Boundary control for optimal mixing via Stokes flows and numerical implementation

TL;DR

This paper develops numerical methods for boundary control in Stokes flows to optimize mixing, demonstrating that boundary control alone can effectively enhance mixing efficiency in incompressible fluids.

Contribution

It introduces gradient descent algorithms with variational and algorithmic differentiation methods for boundary control optimization in fluid mixing.

Findings

Boundary control achieves mixing comparable to internal methods.

VF method is more efficient than AD for large control basis.

Mixing decay follows a power law over time.

Abstract

This work develops scientific computing techniques to further the exploration of using boundary control alone to optimize mixing in Stokes flows. The theoretical foundation including mathematical model and the optimality conditions for solving the optimal control has been established by Hu and Wu in a series of work. The scalar being mixed is purely advected by the flow and the control is exerted tangentially on the domain boundary through the Navier slip conditions. The control design is motivated by the physical observations that the moving or rotating walls accelerate mixing. A gradient descent-based optimization algorithm is designed. A critical problem is the computation of the Gateaux derivative or the gradient of the cost functional. Two methods are proposed: one is based on the Variational Formula (VF) and one utilizes Algorithmic Differentiation (AD). The convergence of the algorithm is studied and various designs of boundary control using cosine and sine functions with time segmentation are computed. The algorithm has a first order convergence rate and the VF method is more efficient by taking only one third of the time as the AD method when the dimension of control basis is large. The numerical implementations show that the boundary control produces similar mixing results as internal mixings in the existing literature. The mixing effect becomes better when more diverse basis control functions and more time segmentation are utilized. It is shown that the mixing decay rate in time follows power rules, approximately. The numerical study in this work suggests that boundary control alone could be an effective strategy for mixing in incompressible fluid flows.