Mechanically active stress model for linear compressible nth-gradient viscous fluids

Selected results from the literature on first-, second- and third-gradient models for linear compressible viscous fluids (with constant phenomenological coefficients) are reinterpreted to propose a phenomenological model for nth-gradient fluids. The constitutive equations for the stress tensors are formulated in a reduced form involving only mechanically active second-rank tensors. Under adiabatic conditions, with negligible temperature variations, the continuity, momentum and barotropic equations are derived, together with expressions for the reduced kinetic energy dissipation function and the speed of sound. Within the framework of the proposed model, the role of thermodynamic pressure is examined and a discussion of mechanical and viscous pressures is provided. The issue of boundary conditions is also addressed. The strengths, limitations and critical aspects of the proposed model are highlighted. To illustrate the non-local effects associated with nth-gradient fluids, the one-dimensional potential flow of longitudinal waves is examined.