Shock Waves in Dense Hard Disk Fluids

TL;DR

This study analyzes shock wave behavior in dense hard disk fluids using both analytical hydrodynamics and molecular dynamics simulations, revealing how density affects compressibility and shock propagation.

Contribution

It provides the first closed-form analytical solutions for shock dynamics in dense hard disk media and validates them with large-scale molecular dynamics simulations.

Findings

Analytical expressions accurately predict shock behavior across phases.

Energy transfer behind shocks is nearly independent of packing fraction.

Good agreement between theory and simulation over a wide density range.

Abstract

Media composed of colliding hard disks (2D) or hard spheres (3D) serve as good approximations for the collective hydrodynamic description of gases, liquids and granular media. In the present study, the compressible hydrodynamics and shock dynamics are studied for a two-dimensional hard-disk medium at both the continuum and discrete particle level descriptions. For the continuum description, closed form analytical expressions for the inviscid hydrodynamic description, shock Hugoniot, isentropic exponent and shock jump conditions were obtained using the Helfand equation of state. The closed-form analytical solutions permitted us to gain physical insight on the role of the material's density on its compressibility, i.e. how the medium compresses under mechanical loadings and sustains wave motion. Furthermore, the predictions were found in excellent agreement with calculations using the Event Driven Molecular Dynamic method involving 30,000 particles over the entire range of compressibility spanning the dilute ideal gas and liquid phases. In all cases, it was found that the energy imparted by the piston motion to the thermalized medium behind the propagating shock was quasi-independent of the medium's packing fraction, with a correction vanishing with increasing shock Mach numbers.