Resonance phenomena and long-term chaotic advection in Stokes flows

Abstract

Creating chaotic advection is the most efficient strategy to achieve mixing in a microscale or in a very viscous fluid, and it has many important applications in microfluidic devices, material processing and so on. In this paper, we present a quantitative long-term theory of resonant mixing in 3-D near-integrable flows. We use the flow in the annulus between two coaxial elliptic counter-rotating cylinders as a demonstrative model. We illustrate that such resonance phenomena as resonance and separatrix crossings accelerate mixing by causing the jumps of adiabatic invariants. We calculate the width of the mixing domain and estimate a characteristic time of mixing. We show that the resulting mixing can be described in terms of a single diffusion-type equation with a diffusion coefficient depending on the averaged effect of multiple passages through resonances. We discuss what must be done to accommodate the effects of the boundaries of the chaotic domain.