Pressure distribution in microtubes with variable cross section

Abstract

A steady compressible isothermal slip gas flow in a microtube is analyzed for low Reynolds numbers. The microtube is with varying cross-section, which from the standpoint of geometry gives three considered cases: convergent microtube, divergent microtube and microtube with constant radius. The gas flow is caused by the pressure difference between the inlet and the outlet cross-section. The solution for pressure and mass flow rate is obtained by macroscopic approach from Navier-Stokes equations with the second order velocity slip boundary condition. Analysis of the order of dimensionless terms in governing equations is possible with the usage of exact relation between Reynolds, Mach and Knudsen numbers. The solution procedure for this flow model is based on perturbation approach, where significant variables are assumed in the form of perturbation series. The first approximation represents the solution for the continuum flow conditions, and the second approximation incorporates the effect of gas rarefaction. The obtained solution is compared with verified results of numerical experiment found in literature and good agreement is achieved. Hence, the reliability of obtained solution and presented method is confirmed.