Rarefied isothermal gas flow in a long circular tube due to oscillating pressure gradient

Abstract

The time-dependent isothermal fully developed rarefied gas flow in a circular tube driven by harmonically oscillating pressure gradient is investigated, based on the linearized unsteady BGK kinetic model equation. The flow is characterized by the gas rarefaction parameter, which is proportional to the inverse Knudsen number and the oscillation parameter, defined as the ratio of the collision frequency over the pressure gradient oscillation frequency. Computational results of the amplitude and the phase angle of the flow rates and the velocity distributions, as well as of the periodic time evolution of these macroscopic quantities, are provided, covering the whole range of the two parameters. The kinetic results properly recover the limiting solutions in the slip and free molecular regimes for low- and high-speed oscillations. At low frequencies, the time-dependent flow becomes quasi-steady and gradually tends to the corresponding steady-steady one, which is reached faster when the flow is more rarefied. As the frequency is increased, the amplitude of the macroscopic quantities is decreased and their phase angle lag with respect to the pressure gradient is increased approaching asymptotically the limiting value of π/ 2. In terms of the gas rarefaction, there is a non-monotonic behavior and the maximum flow rate amplitude may be observed at some intermediate value of the gas rarefaction parameter depending upon the oscillation parameter. At high frequencies, the flow consists of an inviscid piston flow in the core and the frictional Stokes wall layer with a velocity overshoot. These effects, well known in the viscous regime, are also present here in the transition regime and depend on both the gas rarefaction and oscillation parameters. As the gas rarefaction is increased, higher oscillation frequencies are needed to trigger these phenomena. Oscillatory rarefied flows are of main interest in sensors, controllers and resonators, which may be present in various microfluidic applications (e.g., microcooling, microseparators and micropropulsion). © 2017, Springer-Verlag GmbH Germany, part of Springer Nature.