The effect of the Enskog collision terms on the steady shock structure in a hard sphere gas

TL;DR

This paper investigates how Enskog collision terms influence the structure of steady shocks in a hard sphere gas, revealing modifications in shock thickness, fixed point differences, and sound speed compared to classical models.

Contribution

It provides a detailed analysis of the impact of Enskog corrections on shock transitions, including the derivation of simplified equations and the quantification of changes in shock properties.

Findings

Enskog correction reduces the difference between pre- and post-shock states.

Shock thickness increases due to Enskog effects, especially at low Mach numbers.

Enskog-Navier-Stokes equations predict a faster sound speed than classical models.

Abstract

In this work we study the effect of the Enskog collision terms on the steady shock transitions in the supersonic flow of a hard sphere gas. We start by examining one-dimensional, nonlinear, nondispersive planar wave solutions of the Enskog-Navier-Stokes equations, which move in a fixed direction at a constant speed. By further equating the speed of the reference frame with the speed of such a wave, we reduce the Enskog-Navier-Stokes equations into a more simple system of two ordinary differential equations, whose solutions depend on a single scalar spatial variable. We then observe that this system has two fixed points, which are taken to be the states of the gas before and after the shock, and compute the corresponding shock transition in the form of the heteroclinic orbit connecting these two states. We find that the Enskog correction affects both the difference between the fixed points, and the thickness of the transition. In particular, for a given state of the gas before the shock transition, the difference between the fixed points is reduced, while the shock thickness is increased, with the relative impact on the properties of transition being more prominent at low Mach numbers. We also compute the speed of sound in the Enskog-Navier-Stokes equations, and find that, for the same thermodynamic state, it is somewhat faster than that in the conventional Navier-Stokes equations, with an additional dependence on the density of the gas.