  Howard Brenner has recently proposed modifications to the Navier-Stokes
equations that relate to a diffusion of fluid volume that would be significant
for flows with high density gradients. In a previous paper (Greenshields &
Reese, 2007), we found these modifications gave good predictions of the viscous
structure of shock waves in argon in the range Mach 1.0-12.0 (while
conventional Navier-Stokes equations are known to fail above about Mach 2).
However, some areas of concern with this model were a somewhat arbitrary choice
of modelling coefficient, and potentially unphysical and unstable solutions. In
this paper, we therefore present slightly different modifications to include
molecule mass diffusion fully in the Navier-Stokes equations. These
modifications are shown to be stable and produce physical solutions to the
shock problem of a quality broadly similar to those from the family of extended
hydrodynamic models that includes the Burnett equations. The modifications
primarily add a diffusion term to the mass conservation equation, so are at
least as simple to solve as the Navier-Stokes equations; there are none of the
numerical implementation problems of conventional extended hydrodynamics
models, particularly in respect of boundary conditions. We recommend further
investigation and testing on a number of different benchmark non-equilibrium
flow cases.
