A theoretical and experimental study of hyperbolic and degenerate mixing regions in a chaotic Stokes flow

Abstract

We examine the rotor–oscillator flow, a slow viscous flow between long parallel plates driven by the rotation of a slender cylinder (the rotor) and the longitudinal oscillation of one of the plates (the oscillator). For rotor locations of interest to us, this flow exhibits a hyperbolic mixing region, characterized by homoclinic tangling associated with a hyperbolic fixed point, and a degenerate mixing region, characterized by heteroclinic tangling associated with two degenerate fixed points on one of the boundary plates (normally the oscillator). These mixing regions are investigated both theoretically, by applying various dynamical tools to a mathematical model of the flow, and experimentally, by observing the advection of a passive tracer in a specially constructed apparatus. Although degenerate mixing regions have been largely ignored or undervalued in previous research on chaotic mixing, our results demonstrate that more mixing is associated with the degenerate mixing region than the hyperbolic one in many cases. We have also discovered a peculiar phenomenon, which we call Melnikov resonance, involving a rapid fluctuation in the size of the hyperbolic mixing region as the frequency of the oscillator is varied.