Past exam of the kinetic theory course of the University of Oxford / 2025 / 1 / d / Solution

Write $c_i=v_i-u_i$ and introduce the raw third velocity moment $M_{ijk}$ and the third central velocity moment $Q_{ijk}$:The expansion of their product givesThe heat flux is $q_i=Q_{ijj}/2$ for unit mass.

Multiply the BGK model by $v_i{v}_j$. The external force term isTherefore the raw second velocity moment satisfiesThe matching local Maxwellian has ${\Pi }_{ij}^{(0)}=\rho u_i{u}_j+\rho \Theta {\delta }_{ij}$, soTo remove the bulk motion, let $D_t={\partial }_t+u_k{\partial }_k$ be the material derivative. The continuity equation and the momentum conservation equation implyand henceSubtract this identity from the raw second velocity moment equation and insert the expansion of $M_{ijk}$. The terms $u_i{\partial }_k{P}_{jk}$ and $u_j{\partial }_k{P}_{ik}$ cancel by the product rule. The external force terms also cancel, since a common acceleration changes the local mean velocity without directly changing the relative velocities. What remains isThe trace of the relaxation term is zero because $P_{ii}=3\rho \Theta$. Thus the BGK model relaxes anisotropy of the pressure tensor while preserving the local random kinetic energy. The evolution is still a velocity-moment hierarchy, since $Q_{ijk}$ requires further information about $f$.