Boundary conditions for the upwind finite difference Lattice Boltzmann model: Evidence of slip velocity in micro-channel flow

We conduct a systematic study of the effect of various boundary conditions (bounce back and three versions of diffuse reflection) for the two-dimensional first-order upwind finite difference Lattice Boltzmann model. Simulation of Couette flow in a micro-channel using the diffuse reflection boundary condition reveals the existence of a slip velocity that depends on the Knudsen number ε = λ/L, where λ is the mean free path and L is the channel width. For walls moving in opposite directions with speeds ±uw, the slip velocity satisfies uslip = 2εuwall/(1 + 2ε). In the case of Poiseuille flow in a micro-channel, the slip velocity is found to depend on the lattice spacing δs and Knudsen number ε to both first and second order. The best results are obtained for diffuse reflection boundary conditions that allow thermal mixing at a wall located at half lattice spacing outside the boundary nodes.