Lagrangian dynamics of incompressible thermofluids

TL;DR

This paper establishes an exact Lagrangian dynamics framework for incompressible thermofluids using inverse kinetic theory, linking fluid dynamics with classical dynamical systems through a phase-space approach.

Contribution

It introduces a novel phase-space Lagrangian formulation for incompressible fluids based on inverse kinetic theory, unifying configuration-space and molecular dynamics perspectives.

Findings

Exact Lagrangian dynamics derived for incompressible fluids.

Phase-space representation aligns with physical fluid domain.

Potential applications in CFD and applied fluid dynamics.

Abstract

A key aspect of fluid dynamics is the correct definition of the \textit{% phase-space} Lagrangian dynamics which characterizes arbitrary fluid elements of an incompressible fluid. Apart being an unsolved theoretical problem of fundamental importance, the issue is relevant to exhibit the connection between fluid dynamics and the classical dynamical systems underlying incompressible and non-isothermal fluid, typically founded either on: a) a \textit{configuration-space} Lagrangian description of the dynamics of fluid elements; b) a kinetic description of the molecular dynamics, based on a discrete representation of the fluid. The goal of this paper is to show that the exact Lagrangian dynamics can be established based on the inverse kinetic theory (IKT) for incompressible fluids recently pointed out (Ellero \textit{et al.}, 2004-2006, \cite{Ellero2004}). The result is reached by adopting an IKT approach based on a \textit{restricted phase-space representation} of the fluid, in which the configuration space coincides with the physical fluid domain. The result appears of potential importance in applied fluid dynamics and CFD.