Control of mixing in a Stokes’ fluid flow

Mixing problems are common in science and engineering, the aim being to combine fluids as quickly and efficiently as possible. We consider the design of a simple controller to promote mixing in a Stokes’ fluid flow. In this paper, a controlled stirring motion is represented by a velocity field consisting of the superposition of a steady base flow and a second field modulated by a saturating, time-dependent control variable. The problem can be formulated as an optimal control one, but the presence of a nonlinearity in the state dynamics and an input constraint make the construction of a feedback law difficult. The size of the problem means that receding horizon schemes, revolving around real-time optimization of even a simplified model, are currently not feasible for fast applications. To address this problem, we exploit theory for the control of bilinear systems to propose a simple, well-performing, explicit feedback law. There are several interesting design issues associated with applying this approach to a fluid mixing application. We demonstrate these by designing a controller for a two-dimensional fluid mixing problem with simple cellular flows and discuss the relevant implementation decisions. The closed-loop forcing fields have many features that are as expected: regions of the flow with large spatial velocity gradients target regular islands of concentration and the input favors velocity fields with contours aligned perpendicular to the scalar field.