Emergence of Thermoacoustic Waves: A Variational Approach Consistent with Thermodynamics and the Navier-Stokes Problem

TL;DR

This paper develops a variational framework rooted in non-equilibrium thermodynamics to derive thermoacoustic wave equations, addressing dissipative systems and linking thermodynamics with fluid mechanics and the Navier-Stokes problem.

Contribution

It introduces a novel variational approach that incorporates entropy production and nonholonomic constraints, providing a new theoretical foundation for thermoacoustic waves and their relation to Navier-Stokes equations.

Findings

Derivation of wave equations from thermodynamic functionals.

Integration of nonholonomic constraints via vakonomic mechanics.

Potential insights into Navier-Stokes existence and smoothness problem.

Abstract

This article proposes an in-depth investigation into the emergence of thermoacoustic waves from a variational formalism rooted in non-equilibrium thermodynamics. Differing from traditional approaches based on linear simplifications, this work explores the possibility of deriving wave equations for pressure and density through the extremization of thermodynamic functionals, with special attention to entropy production. We address the inherent complexities in applying variational principles to dissipative systems, incorporating the nuances of Prigogine\'s minimum and maximum entropy production principles. Furthermore, we discuss the integration of nonholonomic constraints, such as the Navier-Stokes equations, through concepts of vakonomic mechanics, and explore how thermoacoustic solutions can offer ansatzes valuable for the existence and smoothness problem of Navier-Stokes solutions, one of the Clay Math Institute\'s Millennium Problems. The objective is to provide a robust theoretical foundation that can shed new light on the interconnection between thermodynamics, fluid mechanics, and wave phenomena