  Brenner has recently proposed modifications to the Navier-Stokes equations
that are based on theoretical arguments but supported only by experiments
having a fairly limited range. These modifications relate to a diffusion of
fluid volume that would be significant for flows with high density gradients.
So the viscous structure of shock waves in gases should provide an excellent
test case for this new model. In this paper we detail the shock structure
problem and propose exponents for the gas viscosity-temperature relation based
on empirical viscosity data that is independent of shock experiments. We then
simulate shocks in the range Mach 1.0-12.0 using the Navier-Stokes equations,
both with and without Brenner's modifications. Initial simulations showed
Brenner's modifications display unphysical behaviour when the coefficient of
volume diffusion exceeds the kinematic viscosity. Our subsequent analyses
attribute this behaviour to both an instability to temporal disturbances and a
spurious phase velocity-frequency relationship. On equating the volume
diffusivity to the kinematic viscosity, however, we find the results with
Brenner's modifications are significantly better than those of the standard
Navier-Stokes equations, and broadly similar to those from the family of
extended hydrodynamic models that includes the Burnett equations. Brenner's
modifications add only two terms to the Navier-Stokes equations, and the
numerical implementation is much simpler than conventional extended
hydrodynamic models, particularly in respect of boundary conditions. We
recommend further investigation and testing on a number of different benchmark
non-equilibrium flow cases.
