Small Knudsen Numbers Research Articles

Acoustic propagation using the direct simulation Monte Carlo method

In the simulation of fluid dynamics, one can either treat the fluid as a continuum or as discrete particles. Although popular for acoustics, the continuum model is limited to small Knudsen numbers (the ratio of mean free path to a length scale). Particle methods are necessary for, but not limited to, problems with Knudsen numbers greater than 0.1, which can occur in shockwaves, microdevices, or rarefied gases. Some well known particle methods include Molecular Dynamics, Monte Carlo methods, and the Lattice Boltzmann Method. The Direct Simulation Monte Carlo (DSMC) method describes gas flows through direct physical modeling of particle motions and collisions. DSMC can model linear and nonlinear acoustics, as well as the details of viscous dissipation. DSMC results will be shown and compared with continuum theory. A DSMC method has been implemented for 1-dimensional linear and nonlinear acoustics problems on parallel computers using object-oriented C++ and the message passing interface (MPI). The code has been run on large Beowulf clusters. The results have been compared to solutions to the linear wave equation. The complete paper will include two-dimensional solutions as well as comparisons to nonlinear continuum results. [Work supported by Consortium for Education in Many-Body Applications, Grant No. NSF-DGE-9987589.]

Inverted velocity profile in the cylindrical Couette flow of a rarefied gas.

The cylindrical Couette flow of a rarefied gas is investigated, under the diffuse-specular reflection condition of Maxwell's type on the cylinders, in the case where the inner cylinder is rotating whereas the outer cylinder is at rest. The inverted velocity profile for small accommodation coefficients, pointed out by Tibbs, Baras, and Garcia [Phys. Rev. E 56, 2282 (1997)] on the basis of a Monte Carlo simulation, is investigated extensively by means of a systematic asymptotic analysis for small Knudsen numbers as well as the direct numerical analysis of the Boltzmann equation, and the parameter range in which the phenomenon appears is clarified.

Kinetic boundary conditions in the lattice Boltzmann method.

Derivation of the lattice Boltzmann method from the continuous kinetic theory [X. He and L. S. Luo, Phys. Rev. E 55, R6333 (1997); X. Shan and X. He, Phys. Rev. Lett. 80, 65 (1998)] is extended in order to obtain boundary conditions for the method. For the model of a diffusively reflecting moving solid wall, the boundary condition for the discrete set of velocities is derived, and the error of the discretization is estimated. Numerical results are presented which demonstrate convergence to the hydrodynamic limit. In particular, the Knudsen layer in the Kramers' problem is reproduced correctly for small Knudsen numbers.

Hydrodynamics of thermal granular convection

A hydrodynamic theory is formulated for buoyancy-driven ("thermal") granular convection, recently predicted in molecular dynamic simulations and observed in experiment. The limit of a dilute flow is considered. The problem is fully described by three scaled parameters. The convection occurs via a supercritical bifurcation, the inelasticity of the collisions being the control parameter. The character of hydrodynamic modes of the system is discussed. The theory is expected to be valid for small Knudsen numbers and nearly elastic grain collisions.

Poiseuille-type flow of a rarefied gas between two parallel plates driven by a uniform external force.

A unidirectional flow of a rarefied gas between two parallel plates driven by a uniform external force is investigated on the basis of kinetic theory with special interest in the behavior in the near continuum regime. The Bhatnagar-Gross-Krook (BGK) model of the Boltzmann equation and the diffuse reflection boundary condition are employed as the basic system. First, a systematic asymptotic analysis of the basic system for small Knudsen numbers is carried out, and a system of fluid-dynamic-type equations and their boundary conditions are derived up to the second order in the Knudsen number. Then, an accurate numerical analysis of the original BGK system is performed for a wide range of the Knudsen number by means of a finite-difference method. The behavior of the gas, such as the non-Navier-Stokes effects in the near continuum regime, is clarified on the basis of the fluid-dynamic-type system as well as the numerical solution of the BGK system.

Numerical Analysis of the Influence of Thermal Stresses on the Nonlinear Thermomolecular Pressure Difference Effect

On the basis of a numerical analysis of the non-Navier-Stokes gas-dynamic equations for slow non-isothermal gas flows, the nonlinear thermomolecular pressure difference effect due to a large temperature gradient along the lateral surface of a capillary is investigated. It is shown that the magnitude of the effect is substantially different from the values calculated using the Navier-Stokes equations. For two models of molecular interaction (Maxwell molecules and hard spheres), the possibility of a quasi-one-dimensional interpretation of the effect for experimental estimation purposes is demonstrated. The solutions of the relaxation kinetic equation for flow in a plane capillary at small Knudsen numbers and the gas-dynamic equations for slow non-isothermal flows are compared and the range of their applicability is estimated.

Concentration effects on photophoresis of aerosol spheres

The photophoretic motion of a homogeneous suspension of identical spherical particles of arbitrary thermal conductivity and surface properties exposed to a radiative flux is considered under conditions of small Knudsen, Reynolds, and Peclet numbers. The effects of interaction of the individual particles are taken into explicit account by employing a unit cell model which is known to provide good predictions for the sedimentation of monodisperse suspensions of spherical particles. The appropriate equations of conservation of energy and momentum are solved for each cell, in which a spherical particle is envisaged to be surrounded by a concentric shell of suspending fluid, and the photophoretic velocity of the particle is calculated for various cases. Analytical expressions of this mean particle velocity are obtained in closed form as monotonically decreasing functions of the non-vanishing volume fraction of the particles. The normalized photophoretic velocity increases with an increase in the relative conductivity of the particles for a given particle concentration.

Kinetic Phenomena in the Diffusion of Gases in Capillaries and Porous Bodies

The flow of gases and gaseous mixtures in capillaries and porous bodies were considered. Phenomenological transfer equations were derived using methods of non-equilibrium thermodynamics. Integral and local equations of non-equilibrium thermodynamics were constructed and phenomenological relations of two types were obtained. Different approaches to the calculation of kinetic coefficients were indicated. Methods of kinetic theory based on Boltzmann's equation and the dusty gas model were analyzed separately. Results of the calculations of kinetic coefficients for small and large Knudsen numbers were reported with allowance for incomplete accommodation of molecules on the surface of channel walls. A number of kinetic effects exhibiting during the flow of gases and gaseous mixtures through the capillaries and porous bodies was discussed.

Thermophoretic force measurements of aggregates of micro-spheres

Direct measurements of the thermophoretic force on spherical and non-spherical particles have been made using an electrodynamic balance mounted in a vacuum chamber. The non-spherical particles consisted of aggregates of polystyrene latex spheres. By varying the system pressure the force was measured over a wide range of Knudsen numbers ranging from the near-continuum regime to the free-molecule regime. The gases were monoatomic helium and diatomic nitrogen. The data were analyzed and normalized to determine the thermophoretic shape factor that relates the force on a non-spherical particle to that on a sphere of the same volume. The shape factors and the thermophoretic forces are compared with available theories. The measured thermophoretic forces are shown to agree with theories for the Knudsen regime and the free-molecule regime, and the thermophoretic shape factors are shown to be consistent with theories based on the limiting cases of large and small Knudsen numbers.

Fluid flow in nanopores: An examination of hydrodynamic boundary conditions

Steady-state Poiseuille flow of a simple fluid in carbon slit pores under a gravity-like force is simulated using a realistic empirical many-body potential model for carbon. In this work we focus on the small Knudsen number regime, where the macroscopic equations are applicable, and simulate different wetting conditions by varying the strength of fluid–wall interactions. We show that fluid flow in a carbon pore is characterized by a large slip length even in the strongly wetting case, contrary to the predictions of Tolstoi’s theory. When the surface density of wall atoms is reduced to values typical of a van der Waals solid, the streaming velocity profile vanishes at the wall, in accordance with earlier findings. From the velocity profiles we have calculated the slip length and by analyzing temporal profiles of the velocity components of particles colliding with the wall we obtained values of the Maxwell coefficient defining the fraction of molecules thermalized by the wall.

THE GHOST EFFECT IN THE CONTINUUM LIMIT FOR A VAPOR–GAS MIXTURE AROUND CONDENSED PHASES: ASYMPTOTIC ANALYSIS OF THE BOLTZMANN EQUATION

A binary mixture of a vapor and a noncondensable gas around arbitrarily shaped condensed phases of the vapor is considered. Its steady behavior in the continuum limit (the limit where the Knudsen number vanishes) is investigated on the basis of kinetic theory in the case where the condensed phases are at rest, and the mixture is in a state at rest with a uniform pressure at infinity when an infinite domain is considered. A systematic asymptotic analysis of the Boltzmann equation with kinetic boundary condition is carried out for small Knudsen numbers, and the system of fluid-dynamic type equations and their appropriate boundary conditions that describes the behavior in the continuum limit is derived. The system shows that the flow of the mixture vanishes in the continuum limit, but the vanishing flow gives a finite effect on the behavior of the mixture in this limit. This is an example of the ghost effect discovered recently by Sone and coworkers [e.g., Y. Sone et al., Phys. Fluids 8, 628 and 3403 (1996); Y. Sone, in Rarefied Gas Dynamics, edited by C. Shen (Peking University Press, Beijing, 1997), p. 3]. It is shown that there are several new source factors of the ghost effect that are peculiar to a gas mixture, i.e., that originate from the nonuniformity of the concentration.

Mass accommodation and uptake coefficients — a quantitative comparison

An analytical formula, where the uptake coefficient is given as a function of mass accommodation coefficient, has been derived. The uptake coefficient is seen always to be smaller than or equal to the mass accommodation coefficient. The independence of the uptake coefficient on mass accommodation coefficient for small Knudsen number is demonstrated. If the boundary conditions, particularly the vapour pressure at the droplet surface, are not well defined, significant errors in the value of the mass accommodation coefficient can be expected. Values obtained in not well-defined conditions are typically smaller than the real value.

Stationary States before Sonoluminescence: Discrete Kinetic Approach

Orientation-free discrete velocity model and diffuse-reflection boundary conditions are adopted together to solve the stationary noble-gas transport problem within a bubble in microdomains. Once the gas inside the bubble is dilute enough, we observe unusual very-high velocity (or pressure) and temperature fields for noble gases (corresponding to rather small Knudsen number: Kn ≤ 0.01, which is defined as the ratio of the mean free path of the noble gas to the bubble diameter) inside the bubble. The transport coefficients, or gas-velocity and gas-temperature fields (depending on Kn) are not uniform inside the bubble and there are sudden drops near the bubble wall. The latter, perhaps, is due to the heat-conducting bubble-wall. These results might give us some insights related to the initial state of the noble gases within the bubble before the on-set of sonoluminescence.

On evaporation of a spherical particle at arbitrary Knudsen numbers

The description of drop growth in certain devices or under natural conditions (e.g., in clouds), as well as of thermophoresis and diffusiophoresis phenomena, needs a quantitative theory capable of treating the gas flow around a particle immersed in gas aerosol in a wide range of Knudsen numbers. This theory must be based on solving the Boltzmann kinetic equation. To date, there exists a number of approaches related to solving the Boltzmann equation for arbitrary Knudsen numbers. Among them, we note the Lees method [1, 2] with certain modifications [3, 4] and a method based on solving integro-moment equations by the variational Bubnov–Galerkin method [5–7]. Each of these methods have advantages and shortcomings. For example, the Lees method describes gas flows at large Knudsen numbers (Kn @ 1) well but fails at small Knudsen numbers (Kn ! 1). In the intermediate range of Knudsen numbers (Kn ~ 1), the accuracy of the method is, generally speaking, unknown. The integro-moment Bubnov–Galerkin method is variational and makes it impossible to find the distribution function. In this study, we use the method proposed previously in [8] for solving the Boltzmann equation at arbitrary Knudsen numbers. This method can be considered a natural generalization of the method of half-space moments with allowance for the ideas applied in the Lees method [1, 2]. The essence of the method is the following. A gas flow is described by a distribution function that has a discontinuity in the velocity space (as in the method of half-space moments [9]). However, as in the Lees method, the discontinuity takes place in the influence cone (Fig. 1) rather than in the half-plane vn = 0 (vn is the molecule velocity component normal to the particle surface). The procedure of constructing a system of moment equations is similar to the method of halfspace moments [9]. Thus, the method proposed com-

Photophoresis of an Aerosol Sphere in a Spherical Cavity

This study examines the quasisteady photophoretic motion of a spherical aerosol particle with arbitrary thermal conductivity and surface properties, located at the center of a spherical cavity and exposed to a radiative flux. Assuming a small Knudsen number, the fluid flow is described by a continuum model with a temperature jump, thermal creep, and frictional slip on the solid surfaces. In the limit of the small Peclet and Reynolds numbers, appropriate equations of conservation of energy and momentum are solved for the system and the photophoretic velocity of the particle is obtained in closed form. The normalized photophoretic mobility increases with the relative conductively of the particle. The boundary effect of the cavity wall on the photophoresis of an aerosol particle is generally quite significant in appropriate situations. In practical aerosol systems, the boundary effect on photophoresis is much weaker than on gravitational field driven motion.

Molecular-dynamics simulation of two-dimensional thermophoresis

A numerical technique is presented for the thermal force exerted on a solid particle by a gaseous medium between two flat plates at different temperatures, in the free molecular or transition flow. This is a two-dimensional molecular-dynamics simulation of hard disks in a inhomogeneous thermal environment. All steady-state features exhibited by the compressible hard-disk gas are shown to be consistent with the expected behaviors. Moreover the thermal force experienced by a large solid disk is investigated, and compared to the analytical case of cylinders moving perpendicularly to the constant temperature gradient for an infinite Knudsen number and in an infinite medium. We show precise examples of how this technique can be used simply to investigate more difficult practical problems, in particluar the influence of nonlinear gradients for large applied differences of temperature, of proximity of the walls, and of smaller Knudsen numbers.

Flow of normal 3He in aerogel

We consider the flow of normal 3He in aerogel in the temperature regime T c⩽ T⪡ T F. A recent phenomenological treatment that predicted the transition from Drude to Hagen–Poiseuille flow at low T and low aerogel concentration, was restricted to small Knudsen numbers λ ∗/d ( λ ∗= quasiparticle mean free path of 3He in aerogel, d= sample dimension), the so-called slip regime. In this contribution we lift this restriction and present a variational calculation, valid at arbitrary Knudsen numbers. We calculate the slip length of liquid 3He in aerogel and predict the temperature dependence of the flow resistance in this general situation.

Asymptotic theory of the Boltzmann system, for a steady flow of a slightly rarefied gas with a finite Mach number: General theory

A steady rarefied gas flow with Mach number of the order of unity around a body or bodies is considered. The general behaviour of the gas for small Knudsen numbers is studied by asymptotic analysis of the boundary-value problem of the Boltzmann equation for a general domain. The effect of gas rarefaction (or Knudsen number) is expressed as a power series of the square root of the Knudsen number of the system. A series of fluid-dynamic type equations and their associated boundary conditions that determine the component functions of the expansion of the density, flow velocity, and temperature of the gas is obtained by the analysis. The equations up to the order of the square root of the Knudsen number do not contain non-Navier–Stokes stress and heat flow, which differs from the claim by Darrozes (in Rarefied Gas Dynamics, Academic Press, New York, 1969). The contributions up to this order, except in the Knudsen layer, are included in the system of the Navier–Stokes equations and the slip boundary conditions consisting of tangential velocity slip due to the shear of flow and temperature jump due to the temperature gradient normal to the boundary.

Space-Time Discretization of Series Expansion Methods for the Boltzmann Transport Equation

The approximate solution of the Boltzmann transport equation via Galerkin-type series expansion methods leads to a system of first order differential equations in space and time for the expansion coefficients. This system is extremely stiff close to the fluid dynamical regime (for small Knudsen numbers), and exhibits a mildly dispersive behavior, due to the acceleration of waves by the external force (the electric field). In this paper a class of difference methods is presented and analyzed which represent a generalization of the well-known Scharfetter--Gummel exponential fitting approach for the drift-diffusion equations. It is shown that, by using appropriate operator splitting methods for the time discretization, one obtains stability properties which are only mildly dependent on the Knudsen number and essentially independent of the size of the electric field.

Burnett description for plane Poiseuille flow.

Two recent works have shown that at small Knudsen number (K) the pressure and temperature profiles in plane Poiseuille flow exhibit a different qualitative behavior from the profiles obtained by the Navier-Stokes equations. Tij and Santos [J. Stat. Phys. 76, 1399 (1994)] used the Bhatnagar-Gross-Kook model to show that the temperature profile is bimodal and the pressure profile is nonconstant. Malek-Mansour, Baras, and Garcia [Physica A 240, 255 (1997)] qualitatively confirmed these predictions in computer experiments using the direct simulation Monte Carlo method (DSMC). In this paper we compare the DSMC measurements of hydrodynamic variables and non-equilibrium fluxes with numerical solutions of the Burnett equations. Given that they are in better agreement with molecular-dynamics simulations [E. Salomons and M. Mareschal, Phys. Rev. Lett. 69, 269 (1992)] of strong shock waves than Navier-Stokes [F. J. Uribe, R. M. Velasco, and L. S. García-Colín, Phys. Rev. Lett. 81, 2044 (1998)], and that they are second order in Knudsen number suggests that the Burnett equations may provide a better description for large K. We find that for plane Poiseuille flow the Burnett equations do not predict the bimodal temperature profile but do recover many of the other anomalous features (e.g., nonconstant pressure and nonzero parallel heat flux).