UCSB Applied Math/PDE/Data Science Seminar

Abstract: Elastic instabilities are abundant in natural and engineered structures across a wide range of scales, from supercoiled DNA and folded tissues to the waves of leaves and the petals of flowers. While much progress has been made over the last two centuries in understanding and predicting the equilibrium shapes of stressed materials, the non-equilibrium dynamics of buckling and wrinkling phenomena continues to pose interesting theoretical and computational challenges. In this talk, I will discuss our recent joint theoretical and experimental efforts to understand the evolution of elastic patterns that emerge from elastic and fluid dynamical instabilities. In the first part, by considering the evolution of wrinkle patterns of confined elastic membranes on fluid surfaces, I will show how integral constraints can slow down pattern selection dynamics, causing departures from self-similar behaviors frequently observed in fluid mechanics. In the second part, I will demonstrate how non-trivial buckling patterns may emerge under rapid quenching. Specifically, I will demonstrate how tuning external control parameter enables the targeted selection of specific buckling modes. This phenomenon, which is reminiscent of the Kibble-Zurek mechanism in continuous non-equilibrium phase transitions, promises novel approaches to dynamical pattern design.

Host: Björn Birnir

Abstract: In this talk, I will present the Nernst-Planck-Boussinesq (NPB) system, a coupled model that combines the Nernst-Planck equations with the Boussinesq approximation to describe ionic electrodiffusion in an incompressible fluid under non-isothermal conditions. A key feature of the NPB system is the nonlinear structure of the electromigration term, which is influenced by the reciprocal of the temperature. This distinguishes the NPB system from other electrodiffusion models, such as the Nernst-Planck-Navier-Stokes system, and introduces unique mathematical challenges. I will discuss several analytical properties of the system, focusing on the global existence of weak solutions in three dimensions, as well as the long-time behavior of these solutions, including their exponential convergence to steady states. This is a joint work with Elie Abdo and Ruimeng Hu.

Host: Ruimeng Hu