On the optimization of mixing protocol in a certain class of three-dimensional Stokes flows

Abstract

Mixing in a special class of three-dimensional, non-inertial periodic flows is studied numerically. In the type of flow considered here, the cross-sectional velocity components are independent of the axial flow and the axial flow is independent of the axial coordinate. Using the eccentric helical annular mixer as a prototype, we consider the counter-rotating case with steady rotation of the outer cylinder and sinusoidal modulation of the inner one. Apart from the mixer geometry, the behavior of the system is governed by two dimensionless parameters obtained by scaling the cross-sectional stirring protocol with respect to the characteristic residence time of the fluid in the mixer. The first parameter is related to the average number of turns of the outer cylinder and the second one is related to the average number of modulation periods of the inner cylinder. The convection-diffusion equation is solved numerically, with temperature as a passive scalar, at high Péclet number. For a given three-dimensional mixer geometry and axial flow rate we show that there is an optimum modulation frequency for which the exit standard deviation of the temperature field is a minimum. Lagrangian simulations at infinite Péclet number and the use of other tools to study mixing, such as stretching calculations and tracer tracking methods, confirm that the optimized protocol does result in very effective mixing.