Burnett equations

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${\displaystyle \mathrm {Kn} }$:

$${\displaystyle f(r,c,t)=f^{(0)}(c|n,u,T)\left[1+\mathrm {Kn} \phi ^{(1)}(c|n,u,T)+\mathrm {Kn} ^{2}\phi ^{(2)}(c|n,u,T)+\cdots \right]}$$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 ${\displaystyle \phi ^{(1)},\phi ^{(2)},\dots }$ are higher-order corrections that account for non-equilibrium effects, with each subsequent term incorporating higher powers of the Knudsen number.

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.