Resonances and Mixing in Confined Time-dependent Stokes Flows: The experiments, Numerics, and Analytics

Abstract

Mixing in Stokes flows is notoriously difficult to achieve. With characteristic scales of the flows being too small for the turbulence to be present, and too large for the molecular diffusion to be significant, the chaotic advection presents almost the only mechanism that can lead to mixing. Unfortunately for mixing, the intrinsic symmetries of the flow create invariant surfaces that act as barriers to mixing. Thus, a key to efficient mixing is to add to the original (symmetric) flow a certain kind of perturbation that destroys those symmetries. In this dissertation, two ways of obtaining mixing in 3D near-integrable bounded time -dependent Stoke Flows are studied: resonances and separatrix crossings. First, I illustrate that the resonances between different components of the original flow and the perturbation may break the invariant surfaces, paving a way to the large-scale mixing. Theoretical estimations are compared against the results of numerical simulations, as well as 3D particle tracking velocimetry (3D-PTV) experimental results. Second, chaotic advection and mixing due to quasi-random jumps of the adiabatic invariant (AI) occurring when a streamline crosses the separatrix surfaces is studied. Analytical expressions for the change in the AI near the separatrix surfaces are derived and compared with numerical simulations.