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Extra info for CAS-CERN Accelerator School - Applied Geodesy for Particle Accelerators
In this log-log plot, we can easily identify three distinct ﬂow regimes, although the corresponding values at the boundaries between the diﬀerent ﬂow regimes, are somewhat diﬀerent from the aforementioned ones (Schaaf and Chambre, 1961). 6 and the transition regime up to Kn = 17. It is interesting to notice the very slow variation of mass ﬂowrate in terms of the pressure diﬀerence in the transition regime. 13 also suggests that a nonlinear pressure drop exists in this rareﬁed pipe ﬂow. This was also veriﬁed in the slip ﬂow experiments of (Sreekanth, 1969) in pipes with a diameter of 2 inches at low pressures.
16 1. Basic Concepts and Technologies • At what scales will the statistical ﬂuctuations be signiﬁcant? It turns out that sampling a volume that contains 10,000 molecules results in 1% statistical ﬂuctuations in the averaged quantities. 7 × 10−22 m3 . If we try to measure the macroscopic gradients (like velocity, density, and temperature) in threedimensional space, one side of our sampling volume will be about 65nm. A key nondimensional parameter for gas microﬂows is the Knudsen number, which is deﬁned as the ratio of the mean free path over a characteristic geometric length or a length over which very large variations of a macroscopic quantity may take place.
We then discuss techniques of coupling DSMC with continuum descriptions, and we analyze microﬁlters as an example. We also include an overview of the Boltzmann equation as well as benchmark solutions in the slip ﬂow and transitional regimes, that could be useful to the reader for validation of experimental and numerical results. Finally, we present the lattice Boltzmann method, which solves the Boltzmann equation fast but in a constrained subspace. This method too could be a very eﬀective tool for simulating ﬂows in complex microgeometries.