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 $Kn$:

$f(r,c,t)=f^{(0)}(c|n,u,T)[1+K_n{\varphi }^{(1)}(c|n,u,T)+K_n^2{\varphi }^{(2)}(c|n,u,T)+\cdots ]$

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 ${\varphi }^{(1)},{\varphi }^{(2)},$ 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 ${\varphi }^{(2)}$. 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:

${𝐮}_t+(𝐮\cdot \mathrm{\nabla })𝐮+\mathrm{\nabla }p=\mathrm{\nabla }\cdot (\nu \mathrm{\nabla }𝐮)+\text{higher-order terms}$

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) $\sqrt{\tau }\frac{d{u}^s}{ds}-\frac{9}{8}{\alpha }_1{u}^{*}(\frac{d{u}^{*}}{ds}{)}^2=\frac{\tau }{u^{*}}-{\tau }_0-1+u^{*}$

Eq. (2) $\frac{45}{16}\sqrt{\tau }\frac{d\tau }{ds}+\frac{9}{4}{\gamma }_1\tau (\frac{d{u}^{*}}{ds}{)}^2-\frac{9}{4}\mathrm{\Psi }{u}^{*}\frac{d\tau }{ds}\frac{d{u}^{*}}{ds}=\frac{3}{2}(\tau -{\tau }_0)-\frac{1}{2}(1-u^{*}{)}^2-{\tau }_0(1-u^{*})$

Derivation

Template:Expand section 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

• Template:Cite journal

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