# [Physics] Could Navier-Stokes equation be derived directly from Boltzmann equation

approximationsfluid dynamicsnavier-stokes;statistical mechanics

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.