The refined inviscid stability condition and cellular instability of viscous shock waves

TL;DR

This paper links the violation of a refined stability condition for viscous shock waves to the emergence of complex cellular instabilities in duct flows, revealing a cascade of bifurcations as the cross-section enlarges.

Contribution

It introduces a refined stability condition for viscous shocks and demonstrates its violation leads to cellular instability and bifurcation cascades in duct flows.

Findings

Violation of the refined stability condition causes multidimensional cellular instability.

Large cross-section ducts exhibit a cascade of Hopf bifurcations.

The refined stability condition is numerically computable.

Abstract

Combining work of Serre and Zumbrun, Benzoni-Gavage, Serre, and Zumbrun, and Texier and Zumbrun, we propose as a mechanism for the onset of cellular instability of viscous shock and detonation waves in a finite-cross-section duct the violation of the refined planar stability condition of Zumbrun--Serre, a viscous correction of the inviscid planar stability condition of Majda. More precisely, we show for a model problem involving flow in a rectangular duct with artificial periodic boundary conditions that transition to multidimensional instability through violation of the refined stability condition of planar viscous shock waves on the whole space generically implies for a duct of sufficiently large cross-section a cascade of Hopf bifurcations involving more and more complicated cellular instabilities. The refined condition is numerically calculable as described in Benzoni-Gavage--Serre-Zumbrun.