Calculation of Knudsen layers and jump conditions using the linearised G13 and R13 moment methods

Abstract

For gas flows in micro-channels the molecular mean free path is often comparable to the channel dimensions and the flow must be modelled using one of the methods of non-continuum gas dynamics. For slip-flow or lower transition regime calculations the Grad 13-moment (G13) method or its extension the Regularised 13-moment (R13) method has been proposed. The R13 method is assumed to be an improvement over the G13 because it can capture, at least to some extent, Knudsen layers close to solid surfaces. This paper investigates the capabilities of the two methods by using them to address the classical problems of viscous slip, temperature jump and thermal creep. For the linearised equations it is possible to obtain analytical solutions and these provide physical insight as well as numerical values. The analysis shows that the Knudsen layer associated with temperature and normal stress decouples from the layer associated with velocity and heat flux. Comparison with more accurate solutions of the kinetic equation shows that the LR13 method underpredicts the velocity slip and temperature jump coefficients by about 10% and only provides a slight improvement over the much simpler LG13 method. Furthermore, the Knudsen layer velocity and temperature profiles bear only a superficial resemblance to more accurate solutions. For reasons that are unclear, the calculation of the thermal creep flow is very much more successful. It is concluded that considerable improvements need to be made before the LR13 method becomes a reliable technique for solving non-continuum flow problems.