Showing papers on "Knudsen number published in 2002"

Kinetic Theory and Fluid Dynamics

Abstract: In this series of talks, I will discuss the fluid-dynamic-type equations that is derived from the Boltzmann equation as its the asymptotic behavior for small mean free path. The study of the relation of the two systems describing the behavior of a gas, the fluid-dynamic system and the Boltzmann system, has a long history and many works have been done. The Hilbert expansion and the Chapman–Enskog expansion are well-known among them. The behavior of a gas in the continuum limit, however, is not so simple as is widely discussed by superficial understanding of these solutions. The correct behavior has to be investigated by classifying the physical situations. The results are largely different depending on the situations. There is an important class of problems for which neither the Euler equations nor the Navier–Stokes give the correct answer. In these two expansions themselves, an initialor boundaryvalue problem is not taken into account. We will discuss the fluid-dynamic-type equations together with the boundary conditions that describe the behavior of the gas in the continuum limit by appropriately classifying the physical situations and taking the boundary condition into account. Here the result for the time-independent case is summarized. The time-dependent case will also be mentioned in the talk. The velocity distribution function approaches a Maxwellian fe, whose parameters depend on the position in the gas, in the continuum limit. The fluid-dynamictype equations that determine the macroscopic variables in the limit differ considerably depending on the character of the Maxwellian. The systems are classified by the size of |fe− fe0|/fe0, where fe0 is the stationary Maxwellian with the representative density and temperature in the gas. (1) |fe − fe0|/fe0 = O(Kn) (Kn : Knudsen number, i.e., Kn = `/L; ` : the reference mean free path. L : the reference length of the system) : S system (the incompressible Navier–Stokes set with the energy equation modified). (1a) |fe − fe0|/fe0 = o(Kn) : Linear system (the Stokes set). (2) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(Kn) (ξi : the molecular velocity) : SB system [the temperature T and density ρ in the continuum limit are determined together with the flow velocity vi of the first order of Kn amplified by 1/Kn (the ghost effect), and the thermal stress of the order of (Kn) must be retained in the equations (non-Navier–Stokes effect). The thermal creep[1] in the boundary condition must be taken into account. (3) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(1) : E+VB system (the Euler and viscous boundary-layer sets). E system (Euler set) in the case where the boundary is an interface of the gas and its condensed phase. The fluid-dynamic systems are classified in terms of the macroscopic parameters that appear in the boundary condition. Let Tw and δTw be, respectively, the characteristic values of the temperature and its variation of the boundary. Then, the fluid-dynamic systems mentioned above are classified with the nondimensional temperature variation δTw/Tw and Reynolds number Re as shown in Fig. 1. In the region SB, the classical gas dynamics is inapplicable, that is, neither the Euler

Lattice-Boltzmann Simulations of Fluid Flows in MEMS

Abstract: The lattice Boltzmann model is a simplified kinetic method based on the particle distribution function. We use this method to simulate problems in MEMS, in which the velocity slip near the wall plays an important role. It is demonstrated that the lattice Boltzmann method can capture the fundamental behaviors in micro-channel flow, including velocity slip, nonlinear pressure drop along the channel and mass flow rate variation with Knudsen number. The Knudsen number dependence of the position of the vortex center and the pressure contour in micro-cavity flows is also demonstrated.

Kinetic boundary conditions in the lattice Boltzmann method.

Abstract: 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.

On Fluid/Wall Slippage

Abstract: Certain (nonpolymeric) fluids show an anomalously low friction when flowing against well-chosen solid walls. We discuss here one possible explanation, postulating that a gaseous film of small thickness h is present between fluid and wall. When h is smaller than the mean free path l of the gas (Knudsen regime), the Navier length b is expected to be independent of h and very large (micrometers).

Effect of Pore Structure, Randomness and Size on Effective Mass Diffusivity

Abstract: m coatings is analyzed using 2-D network models of connecting arms anderified experi- mentally for a multiple pore length scale coating layer. The network model includes ( effects ofariation in the lattice randomness Voronoi tessellation in the form of Delau- )( ) nay lattice triangulation , pore coordination number, pore size Knudsen effect , and : pore-size distribution on the predicted D. The effect of pore poisoning, resulting in a m pore blockage, is analyzed. Correlations for the porosity and pore-blockage dependency : () of D , as well as relationships for the pore size low-dimensionality and multiple m pore length scale effects, are also discussed. An experiment performed on a catalytic () conerter washcoat segment represented by three pore length scales placed on an oth- erwise impermeable wall of an electrochemical sensor shows a good agreement with the : predicted D based on a multiple pore length scale medium with parallel diffusion m paths.

Constant-Wall-Temperature Nusselt Number in Micro and Nano-Channels

Abstract: We investigate the constant-wall-temperature convective heat-transfer characteristics of a model gaseous flow in two-dimensional micro and nano-channels under hydrodynamically and thermally fully developed conditions Our investigation covers both the slip-flow regime 0≤Kn≤01, and most of the transition regime 01

Comparison of Kinetic Theory and Hydrodynamics for Poiseuille Flow

Abstract: Comparison of particle (DSMC) simulation with the numerical solution of the Navier–Stokes (NS) equations for pressure-driven plane Poiseuille flow is presented and contrasted with that of the acceleration-driven Poiseuille flow. Although for the acceleration-driven case DSMC measurements are qualitatively different from the NS solution at relatively low Knudsen number, the two are in somewhat better agreement for pressure-driven flow.

Gaseous mixture flow through a long tube at arbitrary Knudsen numbers

Abstract: The mass flow, heat flux, and diffusion flux of rarefied gas mixture through a tube caused by gradients of pressure, temperature, and concentration were calculated over a wide range of the Knudsen number on the basis of the kinetic equation. The thermodynamic fluxes are presented in the form that allows us to prove the Onsager relations and then to reduce the number of kinetic coefficients determining the solution down to six. The numerical values of the kinetic coefficients are tabulated and the velocity profiles are given in figures.

Computations of the Flow and Heat Transfer in Microdevices Using DSMC With Implicit Boundary Conditions

Abstract: The heat transfer and the fluid dynamics characteristics of subsonic gas flows through microchannels are examined using the direct simulation Monte Carlo (DSMC) method. A simple implicit treatment for the low-speed inflow and outflow boundaries for the DSMC of the flows in microelectromechanical systems (MEMS) is used. Micro-Couette flows and micro-Poiseuille flows are simulated with the value of the Knudsen numbers ranging between 0.06 and 0.72. Where appropriate, the calculated velocity slip and temperature distribution are compared with analytical solutions derived from the Navier-Stokes equations with slip-boundary conditions. A patterned microstructure with nonuniform surface temperature is also simulated

Stokes-Fourier and Acoustic Limits for the Boltzmann Equation: Convergence Proofs

Abstract: We establish a Stokes-Fourier limit for the Boltzmann equation considered over any periodic spatial domain of dimension 2 or more. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that globally in time converge weakly to a unique limit governed by a solution of Stokes-Fourier motion and heat equations provided that the fluid moments of their initial fluctuations converge to appropriate L 2 initial data of the StokesFourier equations. Both the motion and heat equations are recovered in the limit by controlling the fluxes and the local conservation defects of the DiPerna-Lions solutions with dissipation rate estimates. The scaling of the fluctuations with respect to Knudsen number is essentially optimal. The assumptions on the collision kernel are little more than those required for the DiPerna-Lions theory and that the viscosity and heat conduction are finite. For the acoustic limit, these techniques also remove restrictions to bounded collision kernels and improve the scaling of the fluctuations. Both weak limits become strong when the initial fluctuations converge entropically to appropriate L 2 initial data. c 2002 John Wiley & Sons, Inc.

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

Abstract: 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.

Rayleigh-Bénard flow of a rarefied gas and its attractors. I. Convection regime

Abstract: In this paper we investigate the long time behavior (final state) of the Rayleigh–Benard (RB) flow of a rarefied monatomic gas for a set of the nondimensional Knudsen and Froude numbers in the intervals Kn∈[1.0×10−3,4×10−2], Fr∈[1.0×10−1,1.5×103]. For the most part of the computations the third nondimensional parameter, the ratio of the cold and hot wall temperatures is fixed to Tc/Th=0.1, corresponding to a large temperature difference (Th serves as reference temperature), for which the RB system is believed to reach most of the possible final states (attractors). The low Knudsen numbers allow the problem to be investigated numerically by using two completely different methods: direct simulation Monte Carlo (DSMC) method (molecular approach) and finite difference (FD) method (continuum approach based on the model of compressible viscous heat conducting gas with state-dependent transport coefficients). As a result the effect of rarefaction on the onset of convection in the two-dimensional case is studied ...

Performance analysis and optimization considerations for a Knudsen compressor in transitional flow

Abstract: The Knudsen compressor can be applied as either a vacuum pump or compressor for gases. Earlier investigations have indicated that there are several interesting potential applications of the Knudsen compressor because it has no moving parts and requires no lubricants or supplementary working fluids. However, its energy efficiency tends to be low, so that careful optimization is necessary. An important aspect of the optimization is an understanding of the Knudsen compressor’s operating characteristics in the transitional flow regime of rarefied gas dynamics. This article presents a formulation of Knudsen compressor operation in transitional, rarefied flow. In certain simplified but meaningful situations the formulation provides essentially analytical results for the sensitivity of key performance indicators, such as the energy use and device volume per unit of upflow, to changes in operating and geometric parameters. A numerical study of more complicated situations, using the most general form of the formulation developed here, is substantiated by the analytical investigation. The numerical results also extend the understanding of the Knudsen compressor’s performance characteristics to conditions that cannot be addressed by the simplified analytical form. Specifically, minimization of the device’s volume per unit of upflow is found when the entire cascade operates in transitional flow, which can only be studied using the complete formulation. The results make clear that operation in the transitional flow regime can lead to very significant (factor of 5 to 10) reductions of energy use and device volume for a given task.

Thermal boundary layers in sweeping gas membrane distillation processes

Abstract: The sweeping gas membrane distillation process was analyzed in a plate and frame membrane module, and a method clarifying the contribution of each boundary layer separately was developed The temperature polarization coefficients of the fluid phases as well as the overall one have been defined, and their effects on the permeate flux were studied Also studied is the influence of experimentally relevant parameters, such as the inlet temperatures or circulation velocities of the fluids Results from two porous and hydrophobic membranes in different experimental conditions were interpreted on the basis of a combined Knudsen/molecular diffusion flow model The theory agreed well with the experiment

Knudsen forces on microcantilevers

Abstract: When two surfaces at two different temperatures are separated by a distance comparable to a mean-free path of the molecules of the ambient medium, the surfaces experience Knudsen force. This mechanical force can be important in microelectromechanical systems and in atomic force microscopy. A theoretical discussion of the magnitude of the forces and the conditions where they can be encountered is discussed. A potential application of the Knudsen force in designing a cantilever-based vacuum gauge is discussed.

Squeeze film damping effect on a MEMS torsion mirror

Abstract: In this paper we discuss the dynamical characteristics of a torsion mirror in microelectromechanical systems. The squeeze film is modeled using the so-called modified molecular gas film lubrication (MMGL) equation with the coupling effects of surface roughness and gas rarefaction. The MMGL equation is linearized and then a simple mapping method is utilized to obtain the analytical solution of the transformed two-dimensional diffusion equation. From the numerical analyses, it is shown that the surface roughness parameter (Peklenik number), the gas rarefaction parameter (inverse Knudsen number) and squeeze film damping frequencies significantly affect the dynamic characteristics (spring and damping coefficients) of the torsion mirror.

Sound wave propagation in transition-regime micro- and nanochannels

Abstract: We present an extension of the existing continuum theory for sound wave propagation in dilute gases in “narrow” two-dimensional channels to arbitrary Knudsen numbers; the theory provides predictions for the wavelength and attenuation coefficient as a function of the oscillation frequency. A channel is considered narrow in the context of wave propagation when its height is much smaller than the characteristic diffusion length based on the wave frequency. This criterion is easily satisfied by small scale (transition-regime) channels for most frequencies of interest. Numerical simulations for a dilute monoatomic gas using the direct simulation Monte Carlo are used to verify the theoretical results. Good agreement is found between theory and simulation.

Initial results from the first MEMS fabricated thermal transpiration-driven vacuum pump

Abstract: The success of NASA’s future space missions and the development of portable, commercial instrument packages will depend greatly on miniaturized components enabled by the use of microelectromechanical systems (MEMS). Both of these application markets for miniaturized instruments are governed by the use of MEMS components that satisfy stringent power, mass, volume, contamination and integration requirements. An attractive MEMS vacuum pump for instruments requiring vacuum conditions is the Knudsen Compressor, which operates based on the rarefied gas dynamics phenomenon of thermal transpiration. Thermal transpiration describes the regime where gas flows can be induced in a system by maintaining temperature differences across porous materials under rarefied conditions. This pumping mechanism provides two overwhelmingly attractive features as a miniature vacuum pump-no moving parts and no working fluids or lubricants. Due to favorable power, volume and mass estimates a Knudsen Compressor fabricated using MEMS fabrication techniques (lithography, deep reactive ion etching) and new materials (silicon, aerogel) has been completed. The experimental testing of this MEMS Knudsen Compressor device’s thermal and pumping performance are outlined in this manuscript. Good agreement between experiments and numerical predictions using a transitional flow analysis have also been obtained although simple simulations based on the aerogel’s structure are difficult to perform.

Pulsed field gradient nuclear magnetic resonance study of long-range diffusion in beds of NaX zeolite: Evidence for different apparent tortuosity factors in the Knudsen and bulk regimes

Abstract: The pulsed field gradient nuclear magnetic resonance method is applied to study self-diffusion of ethane in beds of zeolite NaX for displacements which are orders of magnitude larger than the size of individual crystals. Comparison of the measured diffusivities with those calculated using simple gas kinetic theory indicates that for the same bed of NaX crystals the apparent tortuosity factor in the Knudsen regime is significantly larger than that in the bulk regime. This finding is tentatively attributed to the more pronounced geometrical trapping by surface imperfections in the Knudsen than in the bulk regime. Tortuosity factors, which are much larger in the Knudsen regime than in the bulk regime, were also recently obtained by dynamic Monte Carlo simulation of gas diffusion in various porous systems.

The self-preserving size distribution theory. I. Effects of the Knudsen number on aerosol agglomerate growth.

Abstract: Gas-phase synthesis of fine solid particles leads to fractal-like structures whose transport and light scattering properties differ from those of their spherical counterparts. Self-preserving size distribution theory provides a useful methodology for analyzing the asymptotic behavior of such systems. Apparent inconsistencies in previous treatments of the self-preserving size distributions in the free molecule regime are resolved. Integro-differential equations for fractal-like particles in the continuum and near continuum regimes are derived and used to calculate the self-preserving and quasi-self-preserving size distributions for agglomerates formed by Brownian coagulation. The results for the limiting case (the continuum regime) were compared with the results of other authors. For these cases the finite difference method was in good in agreement with previous calculations in the continuum regime. A new analysis of aerosol agglomeration for the entire Knudsen number range was developed and compared with a monodisperse model; Higher agglomeration rates were found for lower fractal dimensions, as expected from previous studies. Effects of fractal dimension, pressure, volume loading and temperature on agglomerate growth were investigated. The agglomeration rate can be reduced by decreasing volumetric loading or by increasing the pressure. In laminar flow, an increase in pressure can be used to control particle growth and polydispersity. For D(f)=2, an increase in pressure from 1 to 4 bar reduces the collision radius by about 30%. Varying the temperature has a much smaller effect on agglomerate coagulation.

Heat transfer in the transition regime: solution of boundary value problems for Grad's moment equations via kinetic schemes.

Abstract: This paper presents a systematic approach to the calculation of heat transfer in rarefied gases (Knudsen numbers between 0.01 and 1) by means of Grad's moment method with high moment numbers, based on the Boltzmann equation with linearized collision term. The problem of describing boundary conditions for the moments is solved by the use of the so-called kinetic schemes that allow the implementation of the boundary condition for the Boltzmann equation. The results, obtained with up to 48 one-dimensional moment equations, exhibit temperature jumps at the walls with adjacent Knudsen boundary layers. For given wall temperatures and Knudsen number, the results change with the number of moments, and converge if the number of moments is increased.

Diffusion limit of the lorentz model: asymptotic preserving schemes

Abstract: This paper deals with the diffusion limit of a kinetic equation where the collisions are modeled by a Lorentz type operator. The main aim is to construct a discrete scheme to approximate this equation which gives for any value of the Knudsen number, and in particular at the diffusive limit, the right discrete diffusion equation with the same value of the diffusion coefficient as in the continuous case. We are also naturally interested with a discretization which can be used with few velocity discretization points, in order to reduce the cost of computation.

Heat transfer in a gas mixture between two parallel plates: Finite-difference analysis of the Boltzmann equation

Abstract: The problem of heat transfer and temperature distribution in a binary mixture of rarefied gases between two parallel plates with different temperatures is investigated on the basis of kinetic theory. Under the assumption that the gas molecules are hard spheres and undergo diffuse reflection on the plates, the Boltzmann equation is analyzed numerically by means of an accurate finite-difference method, in which the complicated nonlinear collision integrals are computed efficiently by the deterministic numerical kernel method. As a result, the overall quantities (the heat flow in the mixture, etc.) as well as the profiles of the macroscopic quantities (the molecular number densities of the individual components, the temperature of the total mixture, etc.) are obtained accurately for a wide range of the Knudsen number. At the same time, the behavior of the velocity distribution function is clarified with high accuracy.

Rayleigh–Bénard flow of a rarefied gas and its attractors. II. Chaotic and periodic convective regimes

Abstract: This second part of our study of the Rayleigh–Benard flow of a rarefied gas deals with the analysis of the convection regimes for near-continuum conditions at a Knudsen number Kn=0.001. Here a set of qualitatively new regimes and final states have been found for different Froude (Fr) numbers and a fixed temperature ratio Tc/Th=0.1. In contrast to paper I [Phys. Fluids 14, 2255 (2002)] where we were mainly interested in the long time behavior, or, in other words, in the flow attractors, in this paper we extend the analysis of the most interesting cases to the full transition period from the initial state to the established flow regime. Two transient periods (to a chaotic attractor found for Fr=1.0 and to a stable two-roll configuration for Fr=2.0) are computed by using two different approaches: a direct simulation Monte Carlo method (molecular approach) and a finite difference method solving the system of equations for a compressible viscous heat-conducting gas (continuum approach). The comparison of the s...

Nonlinear Filtering for Low-Velocity Gaseous Microflows

Abstract: Gaseous flows in microfluidic devices are often characterized by relatively high Knudsen numbers. For such flows, the continuum approximation is not valid, and direct simulation Monte Carlo (DSMC) may be used to find an appropriate solution. For low-velocity flows, where the fluid velocity is much smaller than the mean molecular velocity, large statistical fluctuations in the solution mean that the features of the flow may be obscured by noise in the solution. The use of a high-order, nonlinear monotone convection algorithm, flux-corrected transport (FCT), as a filter to extract the solution from the noisy DSMC calculation is described. The diffusion, antidiffusion, and flux-limiting properties of FCT are discussed in terms of their filtering properties. The effects of filtering with FCT are demonstrated for a series of test problems, including a square wave with superimposed random noise, and low-and high-velocity and low- and high-Knudsen-number microchannel flows

Calculations of the near-wall thermophoretic force in rarefied gas flow

Abstract: The thermophoretic force on a near-wall, spherical particle in a rarefied, monatomic gas flow is calculated numerically. The rarefied gas flow is calculated with the Direct Simulation Monte Carlo (DSMC) method, which provides the molecular velocity distribution. The force is calculated from the molecular velocity distribution using a force Green’s function. Calculations are performed over a Knudsen-number range from 0.0475 to 4.75 using Maxwell and hard-sphere collision models. Results are presented for the thermophoresis parameter, ξ, a dimensionless quantity proportional to the thermophoretic force. The spatial profiles of ξ show a clear progression from free-molecular conditions (ξ is constant throughout the domain) to near-continuum conditions (ξ is constant in the interior but increases in the Knudsen layers). For near-continuum conditions, the DSMC calculations and Chapman–Enskog theory are in excellent agreement in the interior, suggesting that their velocity distributions are similar in this region. For all conditions examined, ξ lies between the continuum and free-molecular limits, which differ by only 10%. Moreover, the near-wall ξ values differ from the interior values by less than 5% for a fully diffuse wall, in sharp contrast with most previous studies. An approximate theory for the wall effect is presented that agrees reasonably well with the calculations.