Calculation of the Darcy friction coefficient using the ellipsoidal statistical model

A method is proposed for calculating the Darcy friction coefficient in a flat channel formed by two infinite parallel plates, depending on the values of the Reynolds number and the rarefaction parameter of the gas in the channel. Maxwell's diffuse reflection model is used as a model for the reflection of gas molecules from the channel walls. The isothermal gas flow is due to the presence of a small absolute pressure gradient directed along the channel walls. The proposed method is based on solving the linearized ellipsoidal-statistical Boltzmann kinetic equation with homogeneous boundary conditions using the Chebyshev polynomial approximation. An unknown function that interpolates the solution of this equation is represented as a partial sum of a series in Chebyshev polynomials of the first kind. Using the properties of finite sums for an orthonormal system of Chebyshev polynomials, the problem is reduced to a system of linear algebraic equations with respect to the values of the desired function at the interpolation nodes. The analysis of the obtained results was carried out.