# Burnett equations

In continuum mechanics, a branch of mathematics, the Burnett equations are 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.
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## Series expansion
### Series expansion approach

The series expansion technique used to derive the Burnett equations involves expanding the distribution function $f$ in the Boltzmann equation as a power series in the Knudsen number $Kn$:
$f(r,c,t)=f^{(0)}(c|n,u,T)\left[1+K_{n}\phi ^{(1)}(c|n,u,T)+K_{n}^{2}\phi ^{(2)}(c|n,u,T)+\cdots \right]$
Here, $f^{(0)}(c|n,u,T)$ represents the Maxwell-Boltzmann equilibrium distribution function, dependent on the number density $n$, macroscopic velocity $u$, and temperature $T$. The terms $\phi ^{(1)},\phi ^{(2)},$ etc., are higher-order corrections that account for non-equilibrium effects, with each subsequent term incorporating higher powers of the Knudsen number $Kn$.

### Derivation

The first-order term $f^{(1)}$ in the expansion gives the Navier-Stokes equations, which include terms for viscosity and thermal conductivity. To obtain the Burnett equations, one must retain terms up to second order, corresponding to $\phi ^{(2)}$. The Burnett equations include additional second-order derivatives of velocity, temperature, and density, representing more subtle effects of non-equilibrium gas dynamics.
The Burnett equations can be expressed as:
$\mathbf {u} _{t}+(\mathbf {u} \cdot \nabla )\mathbf {u} +\nabla p=\nabla \cdot (\nu \nabla \mathbf {u} )+{\text{higher-order terms}}$
Here, the "higher-order terms" involve second-order gradients of velocity and temperature, which are absent in the Navier-Stokes equations. These terms become significant in situations with high Knudsen numbers, where the assumptions of the Navier-Stokes framework break down.

## 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 ${\frac {\partial {f}}{\partial {t}}}+c_{k}\partial {f}{x_{k}}+F_{k}\partial {f}{c_{k}}=J(f,f_{1})$