On the stability of shocks with particle pressure

TL;DR

This paper analyzes the linear stability of shocks with particle pressure, revealing new particle-dominated modes, constructing a global oscillation mode, and demonstrating stability through numerical solutions.

Contribution

It introduces a comprehensive stability analysis including particle pressure effects and diffusion, identifying new perturbation modes and constructing a global shock oscillation mode.

Findings

Identified two new particle-pressure dominated modes.

Constructed a global corrugational eigenmode for the shock.

Numerical solutions show the shock is stable under these conditions.

Abstract

We perform a linear stability analysis for corrugations of a Newtonian shock, with particle pressure included, for an arbitrary diffusion coefficient. We study first the dispersion relation for homogeneous media, showing that, besides the conventional pressure waves and entropy/vorticity disturbances, two new perturbation modes exist, dominated by the particles' pressure and damped by diffusion. We show that, due to particle diffusion into the upstream region, the fluid will be perturbed also upstream: we treat these perturbation in the short wavelength (WKBJ) regime. We then show how to construct a corrugational mode for the shock itself, one, that is, where the shock executes free oscillations (possibly damped or growing) and sheds perturbations away from itself: this global mode requires the new modes. Then, using the perturbed Rankine-Hugoniot conditions, we show that this leads to the determination of the corrugational eigenfrequency. We solve numerically the equations for the eigenfrequency in the WKBJ regime for the models of Amato and Blasi (2005), showing that they are stable. We then discuss the differences between our treatment and previous work.