Nonlinear Constitutive equations derived for fluids obeying an ideal gas, a Tait-Kirkwood or a B/A type equation of state

Abstract

A generalized theoretical framework for acoustic, electromagnetic and elastodynamic waves would give fruitful insights into equivalent phenomena in these physical domains and would form a basis to draw up general analytical or numerical solution methods. In this contribution we adopt the structure of Maxwell's equations for electromagnetic fields, encompassing the formulation of two first-order field equations and two first-order constitutive equations, and we apply it to the area of nonlinear acoustics. We derive the constitutive equations of a fluid directly from its thermodynamic equation of state (EOS). In the constitutive equations, the nonlinear medium behaviour of the fluid is described by a pressure-dependent density and compressibility. The resulting equations are general, making them valid for phenomena occuring in applications with finite amplitude waves of any magnitude, like waveform distortion or radiation pressure. This paper concerns with obtaining constitutive equations for fluids obeying an ideal gas law, a Tait-Kirkwood EOS or a 2-term Taylor approximation of the EOS employing the B/A nonlinearity parameter. The latter EOS is used in many of the classical model equations of nonlinear acoustics. We show that all three types result in simple expressions for the density and compressibility.