Rheology of compressible and density-variable Newtonian flows: non-Stokes hypothesis and ‘volume diffusion’

Abstract

Stokes’ hypothesis allows for the frequent neglect of the bulk viscosity term related to fluid dilation effects on the viscous stress tensor in Newtonian flows. Configurations in which this hypothesis is not valid widely exist, but they are not always well distinguished. Meanwhile, it was pointed out earlier that the original Navier–Stokes equations may be incomplete, leading to the development of volume diffusion hydrodynamics. This article recalls the form of the Navier–Stokes equations when fluid dilation (or actual fluid density variation) is properly accounted for in the derivation process of continuum flow equations. A thermodynamic framework to construct constitutive equations is proposed. The results, which are deemed to be the exact and complete form of the Navier–Stokes equations, correspond to the volume diffusion continuum hydrodynamic model. Explicit expressions and meanings are derived for the local volume production rate, specifically the local fluid concentration production, which differ from the conventional definitions. This complete form of the Navier–Stokes equations represents natural Burnett regime hydrodynamic equations, as they satisfy all fundamental mechanical principles and exhibit non-negative entropy production. These continuum flow models may be more appropriate to adopt when local gradients in thermodynamic variables, such as density or temperature, impact the dynamics of the flows.

Keywords

• volume diffusion
• density variable flows
• compressible/incompressible flows
• volume velocity
• Navier–Stokes equations
• Stokes’ hypothesis