First-order moment equations of hybrid second-order moment model are obtained by Euler method, while second-order moment equations are deduced by Lagrange equations. Equations for particle fraction and momentum are provided firstly. A Lagrange model with mean Langevin equations is obtained and Reynold stress equation is deduced, so that hybrid second-order moment model is closed without additional approximate assumptions. Wall-jet-flow loaded with solid particles is simulated. It shows that the model is effective.

Two-dimensional and three-dimensional simulations are performed to investigate supersonic cold gas flows in micro-nozzles using continuum-based no-slip and slip models,respectively.The validity of continuum-based models is examined by DSMC method.The study focuses on low Reynolds number effects,three-dimensional effect and propulsive performance of the micro-nozzle flow.It shows that compared to the prediction of propulsive performance,the simulation of local flow fields needs a more stringent model.The no slip N-S equations are able to predict propulsive performance of micro-nozzles with Kn < 0.03.Reynolds number is a key parameter in governing low Reynolds number effect and propulsive performance.The strong viscous losses can be mitigated and better propulsive performance can be achieved at higher chamber pressures.The micro-nozzle with the ratio of etch depth to throat width more than 13 has a good 2D characteristic as Re > 1000.

In a perturbation analysis,a two-dimensional theoretical solution based on Navier-Stokes equations is constructed for gaseous slip flows in a micro-channel with different slip conditions.The micro-channel flow is investigated theoretically and numerically at various inlet pressure ratios,aspect ratios and fluid cases with several slip models.The influence of the rarefaction effect,thermal creep effect and slip conditions is emphasized.Simulations show that Kn is a key parameter in determining the magnitude of rarefaction effects while Re is a key parameter indicating the thermal creep.Excellent agreements with experimental results are observed in both theoretical and numerical results of low-velocity micro-channel flows at very large aspect ratios.