A discretized integral hydrodynamics

TL;DR

This paper develops a discretized integral formulation of hydrodynamics that unifies various fluid models, including Navier-Stokes and thermal conduction, through a discretization approach related to particle-based methods.

Contribution

It introduces an integral form of conservation equations using an interpolant, connecting continuum hydrodynamics with particle-based methods like SPH and DPD.

Findings

Derivation of integral conservation laws from interpolant forms.

Connection between discretized equations and particle dynamics methods.

Framework applicable to viscous flow and thermal conduction with fluctuations.

Abstract

Using an interpolant form for the gradient of a function of position, we write an integral version of the conservation equations for a fluid. In the appropriate limit, these become the usual conservation laws of mass, momentum and energy. We also discuss the special cases of the Navier-Stokes equations for viscous flow and the Fourier law for thermal conduction in the presence of hydrodynamic fluctuations. By means of a discretization procedure, we show how these equations can give rise to the so-called "particle dynamics" of Smoothed Particle Hydrodynamics and Dissipative Particle Dynamics.