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.^[1][2][3]

They were derived by the English mathematician D. Burnett.^[4]

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The series expansion technique used to derive the Burnett equations involves expanding the distribution function ${\displaystyle f}$ in the Boltzmann equation as a power series in the Knudsen number ${\displaystyle Kn}$:

${\displaystyle f(r,c,t)=f^{(0)}(c|n,u,T)\left[1+K_{n}\phi ^{(1)}(c|n,u,T)+K_{n}^{2}\phi ^{(2)}(c|n,u,T)+\cdots \right]}$

Here, ${\displaystyle f^{(0)}(c|n,u,T)}$ represents the Maxwell-Boltzmann equilibrium distribution function, dependent on the number density ${\displaystyle n}$, macroscopic velocity ${\displaystyle u}$, and temperature ${\displaystyle T}$. The terms ${\displaystyle \phi ^{(1)},\phi ^{(2)},}$ etc., are higher-order corrections that account for non-equilibrium effects, with each subsequent term incorporating higher powers of the Knudsen number ${\displaystyle Kn}$.

The first-order term ${\displaystyle f^{(1)}}$ 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 ${\displaystyle \phi ^{(2)}}$. 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:

${\displaystyle \mathbf {u} _{t}+(\mathbf {u} \cdot \nabla )\mathbf {u} +\nabla p=\nabla \cdot (\nu \nabla \mathbf {u} )+{\text{higher-order terms}}}$

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.

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.^[5]

Eq. (1) ${\displaystyle {\sqrt {\tau }}{\frac {du^{s}}{ds}}-{\frac {9}{8}}\alpha _{1}u^{*}({\frac {du^{*}}{ds}})^{2}={\frac {\tau }{u^{*}}}-\tau _{0}-1+u^{*}}$

Eq. (2) ${\displaystyle {\frac {45}{16}}{\sqrt {\tau }}{\frac {d\tau }{ds}}+{\frac {9}{4}}\gamma _{1}\tau ({\frac {du^{*}}{ds}})^{2}-{\frac {9}{4}}\Psi u^{*}{\frac {d\tau }{ds}}{\frac {du^{*}}{ds}}={\frac {3}{2}}(\tau -\tau _{0})-{\frac {1}{2}}(1-u^{*})^{2}-\tau _{0}(1-u^{*})}$^[6]

Starting with the Boltzmann equation ${\displaystyle {\frac {\partial {f}}{\partial {t}}}+c_{k}\partial {f}{x_{k}}+F_{k}\partial {f}{c_{k}}=J(f,f_{1})}$

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

• 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.

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