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 \mathrm{Kn}: 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, 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)},\dots are higher-order corrections that account for
non-equilibrium effects, with each subsequent term incorporating higher powers of the Knudsen number.
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==