J. Perot comment addresses nicely your question. There is also a good discussion of this topic in the classical textbook by Landau and Lifshitz: Fluid Mechanics, and they also give an example of its importance in acoustics.

The convention in fluid mechanics is to write the stress tensor to be composed of an isotropic part (-p*delta_{ii} ) and a deviatoric stress tau=lambda div(u)+mu(grad(u)+ grad(u)^T). In order for the deviatoric stress not to contribute to the the mean normal force on an area element, it is customary to invoke the Stokes hypothesis: lambda=-2/3 mu.

As noted by J. Perot this is a good approximation for monotonic gases, but certainly not air. So to account for bulk viscosity effects, it is customary to write the deviatoric stress tensor as tau=mu(grad(u)+grad(u)^T-2/3*div(u))+mu_b div(u), where now mu_b is called the bulk viscosity( or second coefficient of viscosity). Obviously for an incompressible fluid ( div(u)=0 ) we get the usually form for the deviatoric stress: tau=mu(grad(u)+ grad(u)^T), and mu is the shear viscosity. If mu_b is taken to be zero, and div(u) is not zero ( a compressible fluid), then the implicit assumption is that the Stokes hypothesis has been made: i.e. the deviatoric stress does not contribute to the mean normal stress.

There has been a lot of interest in recent years to measure/predict the bulk viscosity of water and other common liquids! See for example the paper by Durkhin and Goetz, "Bulk viscosity and compressibility measurement using acoustic spectroscopy", J. Chem Phys. 130, 124519(2009). For example, they find that ethanol at 25C has a shear viscosity of 1.074 cP and a bulk viscosity of 1.4 cP at 100 MHz. Typically the bulk viscosity is higher than the shear viscosity.

In short, the topic of bulk viscosity ( second viscosity) has once again become a fascinating topic, with long hooks back to the early works of Stokes(1845) and Saint Vernant (1843). Finally, readers interested in this topic should take the time to read the ingenious thought experiment (with analysis) suggested by GI Taylor for measuring the bulk viscosity of a liquid: GI Taylor,"The Two Coefficients of Viscosity for an Incompressible Fluid Containing Air Bubbles", Proc. Roy Soc. Lond. A, vol. 226, p34-37, 1954

Please see e.g.

Cook et al., JCP, 203, 2, 379385 (2004) for bulk viscosity models.

We used this approach recently: http://dx.doi.org/10.1063/1.4772192

Physically speaking the assumption of neglecting bulk viscosity seems to depend

on the gas in question. Please see:

1: Emanuel G., Bulk viscosity of a dilute polyatomic gas (Phys.Fluids)

2: Kustova E.G., On the role of bulk viscosity and relaxation pressure in nonequilibrium flows (AIP conference proceedings).

It would be interesting in case you look deeper into the topic.

Based on the mentioned papers above, for some gases accounting for bulk viscosity is highly relevant.

It is usually NOT neglected.

The value for a monotomic gas is provably: lambda = -mu*2/3

(from gas kinetic theory).

Which is what most models use (implicitly). If you use

sigma = mu[ grad(u) + grad(u)^T ] - 2/3*mu*[ divg(u) ]

then you just used the second coefficient as the monotomic gas value.

This value also minimizes the entropy produced by viscosity.

It is the minimum allowable. Anything else must be bigger (and can be positive).

It matters for acoustics damping at long distances.

It matters for the internal structure of shocks (if you are trying to get many

points inside a shock).

In air, it turns out that it is probably NOT very constant.

Which means the Newtonian viscosity assumption is not great in that limit.

So not neglected (not 0). But also not much effort is put into getting it right.

J. Perot comment addresses nicely your question. There is also a good discussion of this topic in the classical textbook by Landau and Lifshitz: Fluid Mechanics, and they also give an example of its importance in acoustics.

The convention in fluid mechanics is to write the stress tensor to be composed of an isotropic part (-p*delta_{ii} ) and a deviatoric stress tau=lambda div(u)+mu(grad(u)+ grad(u)^T). In order for the deviatoric stress not to contribute to the the mean normal force on an area element, it is customary to invoke the Stokes hypothesis: lambda=-2/3 mu.

As noted by J. Perot this is a good approximation for monotonic gases, but certainly not air. So to account for bulk viscosity effects, it is customary to write the deviatoric stress tensor as tau=mu(grad(u)+grad(u)^T-2/3*div(u))+mu_b div(u), where now mu_b is called the bulk viscosity( or second coefficient of viscosity). Obviously for an incompressible fluid ( div(u)=0 ) we get the usually form for the deviatoric stress: tau=mu(grad(u)+ grad(u)^T), and mu is the shear viscosity. If mu_b is taken to be zero, and div(u) is not zero ( a compressible fluid), then the implicit assumption is that the Stokes hypothesis has been made: i.e. the deviatoric stress does not contribute to the mean normal stress.

There has been a lot of interest in recent years to measure/predict the bulk viscosity of water and other common liquids! See for example the paper by Durkhin and Goetz, "Bulk viscosity and compressibility measurement using acoustic spectroscopy", J. Chem Phys. 130, 124519(2009). For example, they find that ethanol at 25C has a shear viscosity of 1.074 cP and a bulk viscosity of 1.4 cP at 100 MHz. Typically the bulk viscosity is higher than the shear viscosity.

In short, the topic of bulk viscosity ( second viscosity) has once again become a fascinating topic, with long hooks back to the early works of Stokes(1845) and Saint Vernant (1843). Finally, readers interested in this topic should take the time to read the ingenious thought experiment (with analysis) suggested by GI Taylor for measuring the bulk viscosity of a liquid: GI Taylor,"The Two Coefficients of Viscosity for an Incompressible Fluid Containing Air Bubbles", Proc. Roy Soc. Lond. A, vol. 226, p34-37, 1954

Thank you for all your answers and references helping to understand the second viscosity effect in compressible flows. As mentioned above, for monotonic gases the Stokes hypothesis is applicable and the shear stress do not contribute to the normal force. In the other case when mu_b is taken to be different to zero for non-monotonic gases, how can we explain the contribution of a tangential stress into the normal force. In other words, why does Stokes hypothesis work only for monotonic gases?

Viscosity is due to exchange of momentum and energy between gas particles in collisions. Monatomic particles only carry the kinetic energy of their thermal motion, but polyatomic gas molecules also have internal energy contributions from rotation and vibration. As particles collide, and exchange energy, there are two types of collisions: those where only kinetic energy is exchanged, and those where kinetic and internal energy (of the molecules!) is exchanged. Derivation of the Navier-Stokes law from the Boltzmann equation shows that the bulk viscosity is due only to processes where all energies are exchanged, while shear viscosity is due to all collisions. Viscosities are proportional to the average time between collisions. Since only some collisions contribute to bulk viscosity, but all collisions contribute to shear viscosity, the latter is smaller.

Book recommendation: Gilberto Kremer: An introduction to the Boltzmann equation and transport processes in gases (Springer 2010)

In a gas the bulk viscosity is coming mainly from rotation/vibration exchanges with translation of molecules in a rapid variation of pressure. This effect is negligible in most cases. Note that the continuous approximation description (Navier-Stokes equations) of gases in not valid for shocks at Mach>2 in which calculations give an appreciable effect of bulk viscosity