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arXiv · 2309.08870

Multidimensional stability and transverse bifurcation of hydraulic shocks and roll waves in open channel flow

Zhao Yang·Kevin Zumbrun
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Abstract

We study by a combination of analytical and numerical methods multidimensional stability and transverse bifurcation of planar hydraulic shock and roll wave solutions of the inviscid Saint Venant equations for inclined shallow-water flow, both in the whole space and in a channel of finite width, obtaining complete stability diagrams across the full parameter range of existence. Technical advances include development of efficient multi-d Evans solvers, low- and high-frequency asymptotics, explicit/semi-explicit computation of stability boundaries, and rigorous treatment of channel flow with wall-type physical boundary. Notable behavioral phenomena are a novel essential transverse bifurcation of hydraulic shocks to invading planar periodic roll-wave or doubly-transverse periodic herringbone patterns, with associated metastable behavior driven by mixed roll- and herringbone-type waves initiating from localized perturbation of an unstable constant state; and Floquet-type transverse ``flapping'' bifurcation of roll wave patterns.

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Zhao Yang, Kevin Zumbrun. 2023-09-16. Multidimensional stability and transverse bifurcation of hydraulic shocks and roll waves in open channel flow. https://arxiv.org/abs/2309.08870

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