Small Knudsen Numbers Research Articles

On the order reduction of semi-Lagrangian methods for BGK model of Boltzmann equation

In the classical literature high order semi-Lagrangian schemes proposed for solving the BGK model of Boltzmann equation suffer from the phenomenon of order reduction when the Knudsen number becomes zero, i.e. in the fluid regime. In this brief note we provide local estimates with respect to small Knudsen number, which enables us to understand the order reduction phenomenon.

GKS and UGKS for High-Speed Flows

The gas-kinetic scheme (GKS) and the unified gas-kinetic scheme (UGKS) are numerical methods based on the gas-kinetic theory, which have been widely used in the numerical simulations of high-speed and non-equilibrium flows. Both methods employ a multiscale flux function constructed from the integral solutions of kinetic equations to describe the local evolution process of particles’ free transport and collision. The accumulating effect of particles’ collision during transport process within a time step is used in the construction of the schemes, and the intrinsic simulating flow physics in the schemes depends on the ratio of the particle collision time and the time step, i.e., the so-called cell’s Knudsen number. With the initial distribution function reconstructed from the Chapman–Enskog expansion, the GKS can recover the Navier–Stokes solutions in the continuum regime at a small Knudsen number, and gain multi-dimensional properties by taking into account both normal and tangential flow variations in the flux function. By employing a discrete velocity distribution function, the UGKS can capture highly non-equilibrium physics, and is capable of simulating continuum and rarefied flow in all Knudsen number regimes. For high-speed non-equilibrium flow simulation, the real gas effects should be considered, and the computational efficiency and robustness of the schemes are the great challenges. Therefore, many efforts have been made to improve the validity and reliability of the GKS and UGKS in both the physical modeling and numerical techniques. In this paper, we give a review of the development of the GKS and UGKS in the past decades, such as physical modeling of a diatomic gas with molecular rotation and vibration at high temperature, plasma physics, computational techniques including implicit and multigrid acceleration, memory reduction methods, and wave–particle adaptation.

Experimental study on the conductance of pure and binary gas mixtures

A binary gas mixture conductance measurement apparatus with an independent gas distribution system was established, the constant volume method to measure the conductance of the mixed gas was used. Also, the conductance of pure gas Ar, He, N2 and binary gas mixtures Ar–He, N2–He, Ar–N2 through a short tube under different gas distribution chamber pressures and different molar fractions were measured. Meanwhile, the equivalent single gas method was used to calculate the conductance of binary gas mixtures. To verify the accuracy of this method, the relative error between the calculated and the measurement was carried out. The result shows that the equivalent single gas method is suitable for a small Knudsen number (1/Kn≥10), the molar fraction of lighter species is low and a small difference in the molecular mass of the components in the mixed gas. The uncertainty of conductance measurement is better than 3.1%.

Pore-scale study of non-ideal gas dynamics under tight confinement considering rarefaction, denseness and molecular interactions

Non-ideal gas flow behaviors are investigated by an Enskog-Vlasov type kinetic model considering the simultaneous effects of gas molecule size (volume exclusion) and long-range intermolecular attractions at the molecular level, which corresponds to the real gas equation of state at the macroscopic level. The Knudsen minimum is captured and a local Knudsen maximum may appear if gas molecule sizes or intermolecular attractive forces are considered. Although the Boltzmann equation is applicable to all the flow regimes in rarefied gas dynamics, it is invalid for a dense gas system, such as a tight or a shale gas reservoir. The Boltzmann-BGK model and the Enskog-BGK model overestimate and underestimate the mass flow rate of real gases, respectively, while Guo's model is more accurate to investigate real gas dynamics under tight confinements from a physical perspective. As the channel width increases or the solid fraction decreases, the impact of intermolecular interactions reduces. An anomalous slip regime occurs if both the volume exclusion and long-range intermolecular attraction are considered. Although the rarefaction effect is more prominent at larger Knudsen numbers, the flow at a smaller Knudsen number (a larger solid fraction or channel width) contributes to more practical gas production. The effect of long-range intermolecular attractions on gas dynamics decreases with temperature for real gases. This paper provides a novel insight into the real gas flow characteristics in a dense gas system, such as tight and shale gas reservoirs, from molecular natures of fluids.

Transasymptotics and hydrodynamization of the Fokker-Planck equation for gluons

We investigate the non-linear transport processes and hydrodynamization of a system of gluons undergoing longitudinal boost-invariant expansion. The dynamics is described within the framework of the Boltzmann equation in the small-angle approximation. The kinetic equations for a suitable set of moments of the one-particle distribution function are derived. By investigating the stability and asymptotic resurgent properties of this dynamical system, we demonstrate, that its solutions exhibit a rather different behavior for large (UV) and small (IR) effective Knudsen numbers. Close to the forward attractor in the IR regime the constitutive relations of each moment can be written as a multiparameter transseries. This resummation scheme allows us to extend the definition of a transport coefficient to the non-equilibrium regime naturally. Each transport coefficient is renormalized by the non-perturbative contributions of the non-hydrodynamic modes. The Knudsen number dependence of the transport coefficient is governed by the corresponding renormalization group flow equation. An interesting feature of the Yang-Mills plasma in this regime is that it exhibits transient non-Newtonian behavior while hydrodynamizing. In the UV regime the solution for the moments can be written as a power-law asymptotic series with a finite radius of convergence. We show that radius of convergence of the UV perturbative expansion grows linearly as a function of the shear viscosity to entropy density ratio. Finally, we compare the universal properties in the pullback and forward attracting regions to other kinetic models including the relaxation time approximation and the effective kinetic Arnold-Moore-Yaffe (AMY) theory.

Onset of thermal convection of a weakly rarefied Maxwellian gas: A continuum-slip approach

The onset of the Rayleigh–Bénard convection for a monatomic rarefied gas of Maxwellian type is investigated at small Knudsen numbers. Density variations due to compressibility have been considered without any restriction imposed on the magnitude of temperature difference. The steady-state conduction problem has been solved both analytically and numerically using a continuum approach with velocity-slip and temperature-jump conditions at the solid boundaries. The influence of the Froude number and the Knudsen number is comprehensively studied. To examine the onset of thermal convection, a linear temporal stability analysis is performed for the conduction state and the dispersion relation is calculated by a Chebyshev collocation method. The neutral stability curve obtained in the Froude–Knudsen number plane marks transition to convection from a pure conduction state. The critical wave number for the onset of convection is found to be close to 3.14, agreeing well with the existing findings. The convection only occurs for Knudsen numbers smaller than 0.027. A comparison between the Maxwellian model and the hard-sphere model is given and discussed. The encouraging results suggest the use of more realistic gas models in future studies.

Langevin Dynamics Calculation of Brownian Coagulation Coefficient for Spherical Equal-size Aerosol Particles in Transient Regime

Coagulation coefficient of aerosol particles due to Brownian motion is an important issue to describe change in particle size distribution. Motion of aerosol particles is diffusive in continuous region (small Knudsen number; Kn), or like free molecular motion of gaseous molecular in free molecular region (large Kn). Fuchs (1964) presented an expression of coagulation coefficient in transition regime by a so-called “Flux Matching” method. In his method, transportation of particles inside of the “limiting sphere” is assumed to be like free molecular, or diffusive outside of the sphere. These days, some researchers presented coagulation coefficient of aerosol particles by direct calculation of motion of aerosol particles. They employed Langevin dynamics equation to represent the stochastic motion of aerosol particles. In this study, we developed new model to calculate the coagulation coefficient. Our model employed spherical calculation space in which one scavenging particle is in the center of it: the calculation sphere moves together with the motion of the scavenging particle. The coagulation coefficient can be calculated from the mean time between collisions and the concentration of collision particles. By using the above numerical model, we have calculated the coagulation coefficient of spherical particles of from 4 nm to 100 nm in diameter.

Modeling and computation for non-equilibrium gas dynamics: Beyond single relaxation time kinetic models

Many kinetic relaxation models have been proposed for the study of rarefied flows. Based on the single relaxation time model, a discrete velocity method-based unified gas-kinetic scheme (UGKS) has been constructed. The UGKS models the gas dynamics on the discretized space directly on account of accumulating flow evolution from particle transport and collision within a time step. Under the UGKS framework, a unified gas-kinetic wave-particle (UGKWP) method has been further developed for non-equilibrium flow simulation, where the time evolution of the gas distribution function is composed of analytical wave and individual particles. In the highly rarefied regime, the flow evolution is mainly described by the particle transport and collision. Because of the use of single relaxation time for particle collision, there is a noticeable discrepancy between the UGKWP solution and the full Boltzmann or direct simulation Monte Carlo (DSMC) result, such as the temperature distribution inside a shock layer at high Mach numbers. In this Letter, a modification of the particle collision time according to the particle velocity will be implemented in the UGKWP. As a result, the new model greatly improves the performance of the UGKWP in the capturing of non-equilibrium flows. There is an excellent match between UGKWP and DSMC or Boltzmann solution in the highly rarefied regime. In the continuum flow regime, due to the absence of particles, the modification of the particle collision time will not take effect and the UGKWP will get back to the hydrodynamic Navier–Stokes flow solver with correct dissipative coefficients at small cell Knudsen numbers.

Gas mixture flow, diffusion, and heat transfer in a long tube at moderately small Knudsen numbers

The binary gas mixture flow, diffusion, and heat transfer through a long tube in the near-continuum regime (moderately small Knudsen numbers) are analyzed. The system of linearized third-order moment equations, obtained by Grad’s method, is used. An expression for the total mass flux of a binary gas mixture is deduced by using the extension of the procedure, proposed in the work by Zhdanov [“Slip and barodiffusion phenomena in slow flows of a gas mixture,” Phys. Rev. E 95, 033106 (2017)] for the study of slip phenomena in slow flows of a gas mixture. Relations for diffusion and heat fluxes are determined from the initial system of moment equations, averaged over the channel cross section, and supplied with several moments of the distribution function at the channel wall found with the modified Maxwell method. Analytical formulas for kinetic coefficients of the Onsager matrix, which connect averaged fluxes and gradients of the corresponding thermodynamic quantities, are obtained. It is shown that the employed approach automatically ensures the validity of the reciprocal relations for the cross terms in the Onsager matrix. The results of calculations of the derived kinetic coefficients for several binary mixtures of noble gases (He–Ar, Ne–Ar, and He–Xe) are presented and compared with numerical data found by the discrete velocity method on the basis of the linearized Boltzmann equation with the McCormack model.

High Order Conservative Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation

In this paper, we present a conservative semi-Lagrangian finite-difference scheme for the BGK model. Classical semi-Lagrangian finite difference schemes, coupled with an L-stable treatment of the collision term, allow large time steps, for all the range of Knudsen number. Unfortunately, however, such schemes are not conservative. There are two main sources of lack of conservation. First, when using classical continuous Maxwellian, conservation error is negligible only if velocity space is resolved with sufficiently large number of grid points. However, for a small number of grids in velocity space such error is not negligible, because the parameters of the Maxwellian do not coincide with the discrete moments. Secondly, the non-linear reconstruction used to prevent oscillations destroys the translation invariance which is at the basis of the conservation properties of the scheme. As a consequence the schemes show a wrong shock speed in the limit of small Knudsen number. To treat the first problem and ensure machine precision conservation of mass, momentum and energy with a relatively small number of velocity grid points, we replace the continuous Maxwellian with the discrete Maxwellian introduced by Mieussens. The second problem is treated by implementing a conservative correction procedure based on the flux difference form. In this way we can construct a conservative semi-Lagrangian scheme which is Asymptotic Preserving (AP) for the underlying Euler limit, as the Knudsen number vanishes. The effectiveness of the proposed scheme is demonstrated by extensive numerical tests.

Surface Roughness Effects on Molecular Gas Lubrication

The effects of surface roughness on the molecular gas lubrication characteristics have been studied and it has become clear that the transverse flow, which is neglected in conventional molecular gas lubrication (MGL) equation, appears in the lubrication film as the roughness slope increases. In this paper, the detail of the transverse flow in the wide range of Knudsen number was analyzed by using direct simulation Monte-Carlo (DSMC) method. It was clarified that the transverse flow is caused by the slip flow on the slope of rough surface．The transverse flow exists only around the rough surface in the case of small Knudsen number, though the flow extends in whole area of lubrication film against large Knudsen number. The flow through the lubrication film complementary decreases as the transverse flow increases.

Modeling of gas transport in porous medium: Stochastic simulation of the Knudsen gas and a kinetic model with homogeneous scatterer

Mass transport of the Knudsen gas in a porous medium is investigated on the basis of the kinetic theory of gases. First, the mass flow conductance is computed numerically for various porosities and solid grain sizes by stochastic particle simulations (SPS). Then, a kinetic model with a homogeneous scatterer is introduced, which contains the reference Knudsen number as the sole parameter that characterizes the collision frequency of gas molecules with the micro-structural solid surface. With the aid of the standard asymptotic analyses for small and large Knudsen numbers combined with the percolation theory, the effective reference Knudsen number is identified to reproduce the SPS results for a wide range of porosities.

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.

DSMC study of the radiometric force acting on a thin plate with surface temperatures much higher than the environment temperature

Radiometric plates have potential applications in aerospace science and technology. The radiometric force acting on a circular thin plate with surface temperatures much higher than the environment was studied with the aid of direct simulation Monte Carlo (DSMC). Dimensional analysis showed that the scaled radiometric force depended on the scaled temperatures of both sides of the plate and the Knudsen number. Statistical analysis of the gas–plate interactions demonstrated that the gas around the plate was in a thermal non-equilibrium state. The radiometric force acting on the plate could change its direction along the radius for small Knudsen numbers, which was thought to be a combined effect of rarefied and continuum flow. Furthermore, a dimensionless expression was proposed for the radiometric force for a wide a range of surface temperatures and for radius-based Knudsen numbers equal to or larger than 0.05. The radiometric force varied non-linearly with the temperature difference, and dependence of the radiometric force on the perimeter and the mean free path for small Knudsen numbers was between those predicted by the theories of Einstein and Sexl. This work improves the understanding of radiometric flow at small Knudsen numbers and can be applied to aerospace technology.

Stationary Non equilibrium States in Kinetic Theory

Stationary non equilibrium solutions to the Boltzmann equation, despite their relevance in applications, are much less studied than time dependent solutions, and no general existence theory is yet available, due to serious technical difficulties. Here we review some results on the construction of stationary non equilibrium solutions, in a general domain in contact with a slightly non-homogeneous thermal reservoir, both for finite and small Knudsen number. We will describe different approaches and different techniques developed. The main focus will be on stationary solutions close to hydrodynamics. In particular, we will give an answer to the longstanding open problem of the rigorous derivation of the steady incompressible Navier–Stokes–Fourier system from the Boltzmann theory, in the presence of a small external force and diffuse boundary condition with small boundary temperature variations.

Acoustic wave propagation at nonadiabatic conditions: The continuum limit of a thin acoustic layer

Existing studies on sound propagation in rarefied gases are extended to consider wave transmission at nonadiabatic ambient conditions, where arbitrarily large reference temperature and density gradients prevail. Asymptotic analysis of the acoustic field is carried out in the limit of small Knudsen numbers and high actuation frequencies, for both mechanical and thermal wall excitations.

Research on Static and Dynamic Characteristics of Gas-lubricated Microbearings for Microfluidic Devices

Aerodynamic journal microbearings with microtribological phenomena significantly influence the operating stability of microfluidic devices. The modified Reynolds equations including different rarefaction models are derived and are solved by the partial derivative method and relaxation iteration algorithm. The effects of Knudsen number and bearing parameters on the static and dynamic characteristics of microbearings are investigated in detail. The results show that the rarefaction effect plays a crucial role in the ultra-thin gas film lubrication. The maximum film pressure of Fukui–Kaneko (FK) model is lowest and the result in Boltzmann model is largest in small Knudsen number regions. As the Knudsen number increases further, the curve of FK model is coincident with that of the Boltzmann model in the transition regime. The direct stiffness coefficients of Boltzmann model increase with the increase of eccentricity ratio and aspect ratio, whereas the effect of Knudsen number on the damping coefficients yields a relatively complicated trend.

Generalized Multiscale Finite Element Method for the Steady State Linear Boltzmann Equation

The Boltzmann equation, as a model equation in statistical mechanics, is used to describe the statistical behavior of a large number of particles driven by the same physics laws. Depending on the media and the particles to be modeled, the equation has slightly different forms. In this article, we investigate a model Boltzmann equation with highly oscillatory media in the small Knudsen number regime, and study the numerical behavior of the Generalized Multi-scale Finite Element Method (GMsFEM) in the fluid regime when high oscillation in the media presents. The Generalized Multi-scale Finite Element Method (GMsFEM) is a general approach to numerically treat equations with multi-scale structures. The method is divided into the offline and online steps. In the offline step, basis functions are prepared from a snapshot space via a well-designed generalized eigenvalue problem (GEP), and these basis functions are then utilized to patch up for a solution through DG formulation in the online step to incorporate specific boundary and source information. We prove the wellposedness of the method on the Boltzmann equation, and show that the GEP formulation provides a set of optimal basis functions that achieve spectral convergence. Such convergence is independent of the oscillation in the media, or the smallness of the Knudsen number, making it one of the few methods that simultaneously achieve numerical homogenization and asymptotic preserving properties across all scales of oscillations and the Knudsen number.

High-accuracy hull iteration method for uncertainty propagation in fluid-conveying carbon nanotube system under multi-physical fields

High-accuracy numerical method is of significance in nano-electromechanical systems. This paper focuses on the influence of nano-system uncertainties on the critical flow velocity of fluid-conveying carbon nanotubes (CNTs) under multi-physical fields. A hull iteration method is proposed to estimate the bounds of the critical flow velocity of fluid-conveying CNT. The nonlinear governing equations for a fluid-conveying CNT embedded in an elastic medium and subjected to thermal and magnetic fields are derived by using the nonlocal Euler–Bernoulli beam theory. Considering the limited available information on the uncertainties, the material and geometrical properties of the CNT are described by using unknown-but-bounded parameters. The iterative scheme is presented in the framework of interval mathematics. After the assessment of the proposed method by comparing Monte Carlo simulation and stochastic analysis method, the detailed parametric investigation is carried out to deeply understand the coupled effects of uncertainties and many parameters such as small scale, Knudsen number, elastic medium and magnetic field parameters, as well as temperature changes. Additionally, the effect of geometric nonlinearity on the critical flow velocity is also examined in detail. The results derived in this paper can provide useful insights on the further application of CNTs as nano-channels to convey fluids.

Modeling the confined fluid flow in micro-nanoporous media under geological temperature and pressure

Understanding the fluid flow in confinement space is the important task for CO2 storage and unconventional gas production. The high temperature and pressure in geological formation proposed a challenge to the traditional model, which ignores the influence of temperature and pressure on the confined single-phase flow and two-phase flow. In our work, an analytical model is proposed to characterize the liquid flow and two phase flow in tight porous media with further considering the influence of temperature and pressure on fluid transport in confinement space. The results demonstrate that solid-liquid interactions will rise up the flow resistance of confined water and reduce the liquid permeability of tight porous media. Additionally, as the burial depth increases, the increasing temperature will not only increase the fluidity of bulk water but also weaken the effect of solid-liquid interactions, and the coupled effects can increase the water volume flux in geological formation. For two-phase flow in the formation, the high temperature and pressure result in small Knudsen number and low gas relative permeability branch while the high temperature increases the water fluidity and rises up the water relative permeability branch. The output of gas can increase Knudsen number, promoting the gas flow capacity and rising up the gas relative permeability branch.