Numerical algorithms and simulations of boundary dynamic control for optimal mixing in unsteady Stokes flows

TL;DR

This paper presents a new high-accuracy numerical optimization algorithm for boundary control in unsteady Stokes flows to enhance mixing efficiency, demonstrating the effectiveness of boundary control strategies through simulations.

Contribution

Develops an efficient hybrid numerical method for boundary control optimization in unsteady Stokes flows, improving mixing strategies with diverse control inputs.

Findings

Mixing efficacy improves with multiple control types and time segmentation.

Mix-norm decays exponentially in optimal mixing scenarios.

Boundary control alone can effectively enhance mixing in incompressible flows.

Abstract

This work develops an efficient and accurate optimization algorithm to study the optimal mixing problem driven by boundary control of unsteady Stokes flows, based on the theoretical foundation laid by Hu and Wu in a series of work. The scalar being mixed is purely advected by the flow and the control is a force exerted tangentially on the domain boundary through the Navier slip conditions. The control design has potential applications in many industrial processes such as rotating wall driven mixing, micromixers with acoustic waves, and artificial cilia mixing. The numerical algorithms have high complexity, high accuracy demand, and high computing expense, due to the multiscale nature of the mixing problem and the optimization requirements. A crucial problem is the computation of the G$\^a$teaux derivative of the cost functional. To this end, a hybrid approach based on variational formula and finite difference is built with high accuracy and efficiency to treat various types of control input functions. We have experimented with various optimization schemes including the steepest descent algorithm, the conjugate gradient method and two line search options (backtracking and exact line search). We are able to identify and implement the best combinations. The numerical simulations show that the mixing efficacy is limited when only one single type of control is applied, but can be enhanced when more diverse control types and more time segmentation are utilized. The mix-norm in the optimal mixings decays exponentially. The numerical study in this work demonstrates that boundary control alone could be an effective strategy for mixing in incompressible flows.