Inverse kinetic theory for incompressible thermofluids

TL;DR

This paper demonstrates that inverse kinetic theories can be extended to thermofluids, allowing the derivation of thermodynamic fluid equations within a phase-space framework, thus satisfying the second law of thermodynamics.

Contribution

It introduces an extended phase-space formulation for inverse kinetic theory applicable to non-isentropic incompressible thermofluids, expanding previous theories to include thermodynamic principles.

Findings

Successful formulation of an extended phase-space IKT for thermofluids.

Validation against Fourier and Navier-Stokes equations with Boussinesq approximation.

Ensures compatibility with the second law of thermodynamics.

Abstract

An interesting issue in fluid dynamics is represented by the possible existence of inverse kinetic theories (IKT) which are able to deliver, in a suitable sense, the complete set of fluid equations which are associated to a prescribed fluid. From the mathematical viewpoint this involves the formal description of a fluid by means of a classical dynamical system which advances in time the relevant fluid fields. The possibility of defining an IKT for the 3D incompressible Navier-Stokes equations (INSE), recently investigated (Ellero \textit{et al}, 2004-2007) raises the interesting question whether the theory can be applied also to thermofluids, in such a way to satisfy also the second principle of thermodynamics. The goal of this paper is to prove that such a generalization is actually possible, by means of a suitable \textit{extended phase-space formulation}. We consider, as a reference test, the case of non-isentropic incompressible thermofluids, whose dynamics is described by the Fourier and the incompressible Navier-Stokes equations, the latter subject to the conditions of validity of the Boussinesq approximation.