Free Molecular Flow Regime Research Articles

A design optimization method for rarefied and continuum gas flows

A design optimization method applicable to both rarefied and continuum gas flows with good efficiency is proposed in the framework of gas-kinetic theory. The method follows the methodology of topology optimization where the areas of gas and solid are marked by the material density, based on which a fictitious porosity model is used to reflect the effect of the solid on the gas and mimic the diffuse boundary condition on the gas-solid interface. The formula of this fictitious porosity model is modified to make the model work well from the free-molecular flow regime to the continuum flow regime. To find the optimized material density, a gradient-based optimizer is adopted and the design sensitivity is obtained by the continuous adjoint method. To solve the primal kinetic equation and the corresponding adjoint equation, efficient multiscale numerical schemes are constructed. Several airfoil optimization problems are solved to demonstrate the performance of the present design optimization method in wide ranges of flow speed and gas rarefaction. It is found that the thickness of the optimized airfoil first increases and then decreases as the Knudsen number increases, and on the supersonic condition the optimized airfoil has quite different shapes of leading edge for different degrees of gas rarefaction.

Tilt effect on squeeze-film damping in electrostatic micro-actuators

In this study, we theoretically explore the impact of tilt on the switching dynamics of a MEMS device operated by a pair of electrostatic actuators and subject to pronounced squeeze-film damping in the transition and free molecular flow regimes. Our investigation reveals that inducing tilt, achieved through the application of uneven voltages on the electrodes, introduces additional source terms in the modified Reynolds equation governing the pressure distribution in the squeeze-film gap. These added terms result in an accelerated decay of the squeeze film force, facilitating faster device switching between the initial and target configurations.To analyze the device dynamics, we develop a coupled numerical model and employ evolutionary multi-objective optimization to determine an optimal actuation scheme. This optimization is carried out for scenarios where initially different voltages are applied to the electrodes during a certain pre-defined time interval before transitioning to the steady voltage corresponding to the target gap size at equilibrium. Our findings highlight that the suggested actuation strategy leads to substantial improvements in the switching time of the device. Furthermore, we provide insights into the conditions favoring tilt actuation over parallel-motion actuation where the initially applied voltages are equal.

Rarefied cylindrical Couette problem: Effect of a very sharp transition from weakly rarefied flow near the inner cylinder to free molecular flow at very low average Knudsen numbers

We consider the rarefied cylindrical Couette flow. The outer cylinder rotates at a constant angular velocity, and the inner one is at rest. The Knudsen number, calculated from the average gas density, is much less than one. To find the fluid dynamic variables and fluxes, the direct simulation Monte Carlo method (DSMC) is used. We find the effect of a very sharp decrease in shear stress and energy fluxes transferred to the cylinders if the Knudsen number is sufficiently low and the rotation speed of the outer cylinder is sufficiently high. We show that when the average Knudsen number is sufficiently low, if the velocity is initially high and increases slightly further, the weakly rarefied flow regime near the inner cylinder transitions abruptly to a free molecular flow regime. Thus, the flow abruptly separates from the inner cylinder. We compare the fluxes obtained by solving the Navier–Stokes equations and by the DSMC method. It turns out that the solutions of the Navier–Stokes equations with slip boundary conditions also change abruptly at certain critical values, and these critical values are close to the appropriate DSMC values.

Damping separation of finite open systems in gravity-related experiments in the free molecular flow regime

Damping separation of finite open systems in gravity-related experiments in the free molecular flow regime

Investigating the temperature distribution behavior and flow parameters of argon fluid in a nanochannel with changing dimensions of the obstacle using the molecular dynamics (MD) method

This article, examines the flow of argon inside a nanochannel with respect to the molecular dynamics (MD) in the free molecular flow regime using LAMMPS software. The nanochannel is made of copper featuring a square cross-section and obstacles of varying dimensions and values. In this study, the flow of argon fluid is three-dimensional. To gain a deeper understanding of the effect of solid walls within the nanochannel and their influence on flow behavior, the research is simulated in a nanochannel with all side walls for the 3D model and without side walls for the 2D model. This research assesses the effect of the obstacles’ dimensions and values on the nanochannel wall surface and areas above the wall surface. The total dimensions of all simulated two- and three-dimensional atomic structures with a square cross-section are assumed to be 60 × 60 × 100 Å3. and the presence of square obstacles (with dimensions of 8 × 8 × 8 Å3) and rectangular obstacles (with dimensions of 8 × 18 × 8 Å3) is examined. This study seeks to understand the influence on flow behavior, temperature distribution, density, heat flux, velocity, and thermal conductivity coefficient. This study is simulated using a time step of 1 fs for 10,000 time steps, involving approximately 10,000–15,000 argon and copper atoms. The results of this research indicate that obstacles with structures of P and R and larger dimensions increase the number of solid atoms exhibiting stronger attractive forces. Compared to a smooth nanochannel, the thermal exchange between fluid and solid atoms results in a density increase of 17.5 % and 17.3 %, respectively. On the other hand, in the 3D nanochannel, the sidewalls of the nanochannel have reduced the effect of the presence of R and P obstacles with larger dimensions, which comparing to a smooth nanochannel, have increased the density by 8.21 % and 7.53 %, respectively. The obstacles with different spatial positions (P and R structures) in the two-dimensional nanochannel cause a rise in the thermal conductivity coefficient. The P structure obstacles have a better effect on the thermal conductivity coefficient in the 2D nanochannel compared to the R structure. In the three-dimensional nanochannel, utilizing smaller obstacles proves to be more effective because it results in better atom distribution or temperature distribution due to increased atomic collisions in the central region compared to the wall regions.

Ab initio modelling of transport phenomena in multi-component mixtures of rarefied gases

The direct simulation Monte Carlo method based on ab initio potentials is applied to model transport phenomena in binary, ternary and quaternary mixtures of noble gases. To this end, the Couette and Fourier problems are solved over the whole range of the rarefaction parameter spanning the near free-molecular, transitional, and slip flow regimes. Moreover, analytical solutions are obtained in the limits of the free-molecular and continuous medium regimes of flow. As a result, the shear stress, heat flux through multi-component mixtures, distributions of velocity, temperature, and mole fractions of each species are calculated as a function of the gas rarefaction. An analysis of these data points out the properties which are strongly affected by the chemical composition. It is found that the largest deviations of characteristics of mixtures from those of a single gas are observed for the binary mixture of helium and krypton, while their deviations for the ternary and quaternary mixtures are smaller. A temperature variation in a mixture leads to non-uniform distributions of chemical composition due to the thermal diffusion phenomenon. Such redistributions are analyzed in both problems and their effect on the heat and momentum transfer is evaluated. The reported results can be used to evaluate the main factors determining transport phenomena in practical applications such as vacuum systems, microfluidics, gas separation technology, space research etc.

Laboratory measurement of volatile ice vapor pressures with a quartz crystal microbalance

Nitrogen, carbon monoxide, and methane are key materials in the far outer Solar System where their high volatility enables them to sublimate, potentially driving activity at very low temperatures. Knowledge of their vapor pressures and latent heats of sublimation at relevant temperatures is needed to model the processes involved. We describe a method for using a quartz crystal microbalance to measure the sublimation flux of these volatile ices in the free molecular flow regime, accounting for the simultaneous sublimation from and condensation onto the quartz crystal to derive vapor pressures and latent heats of sublimation. We find vapor pressures to be somewhat lower than previous estimates in literature, with carbon monoxide being the most discrepant of the three species, almost an order of magnitude lower than had been thought. These results have important implications across a variety of astrophysical and planetary environments.

Numerical study of microdevice with surface acoustic waves for separation of gas mixtures

Recently, it was shown that traveling surface acoustic waves (SAWs) can affect gas flow in microchannels. The effect of SAWs was studied in free-molecular flow regime, and it was shown that SAWs can induce separation of gas mixtures. In the present work, this effect is studied for denser flow regimes, which are more interesting from a practical point of view. The problem is studied numerically using own modification of the direct simulation Monte Carlo method on the example of a neon–argon mixture. The main finding is that SAWs still enhance separation of gas mixtures outflowing into vacuum through a microchannel under all studied rates of gas rarefaction up to Kn≈0.1. Another important practical result is that effect is present for wave speeds typical for existing SAWs (≈1000 m/s) and in a wide range of SAW amplitude to channel height ratios. Influence of other practical aspects, such as channel length, masses of species, and available magnitudes of material surface speed, are also briefly discussed.

Drag on a sphere in a slow flow of a binary mixture of rarefied gases

The drag force on a sphere in an isothermal binary mixture of rarefied gases is calculated on the basis of the McCormack model for the linearized Boltzmann equation. The diffuse scattering of the gas–surface interaction law is assumed as the boundary condition. The rigid-spheres model of interatomic interaction potential is used as well as ab initio potentials for helium, argon, and krypton gases. The force is obtained in a wide range of the gas rarefaction, which covers the free molecular, transition, and slip regimes of the gas flow. In the free molecular and slip flow regimes, the problem is solved analytically, while in the transition regime, the system of kinetic equations is solved numerically via the discrete velocity method optimized to overcome the problem of discontinuity of the distribution function of molecular velocities on the convex surface. The calculations are carried out for the mixtures helium–argon and helium–krypton at 300 K. In the slip flow regime, the data available in the literature for the viscous slip coefficient of the helium–argon mixture are used, while for the mixture helium–krypton, it is calculated. The influence of the interatomic interaction potential, molar fraction, and ratio of atomic mass of species in the mixture on the drag force is analyzed.

Inversion in binary gas mixtures in rarefied flow conditions: Direct simulation Monte Carlo solution and comparison with the analytical solutions at free molecular regime

This paper analyzes the mixing of gases in a plane channel at rarefied conditions. The direct simulation Monte Carlo method is employed to simulate gas mixing in parallel mixers working at different Knudsen numbers and having different values of wall accommodation coefficient. Results show that the normal-to-wall component of the mole fraction gradient may have the same sign as the corresponding component of the diffusive mass flux vector near the diffuse solid walls in contrast to the predictions of Fick's law for continuum conditions. This non-continuum behavior, which is called “inversion” in the present study, will become more pronounced at higher Knudsen numbers, whereas it will become less evident for smaller wall accommodation coefficients. To confirm that the observed phenomenon is consistent with the basic physical laws governing the rarefied gas dynamics and it is not an artifact of the numerical method, a new analytical model based on the kinetic theory of gases is developed for the parallel mixers that have diffuse walls and are working in the free-molecular regime. Excellent agreement is observed between the analytical and direct simulation Monte Carlo results in the free molecular flow regime. Both methods predict the occurrence of inversion near the diffuse walls at highly rarefied flow conditions.

Comparison of different Gaussian quadrature rules for lattice Boltzmann simulations of noncontinuum Couette flows: From the slip to free molecular flow regimes

The lattice Boltzmann (LB) method can be formulated directly from the Boltzmann equation with the Bhatnagar–Gross–Krook assumption. This kinetic origin stimulates wide interest in applying it to simulate flow problems beyond the continuum limit. In this article, such a thought is examined by simulating Couette flows from the slip to free molecular flow regimes using the LB models equipped with different discrete velocity spaces, derived from the half-range Gauss Hermite (HGH), Gauss Legendre (GL), Gauss Kronrod (GK), and Gauss Chebyshev first and second quadrature rules. It is found that the conventional HGH-based LB models well describe noncontinuum Couette flows in the slip and weak transition flow regimes. Nonetheless, they suffer from significant errors with the further increasing Knudsen number, even if a large number of discrete velocities have been employed. Their results contrast with those by the LB models derived from the other Gaussian quadrature rules, which have far better accuracy at large Knudsen numbers. In particular, the GL- and GK-based LB models well capture the velocity fields of Couette flows in the strong transition and free molecular flow regimes. These numerical simulations in this article highlight the importance of velocity discretization for the LB simulations at different Knudsen numbers. They reveal that the LB models based on the Gauss Hermite (GH) quadrature rule are not always the best choice for simulating low-speed bounded flows at moderate and large Knudsen numbers; under strong noncontinuum conditions, those non-GH-based LB models proposed in this article have yielded more accurate results.

Reynolds Analogy Coefficient in the Longitudinal Cylindrical Couette Problem: from the Continuous Medium to Free Molecular Flow

The gas Couette flow for a cylindrical geometry of bounding surfaces that move in the longitudinal direction relative to their symmetry axis is considered. For a monoatomic gas, the relation between the shear stress and the energy flux transferred to the longitudinal-flow surface (Reynolds analogy) is studied. In the case of continuous medium and free molecular flow regimes, simple explicit analytical expressions for the Reynolds analogy coefficient are obtained, which depend only on the Eckert number and are independent of the ratio of the cylinder radii. The transitional regime for various values of the Knudsen number is studied using the direct simulation Monte Carlo (DSMC) method. It is shown that, in this case, the Reynolds analogy coefficient at a fixed ratio of the radii and the Knudsen number depends on the relative velocity and temperatures of the surfaces mainly through the Eckert number. A relationship between the linear energy fluxes transferred to the cylindrical surfaces is found.

A Review of Research on the Vacuum Plume

Chemical and electrical thrusters are generally utilized to control the attitude and orbit of spacecraft in aerospace. When they are firing, the exhaust expands into the vacuum environment, known as the vacuum plume. The plume flow can collide with spacecraft surfaces due to sufficient expansion, exerting adverse effects on the spacecraft, such as heating load, force/torque, contamination, and sputtering. Therefore, it is vital to investigate the vacuum plume to ensure the function and safety of the spacecraft. This review introduces the ground test and numerical simulation methods of the vacuum plume for chemical and electrical thrusters. The vacuum environment, invasive, and non-invasive (optical) measurements of the ground test are concluded. Numerical simulation of plume flow and its effects is exampled. The hybrid CFD-DSMC (computational fluid dynamics and direct simulation Monte Carlo) algorithm is employed to simulate the gas plume flow spanning continuum and transitional and free molecular flow regimes for chemical thrusters. By contrast, the PIC-DSMC (particle-in-cell plus direct simulation Monte Carlo) algorithm is used for the plasma plume flow containing charged particles exhausted by electrical thrusters. Moreover, the topics of fast prediction of the vacuum plume, plume–surface interaction, and plume–Lunar/Mars regolith interaction are proposed for future research.

Differences in water and vapor transport through angstrom-scale pores in atomically thin membranes

The transport of water through nanoscale capillaries/pores plays a prominent role in biology, ionic/molecular separations, water treatment and protective applications. However, the mechanisms of water and vapor transport through nanoscale confinements remain to be fully understood. Angstrom-scale pores (~2.8–6.6 Å) introduced into the atomically thin graphene lattice represent ideal model systems to probe water transport at the molecular-length scale with short pores (aspect ratio ~1–1.9) i.e., pore diameters approach the pore length (~3.4 Å) at the theoretical limit of material thickness. Here, we report on orders of magnitude differences (~80×) between transport of water vapor (~44.2–52.4 g m−2 day−1 Pa−1) and liquid water (0.6–2 g m−2 day−1 Pa−1) through nanopores (~2.8–6.6 Å in diameter) in monolayer graphene and rationalize this difference via a flow resistance model in which liquid water permeation occurs near the continuum regime whereas water vapor transport occurs in the free molecular flow regime. We demonstrate centimeter-scale atomically thin graphene membranes with up to an order of magnitude higher water vapor transport rate (~5.4–6.1 × 104 g m−2 day−1) than most commercially available ultra-breathable protective materials while effectively blocking even sub-nanometer (>0.66 nm) model ions/molecules.

Velocity discretization for lattice Boltzmann method for noncontinuum bounded gas flows at the micro- and nanoscale

The lattice Boltzmann (LB) method intrinsically links to the Boltzmann equation with the Bhatnagar–Gross–Krook collision operator; however, it has been questioned to be able to simulate noncontinuum bounded gas flows at the micro- and nanoscale, where gas moves at a low speed but has a large Knudsen number. In this article, this point has been verified by simulating Couette flows at large Knudsen numbers (e.g., Kn=10 and Kn=100) through use of the linearized LB models based on the popular half-range Gauss–Hermite quadrature. The underlying cause for the poor accuracy of these conventional models is analyzed in the light of the numerical evaluation of the involved Abramowitz functions. A different thought on velocity discretization is then proposed using the Gauss–Legendre (GL) quadrature. Strikingly, the resulting GL-based LB models have achieved high accuracy in simulating Couette flows, Poiseuille flows, and lid-driven cavity flows in the strong transition and even free molecular flow regimes. The numerical study in this article reveals an essentially distinct but workable way in constructing the LB models for simulating micro- and nanoscale low-speed gas flows with strong noncontinuum effects.

Unraveling the role of operating pressure in the rapid formation of Cu-BDC MOF via a microdroplet approach

A microdroplet-based spray process has been recently developed for the synthesis of metal–organic frameworks (MOFs) which features a short processing time and scalable manner. In this process, atomized microdroplets of the MOFs precursor solution undergo well-controlled physiochemical transformations with each droplet serving as a microreactor. Droplet transformation is closely related to the flow patterns of the spray system. A conventional spray system is operated at atmospheric pressure (under the continuum flow regime). By lowering the operating pressure, the spray system is transitioned to a free-molecular flow regime, where the transport phenomena of the microdroplet become distinct due to the discrete surrounding gas. It is hypothesized that the MOFs formed under this condition could have unique morphological and structural features. Nevertheless, this topic has rarely been investigated. This work studies the formation of the copper 1,4-benzenedicarboxylate (Cu-BDC) MOF under changing flow regimes by modulating the operating pressure. Specifically, changes in crystal size, morphology, and orientation, as well as surface oxidation state, are observed with decreasing operating pressure. Heat and mass transfer calculations, accounting for the transition of flow regimes, suggest that these variations are related to different evaporation rates of the microdroplets. Our study also suggests that the dissociation of solvent molecules from Cu-BDC under sub-ambient pressure increases open metal sites which lead to a strong CO2 binding affinity. This work illustrates the synthesis of MOFs under substantially low operating pressures and offers many new opportunities in the controlled synthesis of MOFs for selective gas adsorption.

Efficiency improvement of Roots vacuum pump working process

SummaryIn this paper influence of rotor profile type of Roots vacuum pumps on backward leakage through channels of rotor mechanism and pumping speed is considered.Elliptical, involute, and circular profiles are widely used. Backward leakage through channels of rotor mechanism with different rotor profiles is determined from conductance which was calculated by angular coefficients method for free molecular flow regime at different rotors rotating angles. It is shown that backward leakage is maximal at rotors rotating angles 25–45° due to maximal inter‐rotor channel opening for gas flow and is minimal at 0°. Elliptical rotor profile provides minimal backward leakage over one revolution.Numerical modeling of Roots pump working process was carried out on the basis of CFD ANSYS‐CFX complex by dynamic meshing method. SST turbulence model was used for calculations. It is confirmed that maximal pumping speed is provided by Roots pump with elliptical rotor profile.

Towards photophoresis with the generalized Lorenz-Mie theory

Based on the adjoint boundary value problem proposed decades ago by Zulehner and Rohatschek [1], analytic and closed-form expressions for the photophoretic forces exerted by arbitrary-shaped beams on homogeneous and low-loss spherical particles is derived in both the free molecular and slip flow regimes. To do so, the asymmetry vector for arbitrary refractive index particles is explicitly calculated by expanding the internal electromagnetic fields with the aid of the generalized Lorenz-Mie theory (GLMT). The approach here proposed is, to the best of the authors’ knowledge, the first systematic attempt to incorporate the GLMT stricto sensu into the field of photophoresis and might as well be extended, e.g. to spheroids and find important applications, among others, in optical trapping and manipulation of microparticles, in geoengineering, particle levitation, optical trap displays and so on.

Aerothermodynamics of a sphere in a monatomic gas based onab initiointeratomic potentials over a wide range of gas rarefaction: transonic, supersonic and hypersonic flows

Aerothermodynamic characteristics of a sphere such as drag and energy transfer coefficients are calculated for Mach numbers varying from 1 to 10 over a wide range of the gas rarefaction degree spanning the free molecular, transitional and near hydrodynamic flow regimes. The effects of major factors determining the gas flow are studied using the direct simulation Monte Carlo method. To reveal the effect of gas species, the simulations are performed based onab initiointeratomic potentials for helium, neon, argon and krypton, as well as based on the hard sphere model. The impact of the accommodation coefficients is evaluated by applying the Cercignani–Lampis model of gas–surface interaction. The calculations are carried out for several values of the free stream and sphere temperatures. It is found that the effects of gas species on the drag and energy transfer coefficients are approximately 3 % and 6 %, respectively. The accommodation coefficients in the Cercignani–Lampis model strongly affect all aerothermodynamic characteristics. The drag and energy transfer coefficients calculated for different accommodation coefficients vary within 30 % and 200 %, respectively. It is found that the variation of the tangential momentum and normal energy accommodation coefficients can induce an increase of the drag coefficient compared to the case of diffuse gas–surface interaction. In hypersonic flows, the drag coefficient varies within 30 % when the sphere temperature varies from the free stream temperature to the stagnation temperature. The drag and energy transfer coefficients are found to be non-monotonic functions of the free stream temperature.

Low-pressure gas flow properties of sintered stainless steel microporous media

The gas flows through the low permeable porous media are of great interest, especially in vacuum technology for filtering, separation processes, protection, and flow control. They can combine a high mass flow rate and a high level of rarefaction. The transient experimental technique, developed previously for the mass flow rate measurements through the microchannels [Rojas Cardenas et al., Phys. Fluids 23, 031702 (2011)], is generalized to obtain the permeability and conductance directly from the pressure variation measurements. The present experimental methodology, allowing for step by step data verification, leads to higher accuracy than the similar and commonly used method such as “pulse-decay” techniques [W. F. Brace and R. J. Martin, Int. J. Rock Mech. Min. Sci. 5, 415–426 (1968)]. The measured data are fitted according to the exponential function with the pressure relaxation time as a single fitting parameter. The permeability and conductance of the sintered stainless steel microporous medium are measured for different gases. By using the methodology proposed by Johansson et al. [Int. J. Heat Mass Transfer 142, 118412 (2019)], the characteristic pore size of the sample was determined to be equal to 1.5 micrometers. It was found that free molecular flow constant conductance is achieved at a pressure below 30 mbar for helium. This property makes this kind of porous media particularly suitable as a leak element by taking advantage of the constancy of conductance in the free molecular flow regime, for example, for calibration of ionization gauges or mass spectrometers [Yoshida et al., Measurement 45(10), 2452–2455 (2012)].