I am looking for information on the coefficient $\lambda$ in the following formulation of the barotropic compressible Navier-Stokes system:
\begin{align}
& \partial_t \rho + \text{div}(\rho u) = 0,\\
& \rho \frac{D}{Dt} u = – \nabla p + \mu \Delta u + \lambda \nabla (\text{div} u),
\end{align}
where $p := P(\rho)$ for some $P$ sufficiently smooth, $\rho, u$ are the density, velocity respectively, and the coefficients $\mu, \lambda$ satisy:
\begin{align}
\mu \geq 0, \ 2\mu + 3\lambda \geq 0.
\end{align}
This particular formulation is taken from P.G. Lemarié-Rieusset's 'The Navier-Stokes Problem in the 21st Century,' and I understand that it is somewhat simplified compared to more physics-oriented formulations like, say, Landau and Lifschitz's 'Fluid Mechanics'.
I am particularly interested in whether or not the mathematically useful assumption $\lambda = 0$ (without necessarily also forcing $\mu =0$) has any physical basis behind it. I have so far only found mention of Stokes's hypothesis, which wouldn't eliminate the $\nabla (\text{div} u )$ term entirely. Again, using Lemarié-Rieusset's notation, the Stokes hypothesis entails
\begin{equation}
2\mu + 3\lambda = 0,
\end{equation}
which is not what I'm looking for.
Any keywords or names for the $\lambda = 0, \ \mu \neq 0$ assumption that I'm looking for, or sources with info on this matter would be greatly appreciated. In short, I'd like to know if there's any real-life scenario where this assumption makes sense, or if it's a mere "toy model" for convenience in maths.
Best Answer
See Bird, Stewart, and Lightfoot, Transport Phenomena, Chapter 1. They indicate that, to linear terms in the velocity components, the portion of the stress tensor over and above the isotropic pressure is proportional to the "deviatoric" rate of deformation tensor, plus a "dilatational viscosity" term. For monoatomic gases, kinetic theory of gases indicates that the dilatational viscosity is zero, and Bird et al indicate that for most other gases it is negligible. Under these circumstances $\lambda=-\frac{2}{3}\mu$.