Enhanced diffusion for time-periodic alternating shear flows

CDSNS Colloquium
Friday, March 15, 2024 - 3:30pm for 1 hour (actually 50 minutes)
Skiles 254
Kyle Liss – Duke – kyle.liss@duke.eduhttps://scholars.duke.edu/person/Kyle.Liss
Keagan Callis

The dynamics of a passive scalar, such as temperature or concentration, transported by an incompressible flow can be modeled by the advection-diffusion equation. Advection often results in the formation of complicated, small-scale structures and can result in solutions relaxing to equilibrium at a rate much faster than the corresponding heat equation in regimes of weak diffusion. This phenomenon is typically referred to as enhanced diffusion. In this talk, I will discuss a joint work with Tarek Elgindi and Jonathan Mattingly in which we construct an example of a divergence-free velocity field on the two-dimensional torus that results in optimal enhanced diffusion.  The flow consists of time-periodic, alternating piece-wise linear shear flows. The proof is based on the probabilistic representation formula for the advection-diffusion equation, a discrete time approximation, and ideas from hyperbolic dynamics.