Modeling nonequilibrium gas flows within a continuum framework remains a significant challenge, particularly in specifying accurate boundary conditions for macroscopic equations. In this work, we revisit a largely overlooked approach to deriving slip boundary conditions, originally proposed by Patterson and later extended by Shidlovsky–known as the half-flux method. While the original half-flux method provides physical insight and simplicity, it neglects intermolecular collisions within the Knudsen layer, thereby limiting its accuracy. To address this limitation, we propose a high-order half-flux method that incorporates the effects of collisions within the Knudsen layer. Specifically, we distinguish between the velocity slip and temperature jump at the outer edge of the Knudsen layer and those near the wall. The proposed method not only enforces the conservation of momentum and energy flux across the Knudsen layer but also accounts for the approximate conservation of stress and heat flux. We derive slip boundary conditions based on both the Maxwell and Cercignani–Lampis–Lord scattering models and apply them to a range of rarefied flows within the Navier–Stokes–Fourier framework. The resulting numerical predictions show improved agreement with molecular-level simulations, demonstrating the method’s enhanced accuracy and effectiveness in slip boundary modeling.
Luan et al. (Sun,) studied this question.