Free Molecular Regime Research Articles

Computing the aerodynamic drag of fractal aggregates in free-molecular and transition regimes

For fine particles moving in the gas different regimes of aerodynamic drag are distinguished depending on their sizes and dust to gas relative velocities. In the Epstein or free-molecular regime, the drag force depends on the projected area or cross-section of the body, and in the Stokes or transition regime, on its linear size. Finding the linear size and the projected area for nonspherical particles is a non-trivial task. To describe the mobility of some type of nonspherical particles - fluffy aggregates, considered as a set of spheres - monomers, the value Df called fractal dimension is often used. For such aggregates with fixed fractal dimension D0, several authors suggested the approximations of the linear size (called Smoluchowski radius Rs) and projected area PA as a function of N - the number of monomers in the aggregate. These authors validated their approximations on experimental data. On the other hand, new direct numerical simulation (DNS) data on mobility of fractal aggregates have been obtained recently. In the paper we constructed new functions PA(Df,N) and Rs(Df, N) interpolating available from the literature approximations of PA(Df = D0,N) and Rs(Df = D0,N) and minimizing the deviation from recent DNS data. These functions are designed for global simulations of protoplanetary discs dynamics and planet formation, but can be used in different applications.

Membrane sandwich squeeze film pressure sensors

Squeeze film pressure sensors exploit the dynamical modification of the mechanical properties of oscillating elements due to the compression of a fluid in a small gap region and allow for direct and absolute pressure measurements. This Tutorial article discusses the working principles of membrane sandwich squeeze film pressure sensors—i.e., sensors comprised of a parallel arrangement of two large-area, ultrathin suspended films forming a few-micrometer gap, and immersed in a fluid—and focuses on their operation in the free molecular flow regime. The effects of gas pressure on the vibrations of the membrane resonators and their coupled dynamics are discussed in general terms before recent experimental implementations using high tensile stress silicon nitride membranes are presented.

The effect of a solid boundary on the propagation of thermodynamic disturbances in a rarefied gas

We study the effect of a rigid boundary on the propagation of thermodynamic disturbances in a gas under non-continuum conditions. We consider a semi-infinite setup confined by an infinite planar wall and introduce initial gas disturbances in the form of density and temperature inhomogeneities. The problem is formulated for arbitrary small-amplitude perturbations and analyzed in the entire range of gas rarefaction rates, governed by the Knudsen (Kn) number. Our results describe the system relaxation to equilibrium, with specific emphasis on the effect of the solid surface. Analytical solutions are obtained in the free-molecular and near-continuum (based on the Navier–Stokes–Fourier and regularized 13 moment equations) regimes and compared with direct simulation Monte Carlo results. The impact of the solid wall is highlighted by comparing between diffuse (adiabatic or isothermal) and specular boundary reflections. Focusing on a case of an initial temperature disturbance, the results indicate that the system relaxation time shortens with increasing Kn. The isothermal boundary consistently reverberates the weakest acoustic disturbance, as the energy carried by the impinging wave is partially absorbed by the surface. The specular and adiabatic wall systems exhibit identical responses in the continuum limit while departing with increasing Kn due to higher-order moment effects. The unsteady normal force exerted by the gas on the surface is quantified and analyzed.

Scattering Kernel Kinetic Approach of Small Particles Transport in a Free Molecular Regime in Gaseous Media

We investigated through the scattering kernel kinetic concept the transport of small particles in gaseous media in a free molecular regime. The scattering kernel is generally used to provide gas-surface interaction boundary conditions for Boltzmann kinetic equation in flow problems (Maxwell, Phil. Trans. R. Soc. 170, 231, 1879; Sharipov, 2001). We developed a general expression for the transport force explicitly in terms of the scattering kernel and of gas molecule velocity distribution function. From the general expression, first, we could use the first-order Chapman-Enskog solution for the distribution function of gas molecules f(r, v, t) which shows three types of transportation mechanisms, or forces, such that F = FD+FT+FS. Where FD is the drag force on the particle due to its relative motion in the gas, FT is the thermophoretic force due to a gradient of temperature in the gas, and FS is due to shear forces on the particle. Second, it is possible to use any of the several scattering kernel models published in the literature. The traditional simple reflection kernels, specular, diffuse, and Maxwell, which comprises particular cases of the generalized force, are limited and do not fully describe the real reflection mode of molecule-particle interaction. So, we used the general (Struchtrup, Phys. Fluids. 25, 112001, 2013) reflection kernel model to study the drag force on small particles. The results were compared with available experimental data. We concluded that it is necessary to explicitly include impact velocity and particle surface rugosity effects which are predicted by the Struchtrup reflection kernel thermal activation model of molecule-particle interaction.

DSMC investigation of rarefied gas flow over a 2D forward-facing step: Effect of Knudsen number

A Forward-Facing Step (FFS) can be approximated as a simplified geometry for modeling the irregularities around space vehicles, the flow around which is often rarefied. These irregularities lead to flow separation, circulation, and re-attachment. This study aims to examine one of the fundamental surface irregularities that can have a significant effect on the design of space vehicles. In this article, Direct Simulation Monte Carlo (DSMC), an effective method for studying rarefied flows, is used to investigate the role of rarefaction on the non-reacting hypersonic rarefied gas flow over a forward-facing step. A comprehensive analysis of the effects of Knudsen number on the on flow-field properties, i.e., velocity, pressure, temperature, density, and surface properties, i.e., pressure coefficient, skin friction coefficient, and heat transfer coefficient is performed. The selected Knudsen number range from 0.05 to 21.33, covering the various rarefaction regimes. The Mach number of the free-stream flow employed was 25. The sections in the vicinity and downstream of the step are examined in-depth, describing the predominant nonequilibrium aspects. The normalized flow properties increased with Knudsen number and are primarily affected by the interplay of compressibility and viscous dissipation effects. Flow recirculation occurred in the slip and transition regime, whereas it was absent in the free-molecular regime; moreover, the recirculation lengths decreased with rarefaction. The surface properties were found to increase with Knudsen number with peak magnitudes located on the upstream and frontal face of the step.

Photophoresis in the slip-flow and free molecular regimes for arbitrary-index particles

Photophoresis in the slip-flow and free molecular regimes under plane wave illumination is investigated in terms of the asymmetry factor for non-volatile homogeneous low-loss spherical particles with an arbitrary index of refraction. An analytic and closed-form expression for this important factor is derived which can be used to predict photophoretic forces exerted on microparticles of arbitrary size parameters, therefore in any optical (Rayleigh, Mie or geometric) regime. Examples are given in terms of the behavior of both the asymmetry factor and the internal field distribution, showing differences between photophoresis of dielectric, negative index and magnetodielectric particles.

Drag and thermophoresis on a sphere in a rarefied gas based on the Cercignani–Lampis model of gas–surface interaction

Abstract

Effect of Particle Size on Boron Combustion in Air

Mathematical modeling of combustion of micron-sized and nanometer boron particles in air with allowance for changes in the heat and mass transfer mechanism and a decreasing particle size is carried out. The Knudsen number is taken as an indicator of a transition from one state to another: a continuous medium assumption is valid for describing the heat and mass transfer mechanisms in the case where Kn $$<$$ 0.01, the free molecular regime takes place for Kn $$>$$ 10, and a transient state occurs for 0.01 $$<$$ Kn $$<$$ 10. Particle sizes at which different heat and mass transfer regimes occur with respect to boron combustion in air at pressures of 0.1–4 MPa are estimated. The time it takes for boron particles to combust within the framework of the assumption of a continuous medium and in a free molecular mode is determined. It is shown that calculation models for determining the burning time of boron particles with initial sizes close to micron-sized and nanodispersed ones should take into account a change in the heat and mass transfer mechanism with variation in the current particle radius during burnout.

Thermal transpiration in molecular gas

The thermal transpiration of molecular gas is investigated based on the model of Wu et al. [“A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases,” J. Fluid Mech. 763, 24–50 (2015)], which is solved by a synthetic iterative scheme efficiently and accurately. A detailed investigation of the thermal slip coefficient, Knudsen layer function, and mass flow rate for molecular gas interacting with the inverse power-law potential is performed. It is found that (i) the thermal slip coefficient and Knudsen layer function increase with the viscosity index determined by the intermolecular potential. Therefore, at small Knudsen number, gas with a larger viscosity index has a larger mass flow rate; however, at late transition and free molecular flow regimes, this is reversed. (ii) The thermal slip coefficient is a linear function of the accommodation coefficient in Maxwell’s diffuse–specular boundary condition, while its variation with the tangential momentum accommodation coefficient is complicated in Cercignani–Lampis’s boundary condition. (iii) The ratio of the thermal slip coefficients between monatomic and molecular gases is roughly the ratio of their translational Eucken factors, and thus, molecular gas always has a lower normalized mass flow rate than monatomic gas. (iv) In the transition flow regime, the translational Eucken factor continues to affect the mass flow rate of thermal transpiration, but in the free molecular flow regime, the mass flow rate converges to that of monatomic gas. Based on these results, accommodation coefficients were extracted from thermal transpiration experiments of air and carbon dioxide, which are found to be 0.9 and 0.85, respectively, rather than unity used in the literature. The methodology and data presented in this paper are useful, e.g., in the pressure correction of capacitance diaphragm gauge when measuring low gas pressures.

Simplified unified wave-particle method with quantified model-competition mechanism for numerical calculation of multiscale flows.

A quantified model-competition (QMC) mechanism for multiscale flows is extracted from the integral (analytical) solution of the Boltzmann-BGK model equation. In the QMC mechanism, the weight of the rarefied model and the weight of the continuum (aerodynamic and hydrodynamic) model are quantified. Then, a simplified unified wave-particle method (SUWP) is constructed based on the QMC mechanism. In the SUWP, the stochastic particle method and the continuum Navier-Stokes method are combined together. Their weights are determined by the QMC mechanism quantitatively in every discrete cell of the computational domain. The validity and accuracy of the present numerical method are examined using a series of test cases including the high nonequilibrium shock wave structure case, the unsteady Sod shock-tube case with a wide range of Kn number, the hypersonic flow around the circular cylinder from the free-molecular regime to the near continuum regime, and the viscous boundary layer case. In the construction process of the present method, an antidissipation effect in the continuum mechanism is also discussed.

Efficient Method of Moments for Simulating Atmospheric Aerosol Growth: Model Description, Verification, and Application

AbstractThe atmospheric aerosol dynamics model (AADM) has been widely used in both comprehensive air quality model systems and chemical transport modeling globally. The AADM consists of Smoluchowski's coagulation equation (SCE), whose solution undergoing Brownian coagulation in the free molecular regime is a challenge because it is inconsistent with aerosols whose size distribution cannot exactly follow the lognormal size distribution. Thus, a new method for solving the SCE without assuming lognormal size distribution is proposed and developed. The underlying principle of this method is that the hybridization of the well‐established method of moments with the assumed lognormal size distribution (log MOM) and Taylor‐series expansion method of moments (TEMOM) is implemented. This method shows excellent agreement with the sectional method (SM) which is used as reference. The accuracy of these two specific models closely approaches that of the TEMOM, but overcomes the limitation of the classical log MOM. The computational time of this scheme is considerably lower than that of the SM. The new method was successfully implemented to reveal the formation and growth of secondary particles emitted from a vehicle exhaust tailpipe. It was found that the formation of new particles only occurs in the interface region of the turbulent exhaust jet (which is very close to the tailpipe exit), whereas no new particles are formed in the mixture of the exhaust jet plume and the surrounding cold air downstream. The new method is verified as an efficient and reliable numerical scheme for studying atmospheric aerosol dynamics.

Numerical analysis of flows past two parallel flat plates in the free-molecular regime

A numerical investigation is presented for a free-molecular flow past two parallel circular flat plates using the test particle Monte Carlo (TPMC) method, with flow shadowing and multiple reflections analyzed. Flow shadowing and multiple reflections have a significant effect on aerodynamic forces, and the distance-to-radius ratio of the two plates has a considerable influence on flow shadowing and multiple reflections. As the distance-to-radius ratio increases, flow shadowing and multiple reflections become weaker. Different distance-to-radius ratios result in different surface pressure distributions.

Mathematical modelling of heat transfer between two non-coaxial cylinders under the effect a constant pressure gradient

The article deals with an isothermal flow of a rarefied gas in the channel, formed by two non-coaxial cylinders. An analytic solution of the Williams equation, describing this flow, has been constructed using the kinetic approach. For the boundary condition on the channel walls, the diffusion model is used. The distribution function for various values of the ratio of radii of cylinders and Knudsen number is constructed. The dependence of the flow on the diameter of the inner cylinder and its position in the duct is investigated. The value of the heat flux through the cross-section of the channel is calculated. The analysis of the obtained results during the transition to free molecular and hydrodynamic regimes is done.

Binary gas mixture outflow into vacuum through an orifice

The results of a numerical study of a binary gas mixture outflow from a source with specified stagnation parameters into vacuum through a round orifice are presented. Silver and helium atoms (with a mass ratio of 26.95) are selected as a mixture species. The near free-molecular, transitional, and near-continuum regimes of the flow are considered with the direct simulation Monte Carlo method used for the computations. The results of simulations show that the rarefaction degree and the mole composition of the mixture have a significant impact on the spatial variation of flow parameters and the flow rates through the orifice. At all degrees of rarefaction, the variation in the dimensionless flow rate (related to the free-molecular flow rate) of the mixture is a non-monotonic function of the mole fraction of the species with a maximum/minimum (for the mass/particle flux). The presence of a light carrier gas (helium) leads to the acceleration, axial focusing, and increase in the flow rate of the heavy species (silver). The velocity slip of light and heavy species is observed at all degrees of rarefaction under consideration. The effect of the increasing density of heavy species near the orifice plane is revealed. The spatial variation of mole fractions of species on the degree of rarefaction is studied. The results of the study are compared to available analytical and experimental data, and the simulation results of pure gas outflow obtained by other authors.

A parallelized tool to calculate the electrical mobility of charged aerosol nanoparticles and ions in the gas phase

Electrical Mobility is a transport property that describes a particle behavior in the gas phase. When dealing with the free molecular regime, ascertaining the shape of a nanoparticle or an ion directly from measurements of mobility becomes quite difficult as the particle no longer can be assumed to have spherical shape. Here we propose an efficient parallelized tool, IMoS, that makes use of all-atom models to calculate the mobility of nanoparticles in a variety of gases. The program allows for different types of calculations that range from the efficient Projection Approximation (PA) algorithm to the 4-6-12 Lennard-Jones potential Trajectory Method. It also includes a diffuse inelastic simulation that achieves Millikan's predicted 1.36 value over PA. When compared to experimental results, the error of the most efficient calculations is shown to be approximately 2–4% on average.

Rarefied gas flow through a rough channel into a vacuum

Rarefied gas flow through a rough channel of finite length into a vacuum was investigated using the direct simulation Monte Carlo method. The non-equilibrium effects at the input and output of the channel were considered by including certain pre-and post-channel regions into the geometry under consideration. The mass flow rate through a short and long channel was computed in a wide range of gas rarefaction from the free molecular regime to the hydrodynamic near. It is shown that the noticeable effect of surface roughness on flow rate is manifested in the free molecular and transition regimes. The analysis is provided for the flow field inside the channel as well as in the upstream and downstream regions. The results obtained are consistent with the theoretical, numerical, and experimental data available in the open literature.

Development of an ultraviolet constant angle Mie scattering detector toward the determination of aerosol growth kinetics in the transition and free molecular regime

The measurement of aerosol growth kinetics at ever smaller sizes toward the transition and free molecular regime is of interest to provide for validation of theoretical predictions. Such measuremen...

Modeling of Cellular Detonation in Gas Suspensions of Two Fractions of Aluminum Nanoparticles

Cellular detonation in suspensions of two fractions of submicron and nano-sized aluminum particles is studied by numerical methods. Approaches of mechanics of heterogeneous media are used. The transition from the continuum to free-molecular regime of the flow around the particles is taken into account in the processes of interphase interaction. Particle combustion is described within the framework of the previously developed model of reduced kinetics. The results calculated for two-dimensional flows in a plane channel reveal significant differences from suspensions of two fractions of microparticles, where degeneration of transverse waves is observed. In the present study, weakly regular and irregular cellular structures are formed, which is also confirmed theoretically on the basis of the acoustic analysis based on the method proposed by Barthel.

Measurement of Heat Transfer from Anodic Oxide Film on Aluminum in High Knudsen Number Flows.

The heat transfer in vacuum depends on the gas–surface interaction. In this study, the heat flux from anodic oxide films on aluminum with different anodizing times through a gas confined between two surfaces with different temperatures was studied. We prepared a non-treated surface, a surface with a normal anodizing time of 30 min, and a surface with 90 min, where the formed film would partially dissolve by long time exposure to the solution. The formation of the films was checked by electrical resistance. Scanning electron microscope (SEM) images were obtained for the three sample surfaces. Even though it was difficult to observe the hexagonal cylindrical cell structures on anodic oxide films, the 30 min sample surface was shown to be rough, and it was relatively smooth and powdery for the 90 min sample surface. The heat fluxes from three sample surfaces were measured from the free-molecular to near free-molecular flow regimes, and analyzed to obtain the energy accommodation coefficients. The heat fluxes were well fitted by the fitting curves. The energy accommodation coefficients for both helium and argon increased by anodizing an aluminum sample surface, while they decreased with increasing the anodizing time up to 90 min indicating the dissolution of the film.

The role of molecule-surface interaction in the diffusiophoresis of a fine aerosol particle

The role of the interaction of gas molecules with a surface on the diffusiophoretic force and velocity of a spherical aerosol particle is studied under the conditions of a free molecular regime. Expressions of these physical quantities are obtained for the arbitrary scattering kernels of different species of molecules. Calculations of the force and velocity of diffusiophoresis are carried out with the use of different scattering kernel models.