I know how to derive Navier-Stokes equations from Boltzmann equation in case where bulk and viscosity coefficients are set to zero. I need only multiply it on momentum and to integrate it over velocities.

But when I've tried to derive NS equations with viscosity and bulk coefficients, I've failed. Most textbooks contains following words: "for taking into the account interchange of particles between fluid layers we need to modify momentum flux density tensor". So they state that NS equations with viscosity cannot be derived from Boltzmann equation, can they?

The target equation is

$$

\partial_{t}\left( \frac{\rho v^{2}}{2} + \rho \epsilon \right) = -\partial_{x_{i}}\left(\rho v_{i}\left(\frac{v^{2}}{2} + w\right) – \sigma_{ij}v_{j} – \kappa \partial_{x_{i}}T \right),

$$

where

$$

\sigma_{ij} = \eta \left( \partial_{x_{[i}}v_{j]} – \frac{2}{3}\delta_{ij}\partial_{x_{i}}v_{i}\right) + \varepsilon \delta_{ij}\partial_{x_{i}}v_{i},

$$

$w = \mu – Ts$ corresponds to heat function, $\epsilon$ refers to internal energy.

Edit. It seems that I've got this equation. After multiplying Boltzmann equation on $\frac{m(\mathbf v – \mathbf u)^{2}}{2}$ and integrating it over $v$ I've got transport equation which contains objects

$$

\Pi_{ij} = \rho\langle (v – u)_{i}(v – u)_{j} \rangle, \quad q_{i} = \rho \langle (\mathbf v – \mathbf u)^{2}(v – u)_{i}\rangle

$$

To calculate it I need to know an expression for distribution function. For simplicity I've used tau approximation; in the end I've got expression $f = f_{0} + g$. An expressions for $\Pi_{ij}, q_{i}$ then are represented by

$$

\Pi_{ij} = \delta_{ij}P – \mu \left(\partial_{[i}u_{j]} – \frac{2}{3}\delta_{ij}\partial_{i}u_{i}\right) – \epsilon \delta_{ij}\partial_{i}u_{i},

$$

$$

q_{i} = -\kappa \partial_{i} T,

$$

so I've got the wanted result.

## Best Answer

I think you were right. The viscous term in the NS equations cannot be derived from the Boltzmann equations. If you derive the conservation laws from the Boltzmann equations using first order approximation, you will get an force term, which should include the pressure, viscous forces and external forces shown in the NS equations.

I think the approximation of the viscous term in the NS equations (viscous stress related to velocity gradient) were constructed from a continuum perspective, with the the tensor form satisfying certain symmetric properties of a stress tensor. See for example "An Introduction to Fluid Dynamics" by G. K. Batchelor for a nice discussion.

However, what I have seen is the derivation of the viscosity by assuming a velocity profile from the linearized Boltzmann equation. It is a question from the textbook "Statistical Physics of Particles" by Kardar, Ch. 3 questions 9.