Macroscopic Equations

At the continuum level use of the BGK approximation (Eq. 31) will obtain the following Euler equations,

$$\begin{eqnarray*}\frac{d\rho}{dt}=-\rho \nabla \cdot \vec{v} \\ \rho \frac{d\ve... ...{\leftrightarrow} {P^{0}}: \stackrel {\leftrightarrow}{\Lambda} \end{eqnarray*}$$

where $\Lambda_{ij} = \frac{1}{2}(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i})$the internal energy, $e=\frac{3}{2}\frac{K}{m}T +\frac{\phi}{\rho}$ and the pressures tensor $P^{0}_{ij} =nKT\delta_{ij} +B_{ij}$ is modified to include the intermolecular forces. The next order correction to the Euler equations entails solving for $\delta P_1 =P_1-P^{eq}$ to determine corrections to the transport equations from viscous effects and thermal conductivity. At this order the additional terms in the Hermite expansion do not contribute to the viscosity, µ, and the thermal conductivity, K_t, so that the usual expressions $\mu=n \lambda KT$ and $K_t=\frac{5}{2}n\lambda K^2T/m$ are obtained. While this result holds for the continuum case, corrections would appear in the lattice Boltzmann methods due to discretization. Finally, if needed, generalization to different Prandtl number can be obtained using the ellipsoidal Equilibrium distribution or multiple relaxation times [25].