Uniform resonant chaotic mixing in fluid flows

Abstract

Laminar flows can produce particle trajectories that are chaotic, with nearby tracers separating exponentially in time. For time-periodic, two-dimensional flows and steady three-dimensional (3D) flows, enhancements in mixing due to chaotic advection are typically limited by impenetrable transport barriers that form at the boundaries between ordered and chaotic mixing regions. However, for time-dependent 3D flows, it has been proposed theoretically that completely uniform mixing is possible through a resonant mechanism called singularity-induced diffusion; this is thought to be the case even if the time-dependent and 3D perturbations are infinitesimally small. It is important to establish the conditions for which uniform mixing is possible and whether or not those conditions are met in flows that typically occur in nature. Here we report experimental and numerical studies of mixing in a laminar vortex flow that is weakly 3D and weakly time-periodic. The system is an oscillating horizontal vortex chain (produced by a magnetohydrodynamic technique) with a weak vertical secondary flow that is forced spontaneously by Ekman pumping--a mechanism common in vortical flows with rigid boundaries, occurring in many geophysical, industrial and biophysical flows. We observe completely uniform mixing, as predicted by singularity-induced diffusion, but only for oscillation periods close to typical circulation times.

Comparative numerical-experimental analysis of the universal impact of arbitrary perturbations on transport in three-dimensional unsteady flows

“…Observations of a similar behavior in a completely different system, an external 3D time-periodic flow driven by a rotating sphere, strongly suggest this is a universal phenomenon [18]. The tube structure is closely related to the capture into resonances in 3D steady and unsteady flows [19][20][21][22].…”

Optimizing the source distribution in fluid mixing

“…Previous work on optimization of mixing has focused on breaking flow symmetries [1][2][3][4], or optimizing quantities associated with chaotic advection, such as Lyapunov exponents or topological entropy [5][6][7][8][9][10]. Recent work has also involved optimizing the norm of the concentration field [11,12], in a manner similar to here.…”

On passage through resonances in volume-preserving systems

“…The later works include [15]. However, the corresponding theory, that gives a quantitative description of either single crossings or the long-time dynamics was not developed.…”