Slip-flow Theory Research Articles

Molecular Mechanics of Liquid and Gas Slip Flow

By taking into account the inhomogeneity introduced by the presence of a solid boundary, slip-flow theory extends the range of applicability of the venerable Navier–Stokes description to smaller scales and into the regime where confinement starts to be important. Due to the inherently atomistic nature of solid–fluid interactions at their interface, slip flow can be described, at least in principle, predictively at this level. This review aims to summarize our current understanding of slip flow at the atomistic level in dilute gases and dense liquids. The discussion extends over the similarities and differences between slip in gases and liquids, characterization and measurement of slip by molecular simulation methods, models for predicting slip, and open questions requiring further investigation.

Modelling sub-micron particle slip flow in liquid chromatography

The profoundly fast flow rates, observed at unexpectedly low column back-pressures, of capillaries packed with colloidal silica-based stationary phases is explained using slip flow theory applied to the particles themselves rather than the capillary walls. An equation is proposed that leverages nanoscale thermodynamics, geometrical considerations and Hagen-Poiseuille flow without the traditional no-slip boundary condition. It is found that the proposed model successfully fits/predicts real-world data taken from the literature, presented in a plot of the observed flow (enhancement) as a function of the (diminishing) mean particle size, so long as the particles do not exceed ~1 μm. This finding lends support to the continued use of colloidal silica packings in liquid chromatographic applications to achieve efficient separations of molecules on the smallest scale.

Consistent lattice Boltzmann modeling of low-speed isothermal flows at finite Knudsen numbers in slip-flow regime: Application to plane boundaries.

The first nonequilibrium effect experienced by gaseous flows in contact with solid surfaces is the slip-flow regime. While the classical hydrodynamic description holds valid in bulk, at boundaries the fluid-wall interactions must consider slip. In comparison to the standard no-slip Dirichlet condition, the case of slip formulates as a Robin-type condition for the fluid tangential velocity. This makes its numerical modeling a challenging task, particularly in complex geometries. In this work, this issue is handled with the lattice Boltzmann method (LBM), motivated by the similarities between the closure relations of the reflection-type boundary schemes equipping the LBM equation and the slip velocity condition established by slip-flow theory. Based on this analogy, we derive, as central result, the structure of the LBM boundary closure relation that is consistent with the second-order slip velocity condition, applicable to planar walls. Subsequently, three tasks are performed. First, we clarify the limitations of existing slip velocity LBM schemes, based on discrete analogs of kinetic theory fluid-wall interaction models. Second, we present improved slip velocity LBM boundary schemes, constructed directly at discrete level, by extending the multireflection framework to the slip-flow regime. Here, two classes of slip velocity LBM boundary schemes are considered: (i) linear slip schemes, which are local but retain some calibration requirements and/or operation limitations, (ii) parabolic slip schemes, which use a two-point implementation but guarantee the consistent prescription of the intended slip velocity condition, at arbitrary plane wall discretizations, further dispensing any numerical calibration procedure. Third and final, we verify the improvements of our proposed slip velocity LBM boundary schemes against existing ones. The numerical tests evaluate the ability of the slip schemes to exactly accommodate the steady Poiseuille channel flow solution, over distinct wall slippage conditions, namely, no-slip, first-order slip, and second-order slip. The modeling of channel walls is discussed at both lattice-aligned and non-mesh-aligned configurations: the first case illustrates the numerical slip due to the incorrect modeling of slippage coefficients, whereas the second case adds the effect of spurious boundary layers created by the deficient accommodation of bulk solution. Finally, the slip-flow solutions predicted by LBM schemes are further evaluated for the Knudsen's paradox problem. As conclusion, this work establishes the parabolic accuracy of slip velocity schemes as the necessary condition for the consistent LBM modeling of the slip-flow regime.

Second-Order Knudsen-Layer Analysis for the Generalized Slip-Flow Theory II: Curvature Effects

Numerical analyses of the second-order Knudsen layer are carried out on the basis of the linearized Boltzmann equation for hard-sphere molecules under the diffuse reflection boundary condition. The effects of the boundary curvature have been clarified in details, thereby completing the numerical data required up to the second order of the Knudsen number for the asymptotic theory of the Boltzmann equation (the generalized slip-flow theory). A local singularity appears as a result of the expansion at the level of the velocity distribution function, when the curvature exists.

A force acting on an oblate spheroid with discontinuous surface temperature in a slightly rarefied gas

AbstractAn oblate spheroid, the respective hemispheroids of which are kept at different uniform temperatures, placed in a rarefied gas at rest is considered. The explicit formula for the force acting on the spheroid (radiometric force) is obtained for small Knudsen numbers. This is a model of a vane of the Crookes radiometer. The analysis is performed for a general axisymmetric distribution of the surface temperature of the spheroid, allowing abrupt changes. Although the generalized slip flow theory, established by Sone (Rarefied Gas Dynamics, vol. 1, 1969, pp. 243–253), is available for general rarefied gas flows at small Knudsen numbers, it cannot be applied to the present problem because of the abrupt temperature changes. However, if it is combined with the symmetry relations for the linearized Boltzmann equation developed recently by Takata (J. Stat. Phys., vol. 136, 2009, pp. 751–784), one can bypass the difficulty. To be more specific, the force acting on the spheroid in the present problem can be generated from the solution of the adjoint problem to which the generalized slip flow theory can be applied, i.e. the problem in which the same spheroid with a uniform surface temperature is placed in a uniform flow of a rarefied gas. The analysis of the present paper follows this strategy.

Effects of curvature on rarefied gas flows between rotating concentric cylinders

The gas flow between two concentric rotating cylinders is considered in order to investigate non-equilibrium effects associated with the Knudsen layers over curved surfaces. We investigate the nonlinear flow physics in the near-wall regions using a new power-law (PL) wall-scaling approach. This PL model incorporates Knudsen layer effects in near-wall regions by taking into account the boundary limiting effects on the molecular free paths. We also report new direct simulation Monte Carlo results covering a wide range of Knudsen numbers and accommodation coefficients, and for various outer-to-inner cylinder radius ratios. Our simulation data are compared with both the classical slip flow theory and the PL model, and we find that non-equilibrium effects are not only dependent on Knudsen number and accommodation coefficient but are also significantly affected by the surface curvature. The relative merits and limitations of both theoretical models are explored with respect to rarefaction and curvature effects. The PL model is able to capture some of the nonlinear trends associated with Knudsen layers up to the early transition flow regime. The present study also illuminates the limitations of classical slip flow theory even in the early slip flow regime for higher curvature test cases, although the model does exhibit good agreement throughout the slip flow regime for lower curvature cases. Torque and velocity profile comparisons also convey that a good prediction of integral flow properties does not necessarily guarantee the accuracy of the theoretical model used, and it is important to demonstrate that field variables are also predicted satisfactorily.

Gas kinetic algorithm for flows in Poiseuille-like microchannels using Boltzmann model equation

The gas-kinetic unified algorithm using Boltzmann model equation have been extended and developed to solve the micro-scale gas flows in Poiseuille-like micro-channels from Micro-Electro-Mechanical Systems (MEMS). The numerical modeling of the gas kinetic boundary conditions suitable for micro-scale gas flows is presented. To test the present method, the classical Couette flows with various Knudsen numbers, the gas flows from short microchannels like plane Poiseuille and the pressure-driven gas flows in two-dimensional short microchannels have been simulated and compared with the approximate solutions of the Boltzmann equation, the related DSMC results, the modified N-S solutions with slip-flow boundary theory, the gas-kinetic BGK-Burnett solutions and the experimental data. The comparisons show that the present gas-kinetic numerical algorithm using the mesoscopic Boltzmann simplified velocity distribution function equation can effectively simulate and reveal the gas flows in microchannels. The numerical experience indicates that this method may be a powerful tool in the numerical simulation of micro-scale gas flows from MEMS.

309 BGK 方程式を用いたマイクロスケール流れの数値シミュレーション

This paper describes a CFD method based on the Boltzmann equation developed for low Mach number micro flows. Reliability and flexibility of the methods are demonstrated for flow simulation about micro channels and micro spheres at the Mach number range from 0.0001 to 0.1 and the Knudsen number range from 0.01 to 0.1. Reliable results without any fluctuation are obtained even for the flow Mach number of 0.0001. Fully developed velocity profiles of the micro channels well agree with those of slip flow model. The drag coefficients of the micro spheres are compared with those of continuum and slip flow theory. It is confirmed that the method is a good tool to analyze micro gas flows.

Constant-Wall-Temperature Nusselt Number in Micro and Nano-Channels1

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⩽0.1, and most of the transition regime 0.1<Kn⩽10, where Kn, the Knudsen number, is defined as the ratio between the molecular mean free path and the channel height. We use slip-flow theory in the presence of axial heat conduction to calculate the Nusselt number in the range 0⩽Kn⩽0.2, and a stochastic molecular simulation technique known as the direct simulation Monte Carlo (DSMC) to calculate the Nusselt number in the range 0.02<Kn<2. Inclusion of the effects of axial heat conduction in the continuum model is necessary since small-scale internal flows are typically characterized by finite Peclet numbers. Our results show that the slip-flow prediction is in good agreement with the DSMC results for Kn⩽0.1, but also remains a good approximation beyond its expected range of applicability. We also show that the Nusselt number decreases monotonically with increasing Knudsen number in the fully accommodating case, both in the slip-flow and transition regimes. In the slip-flow regime, axial heat conduction is found to increase the Nusselt number; this effect is largest at Kn=0 and is of the order of 10 percent. Qualitatively similar results are obtained for slip-flow heat transfer in circular tubes.

Thermal Conduction and Viscous Heating in Microscale Couette Flows

The direct simulation Monte Carlo (DSMC) method is used to investigate heat conduction and viscous heating in microscale Couette flows. When augmented by the slip boundary conditions, the continuum equations are found to predict thermal conductance accurately even in the transition and ballistic flow regime for Couette flows. The viscous heat generation rate, in contrast, is significantly underpredicted by the slip flow theory. We propose an alternative approach based on kinetic theory and present a compact analytic expression for the viscous dissipation rate. [S0022-1481(00)00104-3]

Rarefaction and Compressibility Effects in Gas Microflows

Gas microflows are encountered in many applications of Micro-Electro-Mechanical Systems (MEMS). Computational modeling and simulation can provide an effective predictive capability for heat and momentum transfer in microscales as well as means of evaluating the performance of a new microdevice before hardware fabrication. In this article, we present models and a computational methodology for simulating gas microflows in the slip-flow regime for which the Knudsen number is less than 0.3. The formulation is based on the classical Maxwell/Smoluchowski boundary conditions that allow partial slip at the wall. We first modify a high-order slip boundary condition we developed in previous work so that it can be easily implemented to provide enhanced numerical stability. We also extend a previous formulation for incompressible flows to include compressibility effects which are primarily responsible for the nonlinear pressure distribution in micro-channel flows. The focus of the paper is on the competing effects of compressibility and rarefaction in internal flows in long channels. Several simulation results are presented and comparisons are provided with available experimental data. A specific set of benchmark experiments is proposed to systematically study compressibility, rarefaction and viscous heating in microscales in order to provide validation to the numerical models and the slip-flow theory in general as well as to establish absolute standards in this relatively young field of fluid mechanics.

Simulation of heat and momentum transfer in complex microgeometries

In this article we present a time-accurate computational model based on the slip-flow theory to simulate momentum and heat transport phenomena in complex microgeometries, encountered in typical components of microdevices such as microcapillaries, microvalves, microrotors, and microbearings. In the first part, we present extensions to the classical Maxwell/Smoluchowski slip conditions to include high-order Knudsen number effects as well as to take into account the coupling of momentum and heat transfer through thermal creep and viscous heating effects. The numerical method is based on the spectral element technique; validation of the method is obtained by comparison of the numerical simulation results in simple prototype flows (e.g., channel slip-flows) with analytical results. Reduction of pressure drop in microchannels, reported in similar experimental studies, is investigated using slip-flow theory and simulations. In the second part, we consider model inlet flows and a slip-flow past a microcylinder. The effect of slip-flow on skin friction reduction and associated increase in mass flow rate as well as the variation of normal stresses is investigated as a function of Knudsen number. Finally, the effect of compressibility is examined and possible extensions of the current model to take into account such effect are discussed.

A kinetic analysis of unsteady evaporation and condensation with an Oseen-like approximation

A linearized kinetic model equation with an Oseen-like approximation was applied to study the moderately strong unsteady evaporation and condensation problems. Asymptotic solutions were obtained for the weak (linearized) evaporation and condensation problems, disjoining the fluid dynamic equation with the Oseen approximation from the Knudsen layer equation. The asymptotic solution yielded correct features of the flow due to evaporation or condensation. The kinetic model equation was further applied to study the moderately strong evaporation and condensation problems. The results showed good agreement with the previous results derived from the nonlinear treatments and confirmed that the conventional slip flow theory is applicable for studying the moderately strong evaporation and condensation problems.

Steady gas flows past bodies at small Knudsen numbers

Abstract Steady gas flows at small Knudsen numbers around arbitrary bodies (Asymptotic behavior for small Knudsen numbers of the solution of time-independent boundary value problems of the Boltzmann equation over a general domain) are considered when the Reynolds number of the system is of the order of unity. The generalized slip flow theory developed for the Boltzmann-Krook-Welander equation is extended for the standard Boltzmann equation. From the result, the effect of gas rarefaction on the flow (the relation between Boltzmann and hydrodynamic systems) is clarified, and several features of the force on a closed body in the gas are derived.

Spiral Grooved Visco-vacuum Pump

A spiral grooved vacuum pump which works in the pressure range from 1 to 1000 Pa is studied theoretically and experimentally. The flow in grooves and ridges on the pumping rotor is treated as a slip flow, and Boon and Tal's Narrow groove theory developed for viscoseals is used. The theoretical predictions show good agreement with the experimental results. Suggestions are made regarding the choice of design parameters, i.e., aspect ratio, helix angle, clearance parameter and groove-width ratio.

An Experimental Investigation of Molecular Rarefaction Effects in Gas Lubricated Bearings at Ultra-Low Clearances

The experimental investigation discussed here gives experimental confirmation of the slip-flow theory for modeling hydrodynamic gas bearings with clearances below 0.25 microns. An interferometric technique employing two CW lasers is used to measure the small clearances with an accuracy of 0.025 microns. The effects of molecular rarefaction are studied by operating the slider bearing in different gas media of different mean free paths. Bearings operating at extremely high local Knudsen numbers are studied without approaching excessively high bearing numbers. Experimentally measured trailing edge clearances and pitch angles are compared with theoretical predictions using the modified Reynolds equation with velocity slip boundary conditions. Excellent agreement between experiment and theory is found for clearances as high as 1.60 microns to as low as 0.075 microns with corresponding ambient Knudsen numbers of 0.04 and 2.51, respectively.

The bulk stress in a suspension of spherical particles of condensed phase in its slightly rarefied vapour gas

The rheological behaviour of a dilute suspension of spherical particles of condensed phase dispersed in its own slightly rarefied vapour gas is investigated on the basis of suspension theory (Batchelor 1970) and generalized slip-flow theory for a two-phase system of a gas and its condensed phase derived from the Boltzmann equation. The rarefaction of the gas and the phase-change process at the interfaces of the particles have the effect of reducing the Einstein coefficient of ϕ, volume fraction, in the expression for the effective viscosity in the suspension. In the case in which the pure rarefaction effect alone enters the problem, the coefficient is $\frac{5}{2}(1-2.702\,K)$ , where K is the Knudsen number, a rarefaction parameter defined by K = l / L , l and L being respectively the mean free path of gas molecules and the radius of a spherical particle. When both the rarefaction and the phase-change process are taken into account, this becomes $\frac{5}{2} (1-3.533\,K)$ . These modifications are not small, even at ordinary pressures, when the size of the particles is of the order of microns.

Collisions of Small Cloud Droplets: Gas Kinetic Effects

Abstract Gas kinetic effects in the region between two small approaching cloud droplets were investigated by means of slip flow theory. The resulting correction to the force coefficients derived for creeping flow eliminates the “collision barrier” noted in previous work and thus the need for assuming an abitrary collision criterion. It was found that inclusion of the gas kinetic correction leads to collision efficiencies for droplets 30 μ in radius.

Rarefied Gas Flow on a Semi-Infinite Flat Plate

The rarefied gas flow on a semi-infinite flat plate is studied from the point of view of the kinetic theory of gases. The flow is assumed to be at a low speed, incompressible and at constant temperature. The molecular velocity distribution function is split into two regions. The Krook equation is used as the basic equation which is used to derive the full range moment equations. The derived moment equations are solved in terms of Laplace transformation. The qualitative characters of the flow is analized rather than quantitative characters, since the simplified half-range distribution function, the Krook equation in stead of the Bo1tzmann equation, and the Oseen approximation are used. The results obtained show that the slip flow theory is adequate at the far down stream of the leading edge, and the effect of the free molecule flow has to take into account as the point considered approaches to the leading edge.

Kinetic Theory of Linear Shear Flow

The flow of a monatomic gas between two parallel plates kept at the same temperature and moving in opposite directions is studied. The relative velocity of the plates is much smaller than the speed of sound. The deviation from the equilibrium distribution, φ(c, x), satisfies the linearized Boltzmann equation. The customary boundary conditions are adopted in which a fraction of the molecules is specularly reflected and the rest emitted with a Maxwellian distribution characteristic of the plate. The method consists of setting φ = φ+ for cx > 0 and φ = φ− for cx < 0 so that positive and negative velocities are distinguished. We take φ± = a0±(x)cz + a1±(x)czcx. The space functions are determined by taking half-range velocity moments of the Boltzmann equation. Explicit results for the distribution function, flow velocity and stress are given for a general law of force. Numerical results are worked out for hard sphere molecules. The method treats both microscopic boundary conditions and conservation laws exactly. Precise results are obtained both for the low-pressure region and for the high-pressure coefficient of viscosity. The region of slip flow is analyzed. Maxwell's slip condition is remarkably close to the condition obtained here from the kinetic theory. In this region of pressures, the deviations from the velocity profile of the hydrodynamic slip flow theory are everywhere very small, and are completely negligible at distances greater than ⅛ of a mean free path from a plate.