Approaches to conservative Smoothed Particle Hydrodynamics with entropy

TL;DR

This paper extends Smoothed Particle Hydrodynamics (SPH) to include entropy and temperature dependence, ensuring conservation laws and symplecticity, and demonstrates its application to various fluid systems including heat conduction.

Contribution

It introduces new approaches to incorporate entropy into SPH, maintaining key physical properties and enabling modeling of nonbarotropic fluids and heat conduction.

Findings

Successfully models systems with entropy and temperature dependence.

Demonstrates SPH with entropy on systems with discontinuities.

Extends SPH to include heat flux for hyperbolic heat conduction.

Abstract

Smoothed particle hydrodynamics (SPH) is typically used for barotropic fluids, where the pressure depends only on the local mass density. Here, we show how to incorporate the entropy into the SPH, so that the pressure can also depend on the temperature, while keeping the growth of the total entropy, conservation of the total energy, and symplecticity of the reversible part of the SPH equations. The SPH system of ordinary differential equations with entropy is derived by means of the Poisson reduction and the Lagrange-Euler transformation. We present several approaches towards SPH with entropy, which are then illustrated on systems with discontinuities, on adiabatic and nonadiabatic expansion, and on the Rayleigh-Beenard convection without the Boussinesq approximation. Finally, we show how to model hyperbolic heat conduction within the SPH, extending the SPH variables with not only entropy but also a heat-flux-related vector field.