The Lagrangian dynamics of thermal tracer particles in Navier-Stokes fluids

TL;DR

This paper investigates thermal tracer particles within Navier-Stokes fluids, establishing a statistical framework for their dynamics, which aids in understanding turbulence and resolving the closure problem in hydrodynamic turbulence.

Contribution

It introduces a subset of solutions called thermal tracer particles and develops a statistical description, including Langevin and Fokker-Planck dynamics, for both deterministic and stochastic fluids.

Findings

Thermal tracer particles are uniquely determined by NS fluid fields.

A statistical description of NS fluids using TTPs is formulated.

A proposed solution to the turbulence closure problem is presented.

Abstract

A basic issue for Navier-Stokes (NS) fluids is their characterization in terms of the so-called NS phase-space classical dynamical system, which provides a mathematical model for the description of the dynamics of infinitesimal (or i\textit{deal}) tracer particles in these fluids. The goal of this paper is to analyze the properties of a particular subset of solutions of the NS dynamical system, denoted as \textit{thermal tracer particles} (TTPs), whose states are determined uniquely by the NS fluid fields. Applications concerning both deterministic and stochastic NS fluids are pointed out. In particular, in both cases it is shown that in terms of the ensemble of TTPs a statistical description of NS fluids can be formulated. In the case of stochastic fluids this feature permits to uniquely establish the corresponding Langevin and Fokker-Planck dynamics. Finally, the relationship with the customary statistical treatment of hydrodynamic turbulence (HT) is analyzed and a solution to the closure problem for the statistical description of HT is proposed.