Burnett equations

In continuum mechanics, a branch of mathematics, the Burnett equations is a set of higher-order continuum equations for non-equilibrium flows and the transition regimes where the Navier–Stokes equations do not perform well.

They were derived by the English mathematician D. Burnett.

Series expansion
$$ f(r,c,t) = f^0(c| n,u,T ) 1 + Kn \phi^1 (c|n,u,T) + Kn^2 \phi^2(c|n,u,T) + \cdots$$

Extensions
The Onsager-Burnett Equations, commonly referred to as OBurnett, which form a superset of the Navier-Stokes equations and are second-order accurate for Knudsen number.

Eq. (1) $$ \sqrt{\tau} \frac{du^s}{ds} - \frac{9}{8} \alpha_1 u^* (\frac{du^*}{ds})^2 = \frac{\tau}{u^*} - \tau_0 -1 + u^* $$

Eq. (2) $$ \frac{45}{16} \sqrt{\tau} \frac{d \tau}{ds} + \frac{9}{4} \gamma_1 \tau (\frac{du^*}{ds})^2 - \frac{9}{4} \Psi u^* \frac{d \tau}{ds} \frac{du^*}{ds} = \frac{3}{2} (\tau - \tau_0) - \frac{1}{2} (1-u^*)^2 - \tau_0(1-u^*) $$

Derivation
Starting with the Boltzmann equation $$ \frac{\partial{f}}{\partial{t}} + c_k \partial{f}{x_k} + F_k \partial{f}{c_k} = J(f, f_1) $$