Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur PO, Bangalore 560064, India.
Phys Rev E. 2018 Jan;97(1-1):012912. doi: 10.1103/PhysRevE.97.012912.
The nature of particle-wall interactions is shown to have a profound impact on the well-known "Knudsen paradox" [or the "Knudsen minimum" effect, which refers to the decrease of the mass-flow rate of a gas with increasing Knudsen number Kn, reaching a minimum at Kn∼O(1) and increasing logarithmically with Kn as Kn→∞] in the acceleration-driven Poiseuille flow of rarefied gases. The nonmonotonic variation of the flow rate with Kn occurs even in a granular or dissipative gas in contact with thermal walls. The latter result is in contradiction with recent work [Alam et al., J. Fluid Mech. 782, 99 (2015)JFLSA70022-112010.1017/jfm.2015.523] that revealed the absence of the Knudsen minimum in granular Poiseuille flow for which the flow rate was found to decrease at large values of Kn. The above conundrum is resolved by distinguishing between "thermal" and "athermal" walls, and it is shown that, for both molecular and granular gases, the momentum transfer to athermal walls is much different than that to thermal walls which is directly responsible for the anomalous flow-rate variation with Kn in the rarefied regime. In the continuum limit of Kn→0, the athermal walls are shown to be closely related to "no-flux" ("adiabatic") walls for which the Knudsen minimum does not exist either. A possible characterization of athermal walls in terms of (1) an effective specularity coefficient for the slip velocity and (2) a flux-type boundary condition for granular temperature is suggested based on simulation results.
壁面与颗粒相互作用的本质对众所周知的“Knudsen 佯谬”[或“Knudsen 最小值效应”，指的是随着 Knudsen 数 Kn 的增加，稀薄气体的努塞尔特（Poiseuille）流动的质量流量减小，在 Kn∼O(1)时达到最小值，并随着 Kn 的增加呈对数增长]有着深远的影响。即使在与热壁接触的颗粒或耗散气体中，流动速率也会出现非单调变化。这一结果与最近的工作[Alam 等人，J. Fluid Mech. 782, 99 (2015)JFLSA70022-112010.1017/jfm.2015.523]相矛盾，后者表明在颗粒努塞尔特流动中不存在 Knudsen 最小值，因为在 Kn 值较大时，流动速率会减小。通过区分“热壁”和“非热壁”，可以解决上述难题，并表明对于分子气体和颗粒气体，向非热壁的动量传递与向热壁的传递有很大的不同，这直接导致了在稀薄区域中 Kn 对流动速率变化的反常影响。在 Kn→0 的连续极限下，非热壁与“无流量”（“绝热”）壁密切相关，对于后者，也不存在 Knudsen 最小值。根据模拟结果，提出了一种基于（1）滑移速度的有效反射系数和（2）颗粒温度的通量型边界条件，对非热壁进行特征化的方法。
