Events for 10/12/2021 from all calendars

Nonlinear Partial Differential Equations

Time: 3:00PM - 4:00PM

Location: Zoom

Speaker: Kevin Zumbrun, Indiana University

Title: Stability of roll wave solutions in inclined shallow-water flow

Abstract: We review recent developments in stability of periodic roll-wave solutions of the Saint Venant equations for inclined shallow-water flow. Such waves are well-known instances of hydrodynamic instability, playing an important role in hydraulic engineering, for example, flow in a channel or dam spillway. Until recently, the analysis of their stability has been mainly by formal analysis in the weakly unstable or ``near-onset`` regime. However, hydraulic engineering applications are mainly in the strongly unstable regime far from onset. We discuss here a unified framework developed together with Blake Barker, Mat Johnson, Pascal Noble, Miguel Rodrigues, and Zhao Yang for the study of roll wave stability across all parameter regimes, by a combination of rigorous analysis and numerical computation. The culmination of our analysis is a complete stability diagram, of which the low-frequency stability boundary is, remarkably, given explicitly as the solution of a a cubic equation in the parameters of the solution space.