Lagrangian for Navier-Stokes equations of motion: SDPD approach

TL;DR

This paper investigates whether the SDPD equations of motion can be derived from a Lagrangian framework, finding that it is possible only in the macroscopic limit where particle number tends to infinity.

Contribution

It provides a detailed analysis of the Helmholtz conditions for SDPD, showing the classical Lagrangian derivation is only valid in the continuum limit.

Findings

Classical Lagrangian derivation fails for finite particles.

Lagrangian formulation is valid in the macroscopic limit.

Supports deriving Navier-Stokes equations from SDPD in the continuum limit.

Abstract

The conditions necessary and sufficient for the Smoothed Dissipative Particle Dynamics (SDPD) equations of motion to have a Lagrangian that can be used for deriving these equations of motion, the Helmholtz conditions, are obtained and analysed. They show that for a finite number of SDPD particles the conditions are not satisfied; hence, the SDPD equations of motion can not be obtained using the classical Euler-Lagrange equation approach. However, when the macroscopic limit is considered, that is when the number of particles tends to infinity, the conditions are satisfied, thus providing the conceptual possibility of obtaining the Navier-Stokes equations from the principle of least action.