Relaxation Enhancement by Time-Periodic Flows

TL;DR

This paper investigates how time-periodic incompressible flows can enhance diffusive mixing, demonstrating that time dependence can significantly improve mixing efficiency, especially in flows with certain dynamical properties.

Contribution

It extends previous results to show that time-periodic flows, even with Hamiltonian structure at each fixed time, can be relaxation-enhancing, and provides a general criterion for decay of related semigroups.

Findings

Time-periodic flows can be relaxation-enhancing.

Time dependence of flows can aid mixing.

Extension to nonlinear diffusion models.

Abstract

We study enhancement of diffusive mixing by fast incompressible time-periodic flows. The class of relaxation-enhancing flows that are especially efficient in speeding up mixing has been introduced in [2]. The relaxation-enhancing property of a flow has been shown to be intimately related to the properties of the dynamical system it generates. In particular, time-independent flows $u$ such that the operator $u \cdot \nabla$ has sufficiently smooth eigenfunctions are not relaxation-enhancing. Here we extend results of [2] to time-periodic flows $u(x,t)$ and in particular show that there exist flows such that for each fixed time the flow is Hamiltonian, but the resulting time-dependent flow is relaxation-enhancing. Thus we confirm the physical intuition that time dependence of a flow may aid mixing. We also provide an extension of our results to the case of a nonlinear diffusion model. The proofs are based on a general criterion for the decay of a semigroup generated by an operator of the form $\Gamma+iAL(t)$ with a negative unbounded self-adjoint operator $\Gamma$, a time-periodic self-adjoint operator-valued function $L(t)$, and a parameter $A>>1$.