Showing papers on "Knudsen number published in 2012"

Derivation of transient relativistic fluid dynamics from the Boltzmann equation

Abstract: In this work we present a general derivation of relativistic fluid dynamics from the Boltzmann equation using the method of moments. The main difference between our approach and the traditional 14-moment approximation is that we will not close the fluid-dynamical equations of motion by truncating the expansion of the distribution function. Instead, we keep all terms in the moment expansion. The reduction of the degrees of freedom is done by identifying the microscopic time scales of the Boltzmann equation and considering only the slowest ones. In addition, the equations of motion for the dissipative quantities are truncated according to a systematic power-counting scheme in Knudsen and inverse Reynolds number. We conclude that the equations of motion can be closed in terms of only 14 dynamical variables, as long as we only keep terms of second order in Knudsen and/or inverse Reynolds number. We show that, even though the equations of motion are closed in terms of these 14 fields, the transport coefficients carry information about all the moments of the distribution function. In this way, we can show that the particle-diffusion and shear-viscosity coefficients agree with the values given by the Chapman-Enskog expansion.

Knudsen’s Permeability Correction for Tight Porous Media

Abstract: Various flow regimes including Knudsen, transition, slip and viscous flows (Darcy’s law), as applied to flow of natural gas through porous conventional rocks, tight formations and shale systems, are investigated. Data from the Mesaverde formation in the United States are used to demonstrate that the permeability correction factors range generally between 1 and 10. However, there are instances where the corrections can be between 10 and 100 for gas flow with high Knudsen number in the transition flow regime, and especially in the Knudsen’s flow regime. The results are of practical interest as gas permeability in porous media can be more complex than that of liquid. The gas permeability is influenced by slippage of gas, which is a pressure-dependent parameter, commonly referred to as Klinkenberg’s effect. This phenomenon plays a substantial role in gas flow through porous media, especially in unconventional reservoirs with low permeability, such as tight sands, coal seams, and shale formations. A higher-order permeability correlation for gas flow called Knudsen’s permeability is studied. As opposed to Klinkenberg’s correlation, which is a first-order equation, Knudsen’s correlation is a second-order approximation. Even higher-order equations can be derived based on the concept used in developing this model. A plot of permeability correction factor versus Knudsen number gives a typecurve. This typecurve can be used to generalize the permeability correction in tight porous media. We conclude that Knudsen’s permeability correlation is more accurate than Klinkenberg’s model especially for extremely tight porous media with transition and free molecular flow regimes. The results from this study indicate that Klinkenberg’s model and various extensions developed throughout the past years underestimate the permeability correction especially for the case of fluid flow with the high Knudsen number.

A review on slip models for gas microflows

Abstract: Accurate modeling of gas microflow is crucial for the microfluidic devices in MEMS. Gas microflows through these devices are often in the slip and transition flow regimes, characterized by the Knudsen number of the order of 10−2~100. An increasing number of researchers now dedicate great attention to the developments in the modeling of non-equilibrium boundary conditions in the gas microflows, concentrating on the slip model. In this review, we present various slip models obtained from different theoretical, computational and experimental studies for gas microflows. Correct descriptions of the Knudsen layer effect are of critical importance in modeling and designing of gas microflow systems and in predicting their performances. Theoretical descriptions of the gas-surface interaction and gas-surface molecular interaction models are introduced to describe the boundary conditions. Various methods and techniques for determination of the slip coefficients are reviewed. The review presents the considerable success in the implementation of various slip boundary conditions to extend the Navier–Stokes (N–S) equations into the slip and transition flow regimes. Comparisons of different values and formulations of the first- and second-order slip coefficients and models reveal the discrepancies arising from different definitions in the first-order slip coefficient and various approaches to determine the second-order slip coefficient. In addition, no consensus has been reached on the correct and generalized form of higher-order slip expression. The influences of specific effects, such as effective mean free path of the gas molecules and viscosity, surface roughness, gas composition and tangential momentum accommodation coefficient, on the hybrid slip models for gas microflows are analyzed and discussed. It shows that although the various hybrid slip models are proposed from different viewpoints, they can contribute to N–S equations for capturing the high Knudsen number effects in the slip and transition flow regimes. Future studies are also discussed for improving the understanding of gas microflows and enabling us to exactly predict and actively control gas slip.

A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows II: Multi-Dimensional Cases

Abstract: With discretized particle velocity space, a multi-scale unified gas-kinetic scheme for entire Knudsen number flows has been constructed based on the kinetic model in one-dimensional case (J. Comput. Phys., vol. 229 (2010), pp. 7747-7764). For the kinetic equation, to extend a one-dimensional scheme to multidimensional flow is not so straightforward. The major factor is that addition of one dimension in physical space causes the distribution function to become two-dimensional, rather than axially symmetric, in velocity space. In this paper, a unified gas-kinetic scheme based on the Shakhov model in two-dimensional space will be presented. Instead of particle-based modeling for the rarefied flow, such as the direct simulation Monte Carlo (DSMC) method, the philosophical principal underlying the current study is a partial-different- ial-equation (PDE)-based modeling. Since the valid scale of the kinetic equation and the scale of mesh size and time step may be significantly different, the gas evolu- tion in a discretized space is modeled with the help of kinetic equation, instead of directly solving the partial differential equation. Due to the use of both hydrody- namic and kinetic scales flow physics in a gas evolution model at the cell interface, the unified scheme can basically present accurate solution in all flow regimes from the free molecule to the Navier-Stokes solutions. In comparison with the DSMC and Navier-Stokes flow solvers, the current method is much more efficient than DSMC in low speed transition and continuum flow regimes, and it has better capability than NS solver in capturing of non-equilibrium flow physics in the transition and rarefied flow regimes. As a result, the current method can be useful in the flow simulation where both continuum and rarefied flow physics needs to be resolved in a single com- putation. This paper will extensively evaluate the performance of the unified scheme from free molecule to continuum NS solutions, and from low speed micro-flow to high speed non-equilibrium aerodynamics. The test cases clearly demonstrate that the uni- fied scheme is a reliable method for the rarefied flow computations, and the scheme provides an important tool in the study of non-equilibrium flow.

A Fully-Coupled Free and Adsorptive Phase Transport Model for Shale Gas Reservoirs Including Non-Darcy Flow Effects

Abstract: Accurate modeling of gas through shale-gas reservoirs characterized by nano-meter pores where the effects of various non-Darcy flow regimes and the adsorbed-layer are important is presented and demonstrated by several examples. Quantification of gas transport may be accomplished using the transport equation that is valid for all flow regimes. This equation though needs further modification when transport is through a media where the gas is adsorbed onto the pore wall. In the presence of adsorption, there is a pore pressure dependent loss of porosity and cross-sectional area to free gas transport. The apparent gas permeability correction is accomplished for various flow regimes using the Knudsen number by consideration of the reduction of the cross-sectional area to free gas transport in the presence of adsorption. We show that transport in the adsorbed layer may contribute significantly in the total gas transport in these nanopores. An effective transport model is presented to account for the impact of adsorption through two mechanisms. First, we modify the transport equation to account for the pore-pressure dependent-reduction in the volume available to free gas transport; second, we model transport through the adsorbed layer using Fick’s law of diffusion. The coupled model is then compared to conventional transport models over a wide range of reservoir properties and conditions.

Determination of the Scalar Friction Factor for Nonspherical Particles and Aggregates Across the Entire Knudsen Number Range by Direct Simulation Monte Carlo (DSMC)

Abstract: The friction factor of an aerosol particle depends upon the Knudsen number (Kn), as gas molecule–particle momentum transfer occurs in the transition regime. For spheres, the friction factor can be calculated using the Stokes–Millikan equation (with the slip correction factor). However, a suitable friction factor relationship remains sought-after for nonspherical particles. We use direct simulation Monte Carlo (DSMC) to evaluate an algebraic expression for the transition regime friction factor that is intended for application to arbitrarily shaped particles. The tested friction factor expression is derived from dimensional analysis and is analogous to Dahneke's adjusted sphere expression. In applying this expression to nonspherical objects, we argue for the use of two previously developed drag approximations in the continuum (Kn → 0) and free molecular (Kn → ∞) regimes: the Hubbard–Douglas approximation and the projected area (PA) approximation, respectively. These approximations lead to two calculable geo...

The Collision Rate of Nonspherical Particles and Aggregates for all Diffusive Knudsen Numbers

Abstract: We examine theoretically and numerically collisions of arbitrarily shaped particles in the mass transfer transition regime, where ambiguities remain regarding the collision rate coefficient (collision kernel). Specifically, we show that the dimensionless collision kernel for arbitrarily shaped particles, H, depends solely on a correctly defined diffusive Knudsen number (KnD , in contrast with the traditional Knudsen number), and to determine the diffusive Knudsen number, it is necessary to calculate two combined size parameters for the colliding particles: the Smoluchowski radius, which defines the collision rate in the continuum (KnD →0) regime, and the projected area, which defines the collision rate in the free molecular (KnD →∞) regime. Algorithms are provided to compute these parameters. Using mean first passage time calculations with computationally generated quasifractal (statistically fractal) aggregates, we find that with correct definitions of H and KnD , the H(KnD) relationship found valid for ...

An analytically predictive model for moderately rarefied gas flow

Abstract: In microducts deviation from continuum flow behaviour of a gas increases with rarefaction. When using Navier–Stokes equations to calculate a flow under slightly and moderately rarefied conditions, slip boundary conditions are used which in turn refer to the tangential momentum accommodation coefficient (TMAC). Here we demonstrate that, in the so-called slip and transition regime, the flow in microducts can be reliably described by a consistently non-empirical model without considering the TMAC. We obtain this equation by superposition of convective transport and Fickian diffusion using two-dimensional solutions of Navier–Stokes equations and a description for the Knudsen diffusion coefficient as derived from kinetic theory respectively. For a wide variety of measurement series found in the literature the calculation predicts the data accurately. Surprisingly only size of the duct, temperature, gas properties and inlet and outlet pressure are necessary to calculate the resulting mass flow by means of a single algebraic equation. From this, and taking the discrepancies of the TMAC concerning surface roughness and nature of the gases into account, we could conclude that neither the diffusive proportions nor the total mass flow rates are influenced by surface topology and chemistry at Knudsen numbers below unity. Compared to the tube geometry, the model slightly underestimates the flow rate in rectangular channels when rarefaction increases. Likewise, the dimensionless mass flow rate and the diffusive proportion of the total flow are distinctly higher in a tube. Thus the cross-sectional geometry has a significant influence on the transport mechanisms under rarefied conditions.

Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles

Abstract: This work is devoted to the numerical simulation of the Vlasov equation in the fluid limit using particles. To that purpose, we first perform a micro-macro decomposition as in [3] where asymptotic preserving schemes have been derived in the fluid limit. In [3], a uniform grid was used to approximate both the micro and the macro part of the full distribution function. Here, we modify this approach by using a particle approximation for the kinetic (micro) part, the fluid (macro) part being always discretized by standard finite volume schemes. There are many advantages in doing so: $(i)$ the so-obtained scheme presents a much less level of noise compared to the standard particle method; $(ii)$ the computational cost of the micro-macro model is reduced in the fluid regime since a small number of particles is needed for the micro part; $(iii)$ the scheme is asymptotic preserving in the sense that it is consistent with the kinetic equation in the rarefied regime and it degenerates into a uniformly (with respect to the Knudsen number) consistent (and deterministic) approximation of the limiting equation in the fluid regime.

Effects of nonlocal elasticity and Knudsen number on fluid–structure interaction in carbon nanotube conveying fluid

Abstract: In this paper, we investigate the effect of nano-size of both fluid flow and elastic structure simultaneously on the vibrational behavior of a pinned–pinned and a clamped–clamped nanotube conveying fluid, using both Knudsen number (Kn) and nonlocal continuum theory. Euler–Bernoulli plug flow (EBPF) theory is used for modeling fluid–structure interaction (FSI). It is observed that nonlocal parameter has more effect than Kn on the reduction of critical velocities of a liquid nano-flow. This effect has considerable impact on the reduction of critical velocities for a clamped–clamped beam in comparison with a pinned–pinned one. We concluded that the dimensionless nonlocal parameter, had more impressive effect on the dimensionless critical flow velocity of the second mode divergence and coupled mode flutter instabilities. However, in a gas nano-flow, the situation is totally different and Kn causes more reduction in critical velocities. Furthermore, it is emphasized that ignoring nano-size effects on liquid and gas nano-flow might cause non-conservative design of nano-devices.

Size-dependent thermal conductivity in nanosystems based on non-Fourier heat transfer

Abstract: Based on a phonon hydrodynamic equation, we derive an analytical model to predict the effective thermal conductivity along dielectric thin films or nanowires with smooth wall surface. The model contains only two parameters: bulk thermal conductivity and Knudsen number, no other fitting parameter included. The predictions from the simple model agree approximately with available experimental data in silicon nanostructures and it outperforms other analytical models based on the Boltzmann transport equation.

Coulomb-influenced collisions in aerosols and dusty plasmas.

Abstract: In aerosol and dusty plasma systems, the behavior of suspended particles (grains) is often strongly influenced by collisions occurring between ions and particles, as well as between particles themselves. In determining the collision kernel or collision rate coefficient for such charged entities, complications arise in that the collision process can be completely described neither by continuum transport mechanics nor by free molecular (ballistic) mechanics; that is, collisions are transition regime processes. Further, both the thermal energy and the potential energy between colliding entities can strongly influence the collision rate and must be considered. Flux-matching theory, originally developed by Fuchs, is frequently applied for calculation of collision rate coefficients under these circumstances. However, recent work suggests that crucial assumptions in flux-matching theory are not appropriate to describe transition regime collisions in the presence of potential interactions. Here, we combine dimensional analysis and mean first passage time calculations to infer the collision kernel between dilute charged entities suspended in a light background gas at thermal equilibrium. The motion of colliding entities is described by a Langevin equation, and Coulombic interactions are considered. It is found that the dimensionless collision kernel for these conditions, H, is a function of the diffusive Knudsen number, Kn(D) (in contrast to the traditional Knudsen number), and the potential energy to thermal energy ratio, Ψ(E). For small and large Kn(D), it is found that the dimensionless collision kernels inferred from mean first passage time calculations collapse to the appropriate continuum and free molecular limiting forms, respectively. Further, for repulsive collisions (Ψ(E) negative) or attractive collisions with Ψ(E) 0.5, it is found that flux-matching theory predictions substantially underestimate the collision kernel. We find that the collision process in this regime is governed by the minimum of Kn(D) and Kn(Ψ) (Kn(Ψ) = 3Kn(D)/2Ψ(E)), and based on calculations, propose a function H(Kn(D), Kn(Ψ)) for collision kernel evaluation. The situations for which Ψ(E)>0.5 apply to singly charged nanoparticles and multiply charged submicrometer and supermicrometer particles, and are thus prevalent in both aerosol and dusty plasma environments.

Role of rough surface topography on gas slip flow in microchannels.

Abstract: We conduct a lattice Boltzmann simulation of gas slip flow in microchannels incorporating rough surface effects as characterized by fractal geometry with a focus on gas-solid interaction. The gas slip flow in rough microchannels, which is characterized by Poiseuille number and mass flow rate, is evaluated and compared with smooth microchannels. The effects of roughness height, surface fractal dimension, and Knudsen number on slip behavior of gas flow in microchannels are all investigated and discussed. The results indicate that the presence of surface roughness reduces boundary slip for gas flow in microchannels with respect to a smooth surface. The gas flows at the valleys of rough walls are no-slip while velocity slips are observed over the top of rough walls. We find that the gas flow behavior in rough microchannels is insensitive to the surface topography irregularity (unlike the liquid flow in rough microchannels) but is influenced by the statistical height of rough surface and rarefaction effects. In particular, decrease in roughness height or increase in Knudsen number can lead to large wall slip for gas flow in microchannels.

The effects of Knudsen-dependent flow velocity on vibrations of a nano-pipe conveying fluid

Abstract: In this paper, we investigate the effect of nano-flow on vibration of nano-pipe conveying fluid using Knudsen (Kn) We use Euler–Bernoulli plug-flow beam theory We modify no-slip condition of nano-pipe conveying fluid based on Kn We define a Kn-dependent flow velocity We consider effect of slip condition, for a liquid and a gas flow We reformulate Navier–Stokes equations, with modified versions of Kn-dependent flow velocity We observe that for passage of gas through nano-pipe with nonzero Kn, the critical flow velocities decreased considerably as opposed to those for zero Kn This can show that ignoring Kn effect on a gas nano-flow may cause non-conservative design of nano-devices Furthermore, a more impressive phenomenon happens in the case of clamped-pinned pipe conveying gas fluid While we do not observe any coupled-mode flutter for a zero Kn, we can see the coupled-mode flutter, accompanying the second-mode divergence, for a nonzero Kn

Lattice Boltzmann simulations of pressure-driven flows in microchannels using Navier-Maxwell slip boundary conditions

Abstract: We present lattice Boltzmann simulations of rarefied flows driven by pressure drops along two-dimensional microchannels Rarefied effects lead to non-zero cross-channel velocities, and nonlinear variations in the pressure along the channel Both effects are absent in flows driven by uniform body forces We obtain second-order accuracy for the two components of velocity and the pressure relative to asymptotic solutions of the compressible Navier–Stokes equations with slip boundary conditions Since the common lattice Boltzmann formulations cannot capture Knudsen boundary layers, we replace the usual discrete analogs of the specular and diffuse reflection conditions from continuous kinetic theory with a moment-based implementation of the first-order Navier–Maxwell slip boundary conditions that relate the tangential velocity to the strain rate at the boundary We use these conditions to solve for the unknown distribution functions that propagate into the domain across the boundary We achieve second-order accuracy by reformulating these conditions for the second set of distribution functions that arise in the derivation of the lattice Boltzmann method by an integration along characteristics Our moment formalism is also valuable for analysing the existing boundary conditions It reveals the origin of numerical slip in the bounce-back and other common boundary conditions that impose conditions on the higher moments, not on the local tangential velocity itself

Second-order gaseous slip flow models in long circular and noncircular microchannels and nanochannels

Abstract: This paper significantly extends previous studies to the transition regime by employing the second-order slip boundary conditions. A simple analytical model with second-order slip boundary conditions for a normalized Poiseuille number is proposed. The model can be applied to either rarefied gas flows or apparent liquid slip flows. The developed simple models can be used to predict the Poiseuille number, mass flow rate, tangential momentum accommodation coefficient, pressure distribution of gaseous flow in noncircular microchannels and nanochannels by the research community for the practical engineering design of microchannels and nanochannels. The developed second-order models are preferable since the difficulty and “investment” is negligible compared with the cost of alternative methods such as molecular simulations or solutions of Boltzmann equation. Navier–Stokes equations with second-order slip models can be used to predict quantities of engineering interest such as the Poiseuille number, tangential momentum accommodation coefficient, mass flow rate, pressure distribution, and pressure drop beyond its typically acknowledged limit of application. The appropriate or effective second-order slip coefficients include the contribution of the Knudsen layers in order to capture the complete solution of the Boltzmann equation for the Poiseuille number, mass flow rate, and pressure distribution. It could be reasonable that various researchers proposed different second-order slip coefficients because the values are naturally different in different Knudsen number regimes. It is analytically shown that the Knudsen’s minimum can be predicted with the second-order model and the Knudsen value of the occurrence of Knudsen’s minimum depends on inlet and outlet pressure ratio. The compressibility and rarefaction effects on mass flow rate and the curvature of the pressure distribution by employing first-order and second-order slip flow models are analyzed and compared. The condition of linear pressure distribution is given.

Rarefied gas flow around a sharp edge induced by a temperature field

Abstract: A rarefied gas flow thermally induced around a heated (or cooled) flat plate, contained in a vessel, is considered in two different situations: (i) both sides of the plate are simultaneously and uniformly heated (or cooled); and (ii) only one side of the plate is uniformly heated. The former is known as the thermal edge flow and the latter, typically observed in the Crookes radiometer, may be called the radiometric flow. The steady behaviour of the gas induced in the container is investigated on the basis of the Bhatnagar–Gross–Krook (BGK) model of the Boltzmann equation and the diffuse reflection boundary condition by means of an accurate finite-difference method. The flow features are clarified for a wide range of the Knudsen number, with a particular emphasis placed on the structural similarity between the two flows. The limiting behaviour of the flow as the Knudsen number tends to zero (and thus the system approaches the continuum limit) is investigated for both flows. The detailed structure of the normal stress on the plate as well as the cause of the radiometric force (the force acting on the plate from the hotter to the colder side) is also clarified for the present infinitely thin plate.

Numerical Regularized Moment Method for High Mach Number Flow

Abstract: This paper is a continuation of our earlier work [2] in which a numerical momentmethod with arbitrary order of moments was presented. However, the computationmay break down during the calculation of the structure of a shock wave with Machnumber M 0 >3. In this paper, we concentrate on the regularization of the momentsystems. First, we apply the Maxwell iteration to the infinite moment system anddetermine the magnitude of each moment with respect to the Knudsen number. Afterthat, we obtain the approximation of high order moments and close the moment sys-tems by dropping some high-order terms. Linearization is then performed to obtain avery simple regularization term, thus it is very convenient for numerical implementa-tion. To validate the new regularization, the shock structure of low order systems iscomputed with different number of moments and different shock Mach numbers.Keywords: Boltzmann-BGK equation; Maxwellian iteration; Regularized momentequations 1 Introduction In the field such as high altitude flight and microscopic flows, gas is considered tobe in the transitional regime. In this case, usual fluid models such as Euler equationsand Navier-Stokes-Fourier system will fail when the rarefied effect is significant. Themoment method, which was first proposed by Grad [4], is focused on the description ofthe rarefied gases using a small number of variables. Almost all moment methods arederived from the Boltzmann equation which is regarded to be able to capture the rarefiedeffects accurately. In [2], a special expansion of the distribution functions is adopted tomake it possible the solve the associated Grad-type moment equations numerically withoutthe explicit expressions of the system. Then, we followed [11] and numerically regularizedthe system using the technique of a modified Chapman-Enskog expansion. In [2], it hasbeen verified numerically that a smooth shock structure with Mach number M

Slip Effects on the Unsteady MHD Pulsatile Blood Flow through Porous Medium in an Artery under the Effect of Body Acceleration

Abstract: Unsteady pulsatile flow of blood through porous medium in an artery has been studied under the influence of periodic body acceleration and slip condition in the presence of magnetic field considering blood as an incompressible electrically conducting fluid. An analytical solution of the equation of motion is obtained by applying the Laplace transform. With a view to illustrating the applicability of the mathematical model developed here, the analytic explicit expressions of axial velocity, wall shear stress, and fluid acceleration are given. The slip condition plays an important role in shear skin, spurt, and hysteresis effects. The fluids that exhibit boundary slip have important technological applications such as in polishing valves of artificial heart and internal cavities. The effects of slip condition, magnetic field, porous medium, and body acceleration have been discussed. The obtained results, for different values of parameters into the problem under consideration, show that the flow is appreciably influenced by the presence of Knudsen number of slip condition, permeability parameter of porous medium, Hartmann number of magnetic field, and frequency of periodic body acceleration. The study is useful for evaluating the role of porosity and slip condition when the body is subjected to magnetic resonance imaging (MRI).

Surface–gas interaction effects on nanoscale gas flows

Abstract: Molecular dynamics (MD) method is used to simulate shear driven argon gas flows in the early transition and free molecular flow regimes to investigate surface effects as a function of the surface–gas potential strength ratio (ewf/eff). Results show a bulk flow region and a near wall region that extends three molecular diameters away from the surfaces. Within the near wall region the velocity, density, and shear stress distributions exhibit deviations from the kinetic theory predictions. Increased ewf/eff results in increased gas density, leading toward monolayer adsorption on surfaces. The near wall velocity profile shows reduced gas slip, and eventually velocity stick with increased ewf/eff. Using MD predicted shear stress values and kinetic theory, tangential momentum accommodation coefficients (TMAC) are calculated as a function of ewf/eff, and TMAC values are shown to be independent of the Knudsen number. Presence of this near wall region breaks down the dynamic similarity between rarefied and nanoscale gas flows.

Numerical Investigation of Hypersonic Boundary-Layer Stabilization by Porous Surfaces

Abstract: The acoustic second-mode instability predicted by linear stability theory is compared with direct numerical simulation for a hypersonic flow over various porous walls. The damping effect of the micropores on the second mode is shown by comparison of the two different approaches. In addition to investigating the effect of pore size, the influence of the pore shape is studied by using spanwise grooves and cylindrical pores. Specifically, the comparability of different pore shapes by two definitions of hydraulic diameter is analyzed. The influence of rarefied gas behavior of the flow inside the pores is also investigated by comparing a slip boundary condition with finite Knudsen numbers with a nonslip boundary condition for different radii and pore depths.

Predicting enhanced mass flow rates in gas microchannels using nonkinetic models

Abstract: Different nonkinetic approaches are adopted in this paper towards theoretically predicting the experimentally observed phenomenon of enhanced mass flow rates accompanying pressure-driven rarefied gas flows through microchannels. Our analysis utilizes a full set of mechanically consistent volume-diffusion hydrodynamic equations, allowing complete, closed-form, analytical solutions to this class of problems. As an integral part of the analysis, existing experimental data pertaining to the subatmospheric pressure dependence of viscosity were analyzed. The several nonkinetic approaches investigated were (1) pressure-dependent viscosity exponent model, (2) slip-velocity models, and (3) volume diffusion model. We explored the ability to predict the gas's mass flow rate over the full range of Knudsen numbers, including furnishing a physically sound interpretation of the well-known Knudsen minimum observed in the mass flow rate. Matching of a pressure-dependent viscosity model, one that follows the standard temperature-viscosity power law and its supporting single momentum diffusion mechanism, did not allow an accurate interpretation of the data. Rather, matching of this model with the flow rate was found to mismatch the experimental pressure dependence of the viscosity. An additional transport mechanism model, one based on volume diffusion, offered a comprehensive understanding of the Knudsen minimum, while also resulting in excellent agreement with experimental data well into the transition regime (up to a Knudsen number of 5).

Molecular-dynamics study of Poiseuille flow in a nanochannel and calculation of energy and momentum accommodation coefficients.

Abstract: We report a molecular-dynamics study of flow of Lennard-Jones fluid through a nanochannel where size effects predominate. The momentum and energy accommodation coefficients, which determine the amount of slip and temperature jumps, are calculated for a three-dimensional Poiseuille flow through a nano-sized channel. Accommodation coefficients are calculated by considering a " gravity"- (acceleration field) driven Poiseuille flow between two infinite parallel walls that are maintained at a fixed temperature. The Knudsen number (Kn) dependency of the accommodation coefficients, slip length, and velocity profiles is investigated. The system is also studied by varying the strength of gravity. The accommodation coefficients are found to approach a limiting value with an increase in gravity and Kn. For low values of Kn (<0.15), the slip length obtained from the velocity profiles is found to match closely the results obtained from the linear slip model. Using the calculated values of accommodation coefficients, the first- and second-order slip models are validated in the early transition regime. The study demonstrates the applicability of the Navier-Stokes equation with the second-order slip model in the early transition regime.