Reconciliation of Thermodynamic and Mechanical Pressures and Development of a Frequency-Based Formula for Speed of Sound in Gases

Abstract

Abstract Discrepancy of the thermodynamic and mechanical pressures is a problem at the heart of the current theory of fluid mechanics. The fluid property that leads to this situation is the bulk viscosity, whose effects are zero for incompressible cases and are negligible for most other applications. Therefore, this discrepancy is conventionally ignored in phenomena other than acoustics and shock related ones. However, the flaw in the theory still persists since the late 19th century. In the present study, to improve the existing theory and to come up with a consistent structure in terms of mechanical and thermodynamic pressures, a novel fluid element model is proposed. Unlike the current fluid model that assumes a continuum of fluid, the present model proposes fluid elements that are separated from each other by a thin energy field that manifests itself as the mechanical pressure. Also, unlike the current efforts in explaining bulk viscosity effects through atomic level dynamics, the present model proposes a mesoscale analysis where bulk viscosity is integrated into the fluid element as a damper. Considering all these new features, each fluid element in this new model contains energy in both the wave form and the particle form. Of these two, wave energy is the cause of the thermodynamic pressure. In this manuscript, first, justifications of the mentioned aspects of the new fluid model are given. Then, a speed of sound expression is derived based on the new model involving the bulk viscosity effects. Resultant expression is, then, used for comparison with the findings of previous studies. The proposed formula can also be used to calculate the bulk viscosity of gases at different acoustic frequencies in a way that is more direct than those currently in use.