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ACMS Abstracts: Spring 2016

Stefan Llewellyn Smith (UCSD)

Hollow vortices

Hollow vortices are vortices whose interior is at rest. They posses vortex sheets on their boundaries and can be viewed as a desingularization of point vortices. After giving a history of point vortices, we obtain exact solutions for hollow vortices in linear and nonlinear strain and examine the properties of streets of hollow vortices. The former can be viewed as a canonical example of a hollow vortex in an arbitrary flow, and its stability properties depend on a single non-dimensional parameter. In the latter case, we reexamine the hollow vortex street of Baker, Saffman and Sheffield and examine its stability to arbitrary disturbances, and then investigate the double hollow vortex street. Implications and extensions of this work are discussed.

Tom Solomon (Bucknell)

Experimental studies of reaction front barriers in laminar flows

We present studies of the effects of vortex-dominated fluid flows on the motion of reaction fronts produced by the excitable Belousov-Zhabotinsky reaction. The results of these experiments have applications for advection-reaction-diffusion dynamics in a wide range of systems including microfluidic chemical reactors, cellular-scale processes in biological systems, and blooms of phytoplankton in the oceans. To predict the behavior of reaction fronts, we adapt tools used to describe passive mixing.In particular, the concept of an invariant manifold is extended to account for reactive burning. Burning invariant manifolds (BIMs) are predicted as one-way barriers that locally block the motion of reaction fronts. These ideas are tested and illustrated experimentally in a chain of alternating vortices, a spatially-random flow, vortex flows with imposed winds, and a three-dimensional, nested vortex flow. We also discuss the applicability of BIM theory to the motion of bacteria in fluid flows.

Daniele Cappelletti (KU)

Deterministic and stochastic reaction networks

Mathematical models of biochemical reaction networks are of great interest for the analysis of experimental data and theoretical biochemistry. Moreover, such models can be applied in a broader framework than that provided by biology. The classical deterministic model of a reaction network is a system of ordinary differential equations, and the standard stochastic model is a continuous-time Markov chain. A relationship between the dynamics of the two models can be found for compact time intervals, while the asymptotic behaviours of the two models may differ greatly. I will give an overview of these problems and show some recent development.

Lihui Chai (UCSB)

Semiclassical limit of the Schrödinger-Poisson-Landau-Lifshitz-Gilbert system

The Schrödinger-Poisson-Landau-Lifshitz-Gilbert (SPLLG) system is an effective microscopic model that describes the coupling between conduction electron spins and the magnetization in ferromagnetic materials. This system has been used in connection to the study of spin transfer and magnetization reversal in ferromagnetic materials. In this paper, we rigorously derive the Vlasov-Poisson-Landau-Lifshitz-Glibert system as the semiclassical limit of SPLLG. The major difficulties come from the presence of the spin-magnetization coupling and the discontinuities of the magnetization at the boundary of the material. To overcome these difficulties, we first take the semiclassical limit (vanishing Planck constant) of a smoothed SPLLG system, and then the limit of vanishing smoothing parameter. As a byproduct, we prove the local existence and uniqueness of classical solutions to the smoothed SPLLG system.