Control of mixing in a Stokes’ fluid flow

Abstract

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.

Versatile low-Reynolds-number swimmer with three-dimensional maneuverability

“…Motile bacteria also observed to enhance the mixing at the interface of two streams of fluid [22,23], and a flat rigid surface covered with restrained bacteria can significantly increase the mixing of adjacent fluid particle [24]. While the details of flow pattern produced by the Quadroar is beyond the scope of this article and deserves an independent investigation, we would like to comment that the broad family of rich sweeping patterns, that can be periodic, quasi-periodic and potentially chaotic, can result in a wide range of complex flow fields capable of driving an efficient mixing, which is currently achieved by boundary actuation [25] or externally actuated stirring [26]. Rotating disks of the Quadroar can be considered as micro-impellers [27,28] that not only directly stir the flow, but also propel the Quadroar forward along specific yet complex trajectories to achieve maximum efficiency stirring.…”

Fast fluid heating by adaptive flow reorientation

“…The above limitations motivated a host of efforts to design and create flows for enhancing scalar transport by way of optimal control [18][19][20][21][22][23][24][25][26][27][28]. This overwhelmingly concerns scalar homogenisation in adiabatic systems and the control strategy essentially consists of a priori determining the flow (forcing) that maximally accelerates said homogenisation in terms of an optimality criterion.…”

Modelling for robust feedback control of fluid flows

“…Examples include the lowering of fuel costs and greenhouse gas emissions via the drag reduction of aircraft (Bushnell 2003) and shipping (Corbett & Koehler 2003), optimal mixing of chemical reagents (Couchman & Kerrigan 2010) and wind turbine gust alleviation (Frederick et al 2010), with many more examples stemming from the natural world (Fish & Lauder 2006). Attempts to control fluid flow are typically classified into three broad categories (Gad-el-Hak 2000): passive (e.g.…”