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

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