Rarefied gas flow modeling presents significant challenges in the characterization of nanoscale devices and their applications. An important feature of such flows is the Knudsen layer, which is known to exhibit non-Newtonian viscosity behavior. Significantly, recent research has suggested that the effective viscosity at the surface is about half the standard dynamic viscosity. We examine these claims using numerical solutions of the linearized Boltzmann equation and direct simulation Monte Carlo calculations and discover that (i) the flow exhibits a striking power-law dependence on distance from the solid surface and (ii) the velocity gradient is singular at this surface. This finding contradicts these recent claims and has direct implications for gas flow modeling and the design of nanoscale devices.
Flow of a dilute gas near a solid surface exhibits non-continuum effects that are manifested in the Knudsen layer. The non-Newtonian nature of the flow in this region has been the subject of a number of recent studies suggesting that the so-called 'effective viscosity' at a solid surface is half that of the standard dynamic viscosity. Using the Boltzmann equation with a diffusely reflecting surface and hard sphere molecules, Lilley & Sader discovered that the flow exhibits a striking power-law dependence on distance from the solid surface where the velocity gradient is singular. Importantly, these findings (i) contradict these recent claims and (ii) are not predicted by existing highorder hydrodynamic flow models. Here, we examine the applicability of these findings to surfaces with arbitrary thermal accommodation and molecules that are more realistic than hard spheres. This study demonstrates that the velocity gradient singularity and power-law dependence arise naturally from the Boltzmann equation, regardless of the degree of thermal accommodation. These results are expected to be of particular value in the development of hydrodynamic models beyond the Boltzmann equation and in the design and characterization of nanoscale flows.
SUMMARYIn the direct simulation Monte-Carlo (DSMC) method for simulating rareÿed gas ows, the velocities of simulator particles that cross a simulation boundary and enter the simulation space are typically generated using the acceptance-rejection procedure that samples the velocities from a truncated theoretical velocity distribution that excludes low and high velocities. This paper analyses an alternative technique, where the velocities of entering particles are obtained by extending the simulation procedures to a region adjacent to the simulation space, and considering the movement of particles generated within that region during the simulation time step. The alternative method may be considered as a form of acceptancerejection procedure, and permits the generation of all possible velocities, although the population of high velocities is depleted with respect to the theoretical distribution. Nevertheless, this is an improvement over the standard acceptance-rejection method. Previous implementations of the alternative method gave a number ux lower than the theoretical number required. Two methods for obtaining the correct number ux are presented. For upstream boundaries in high-speed ows, the alternative method is more computationally e cient than the acceptance-rejection method. However, for downstream boundaries, the alternative method is extremely ine cient. The alternative method, with the correct theoretical number ux, should therefore be used in DSMC computations in favour of the acceptance-rejection method for upstream boundaries in high-speed ows.
In most chemistry methods developed for the direct simulation Monte Carlo ͑DSMC͒ technique, chemical reactions are computed as an integral part of the collision simulation routine. In the macroscopic chemistry method developed here, the simulation of collisions and reactions are decoupled in that reactions are computed independently, after the collision routine. The number of reaction events to perform in each cell is calculated using the macroscopic reaction rates k Ϯ and equilibrium constant K*, with local macroscopic flow conditions. The macroscopic method is developed for the symmetrical diatomic dissociating gas. For each dissociation event, a single diatomic simulator particle is selected with a probability based on its internal energy, and is replaced by two atomic particles. For each recombination event, two atomic particles are selected at random, and are replaced by a single diatomic particle. The dissociation energy is accounted for by adjusting the translational thermal energies of all particles in the cell. The macroscopic method gives density profiles in agreement with experimental data for the chemical relaxation region downstream of a strong shock in nitrogen. In the nonequilibrium region within the shock, and along the stagnation streamline of a blunt cylinder in rarefied flow, the macroscopic method gives results in excellent agreement with those obtained using the most common conventional DSMC chemistry method in which reactions are calculated during the collision routine. The number of particles per computational cell has a minimal effect on the results provided by the macroscopic method. Unlike most DSMC chemistry methods, the macroscopic method is not limited to simple forms of k Ϯ and K*. Any forms may be used, and these may be any function of the macroscopic conditions. This is demonstrated by using a two-temperature rate model, and a form of K* with a number density dependence. With the two-temperature model, the macroscopic method gives densities in the post-shock chemical relaxation region that also agree with the experimental data. For a form of K* with a number density dependence, the macroscopic method can accurately reproduce chemical recombination behavior. In a primarily dissociative flow, the number density dependence of K* has very little effect on the flow. The macroscopic method requires slightly less computing time than the most common DSMC chemistry method.
Rarefied gas flows generated by resonating nanomechanical structures pose a significant challenge to theoretical analysis and physical interpretation. The inherent noncontinuum nature of such flows obviates the use of classical theories, such as the Navier-Stokes equations, requiring more sophisticated physical treatments for their characterization. In this Letter, we present a universal dynamic similarity theorem: The quality factor of a nanoscale mechanical resonator at gas pressure P 0 is α times that of a scaled-up microscale resonator at a reduced pressure αP 0 , where α is the ratio of nanoscale and microscale resonator sizes. This holds rigorously for any nanomechanical structure at all degrees of rarefaction, from continuum through to transition and free molecular flows. The theorem is demonstrated for a series of nanomechanical cantilever devices of different size, for which precise universal behavior is observed. This result is of significance for research aimed at probing the fundamental nature of rarefied gas flows and gas-structure interactions at nanometer length scales. Miniaturization of resonant mechanical structures has driven progress across a wide range of advanced technologies, including sensors for mass detection and imaging with atomic resolution [1-3], monitoring of biological processes such as DNA hybridization [4], and mass spectrometry at the molecular scale [5]. While the constituent elastic properties of mechanical resonators remain identical to bulk values upon miniaturization to nanometer length scales [6][7][8][9], operation in fluid environments gives rise to physical phenomena not normally seen at macroscopic levels [10][11][12][13][14]. In liquid, these include transport due to the electrical nature of surfaces [15] and the possibility of violation of the usual no-slip condition [16]. Breach of no-slip is exacerbated for operation in gas, because the relevant molecular length scale is orders of magnitude larger than in liquid [17][18][19]. This leads to failure of the continuum hypothesis at relatively large length scales (microns), obviating the use of classical theories such as the Navier-Stokes equation.Continuum theories are used ubiquitously in the design and characterization of macroscale and microscale mechanical structures. In the context of mechanical resonators in gas, this has provided insight into the fundamental physical processes underpinning their operation. For example, it is known that the quality factor (scaled inverse rate-ofenergy dissipation) of a mechanical resonator decreases as the size of the structure is reduced [10,20]. This is due to growth of the viscous penetration depth, where vorticity is prevalent, relative to the structure size. Yet this knowledge is predicated on the assumption that the gas mean free path λ is much smaller than the size of the structure. The mean free path of air at 1 atm and room temperature is approximately 70 nm [21]. Thus, miniaturization of nanomechanical resonators to several hundred nanometers induces strong noncontinuu...
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