Stokes flow in a mixer with changing geometry

Abstract

A slow-flow mixing device that mimics a natural mixing technique is described. Analytical, numerical and experimental results are presented for the `translating, rotating mixer', which illustrate its mixing effectiveness. In large part, this effectiveness is due to the fact that its geometry changes with time, a feature rare in mathematically tractable slow-flow mixing models. The mixer consists of a large circular cylinder filled with fluid, which is stirred by a circular cylindrical `rod' that moves around in the fluid. The stirring rod may also rotate about its axis. The velocity field is calculated explicitly for the mixer, and its mixing action is simulated numerically. Through a complex-variable formulation of the problem, the energy input required for various mixing protocols may readily be determined, and in turn suggestions for efficient mixing using the device are offered. To validate the analytical and numerical results, tracer-advection experiments are performed, using a simple experimental rig and a variety of mixing protocols, providing encouraging agreement with numerical simulation.