Bulk viscosity in jump conditions for shock waves

Question

One of the jump conditions for shock waves can be derived from the Navier-Stokes equation and it takes the form (see Shu, Gas dynamics, p.212)

$\dfrac{d}{dx}\left(\rho u^2 + P - \frac{4}{3}\mu \frac{du}{dx}\right)=0$,

i.e., the bulk viscosity is neglected.

Why can it be neglected here?

I thought this type of viscosity might be quite important for shock waves because gas is compressed as it passes the shock.

Thanks in advance!

Answer 1

Other names

• volume viscosity
• second viscosity coefficient
• expansion coefficient of viscosity
• coefficient of bulk viscosity

Background

For a compressible, Newtonian liquid, the Navier-Stokes equation is given by: $$ \rho \left[ \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right] = - \nabla \cdot \mathbb{P} + \mu \nabla^{2} \mathbf{v} + \left( \lambda + \frac{4}{3} \mu \right) \nabla \left( \nabla \cdot \mathbf{v} \right) $$ where $\mathbb{P}$ is the pressure tensor, $\mu$ is the dynamic(shear) viscosity coefficient, and $\lambda$ is the coefficient of bulk viscosity.

Answers

the bulk viscosity is neglected. Why can it be neglected here?

We generally do not include bulk viscosity in the Rankine-Hugoniot relations because of what it is. Unlike dynamic and kinematic viscosity, which reflect dissipation of translational flow, bulk viscosity acts as a damping/dissipation term for rotational and vibrational degrees of freedom. Thus, in a monatomic gas (e.g., most plasmas, which is the state of matter found in space) the bulk viscosity is zero.

I thought this type of viscosity might be quite important for shock waves because gas is compressed as it passes the shock.

It seems that in some circumstances, specifically those investigating rotational and/or vibrational degrees of freedom, bulk viscosity is an important parameter [e.g., Dukhin and Goetz, 2009]. Much of shock theory starts with simple, monatomic gases for which $\lambda \rightarrow 0$, thus it is not included. It is true that the Earth's atmosphere is not made of monatomic molecules, so one should include $\lambda$ in the formulation. However, it may be possible to "absorb" it numerically into $\mu$ and get reasonable/reliable results.

In relativistic shocks, I think this parameter would be more important especially for fermions but we are already computationally strained with just MHD simulations. Including particle spin is not even done in most (if not all) PIC simulations.

References

• Dukhin, A.S. and P.J. Goetz "Bulk viscosity and compressibility measurement using acoustic spectroscopy," J. Chem. Phys. 130, pp. 124519, doi:10.1063/1.3095471, 2009.