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.^[1][2][3]

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

Series expansion

Series expansion approach

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 ${\displaystyle Kn}$.

Derivation

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.

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.^[5][6]

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

1

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

2

Derivation

Starting with the Boltzmann equation

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

See also

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