A moment method for low speed microflows

A moment method is proposed to study slow rarefied flows by the linearized Bhatnagar-Gross-Krook (BGK) kinetic model equation. In order to obtain a consistent treatment of boundary conditions, the velocity distribution function is expanded in orthogonal polynomials which are not continuous in the velocity space. The solution of the kinetic equation is then reduced to the solution of a system of differential equations for the expansion coefficients. For one-dimensional problems, the system of moment equations can be easily recast in an hydrodynamic-like form. The method is here applied to isothermal steady boundary driven flows, i.e. the one-dimensional Couette and Poiseuille flows and the two-dimensional cavity flow. The results show that it is possible to obtain excellent approximations of the (virtually) exact solutions of the kinetic model equation by using a small number of moments in a wide range of Knudsen numbers and suggest that it might be possible to obtain a sufficiently accurate description of slow rarefied flows by a small number of moment equations.