Pressure driven flows of rarefied gases through long channels with double trapezoidal cross-sections

Abstract

Steady flows of rarefied single gases through long channels with double trapezoidal cross-section shapes are investigated numerically. The flow is between two large reservoirs having the gas at different pressures. The channels have constant cross-sections along the axial direction. The gas is modeled by the linearized Bhatnagar–Gross–Krook kinetic equation. The diffuse reflection boundary condition is used at the channel walls. The solution of the problem is divided into two stages. At a particular cross-section, the flow is driven by the local pressure gradient. First, the local problem for an arbitrary driving term is solved by using the discrete velocity method. This solution yields the dimensionless flow rate and the velocity profile for a wide range of the gaseous rarefaction. Second, the global flow behavior, i.e. the flow rate and the distribution of the pressure, is deduced for global pressure driven flows on the basis of the conservation of mass. The numerical solution of the kinetic equation is based on the discretization of the spatial and velocity spaces. The spatial space is represented on a rectangular grid. The walls of the channels are aligned parallel to the grid lines or along the diagonal of the grid. Such a choice provides a straightforward calculation of the spatial derivatives. In the interior part of the domain and near the channel walls, second- and first-order finite difference forms are used, respectively. The velocity space is represented by a Gauss–Legendre quadrature. The resulting discrete equations are solved in an iterative manner. The dimensionless flow rates are calculated and tabulated for particular cross-sections in a wide range of the gaseous rarefaction. The flow rate function exhibits the Knudsen minimum. The results are compared to the corresponding ones with other cross-sections. Typical velocity profiles are also shown. Finally, representative results are delivered for global pressure driven flows.