Collisionless Gas Expanding into Vacuum

H IGH-SPEED collisionless, or free-molecular, gas flows passing through small circular or annular holes are fundamental problems with many real applications such as neutral gas expansion out of electric propulsion (EP) devices. Usually, the cold plume flow out of an EP device is modeled by assuming freemolecular flows with a nonzero uniform average exit velocity U0. Even when the average bulk velocity of gas near the orifice is zero, the average velocity at the orifice exit plane is not zero, it corresponds to an outflow with a half-Maxwellian distribution. In the past, analytical studies of similar problems were concentrated on true effusion problems with a zero average exit speed. For example, Liepmann [1] reported the efflux of gases through circular apertures, which is an example of a transition from the gas-dynamic to the gaskinetic regime; Narasimha [2] obtained the exact solutions of density and velocity distributions for a free-molecular effusion flow and the results for a nearly free-molecular effusion flow expanding into vacuum through a circular orifice; and Brook [3] reported the density field of free-molecular flow from an annulus, to study the gas leakage effect from a spacecraft hatch. Other researchers reported many approximate methods or numerical simulations to study rarefied flows through a slit; for example, Rotenberg and Weitzner [4], Hasegawa and Sone [5], Cercignani and Sharipov [6], and Sharipov [7]. Recently, Lilly et al. [8] reported their work onmeasurement and computation ofmass flow andmomentum flux through short tubes in rarefied gas. For the case of free-molecular flows with a nonzero average velocity, the problems are usually very complicated and approximations are often made, such as neglecting the details of the exit geometry or assuming that free-molecular flow are emitted from a point source [9]. In our previous study [10,11], we adopted a relation between velocity directions and geometry locations to investigate freemolecular plume flow problems. This treatment is more general than the solid angle treatment [2]. which was widely used in studying true collisionless effusion flows with a zero average exit speed, but is not applicable to collisionless flows with a nonzero average exit speed. In this study, we further investigate collisionless flows out of a circular or an annular exit with a nonzero average speed. These two cases are very important, not only because of their mathematical significance, but also because of their many direct applications, including spacecraft propulsion. This Note is organized as follows: Section II describes the problems, the corresponding complex exact solutions, and also approximate far-field solutions, which are simpler andmore accurate than existing formulas in the literature; Sec. III compares the analytical results with particle simulation results; and Sec. IV summarizes this study.