--Brenier's results to establish the existence of a global optimal solution. Finally, we derive the corresponding optimality conditions for solving the optimal time control and conduct numerical experiments demonstrating the effectiveness of our control design.