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

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

See also

• Fluid dynamics
• Lars Onsager
• Non-dimensionalization and scaling of the Navier–Stokes equations
• Stokes equations
• Chapman–Enskog theory
• Navier-Stokes equations

Further reading

• García-Colín, L.S.; Velasco, R.M.; Uribe, F.J. (August 2008). "Beyond the Navier–Stokes equations: Burnett hydrodynamics". Physics Reports. 465 (4): 149–189. Bibcode:2008PhR...465..149G. doi:10.1016/j.physrep.2008.04.010.