Showing papers on "Knudsen number published in 2005"

Microfluidics: Fluid physics at the nanoliter scale

Abstract: Microfabricated integrated circuits revolutionized computation by vastly reducing the space, labor, and time required for calculations. Microfluidic systems hold similar promise for the large-scale automation of chemistry and biology, suggesting the possibility of numerous experiments performed rapidly and in parallel, while consuming little reagent. While it is too early to tell whether such a vision will be realized, significant progress has been achieved, and various applications of significant scientific and practical interest have been developed. Here a review of the physics of small volumes (nanoliters) of fluids is presented, as parametrized by a series of dimensionless numbers expressing the relative importance of various physical phenomena. Specifically, this review explores the Reynolds number Re, addressing inertial effects; the Peclet number Pe, which concerns convective and diffusive transport; the capillary number Ca expressing the importance of interfacial tension; the Deborah, Weissenberg, and elasticity numbers De, Wi, and El, describing elastic effects due to deformable microstructural elements like polymers; the Grashof and Rayleigh numbers Gr and Ra, describing density-driven flows; and the Knudsen number, describing the importance of noncontinuum molecular effects. Furthermore, the long-range nature of viscous flows and the small device dimensions inherent in microfluidics mean that the influence of boundaries is typically significant. A variety of strategies have been developed to manipulate fluids by exploiting boundary effects; among these are electrokinetic effects, acoustic streaming, and fluid-structure interactions. The goal is to describe the physics behind the rich variety of fluid phenomena occurring on the nanoliter scale using simple scaling arguments, with the hopes of developing an intuitive sense for this occasionally counterintuitive world.

Slip Correction Measurements of Certified PSL Nanoparticles Using a Nanometer Differential Mobility Analyzer (Nano-DMA) for Knudsen Number From 0.5 to 83

Abstract: The slip correction factor has been investigated at reduced pressures and high Knudsen number using polystyrene latex (PSL) particles. Nano-differential mobility analyzers (NDMA) were used in determining the slip correction factor by measuring the electrical mobility of 100.7 nm, 269 nm, and 19.90 nm particles as a function of pressure. The aerosol was generated via electrospray to avoid multiplets for the 19.90 nm particles and to reduce the contaminant residue on the particle surface. System pressure was varied down to 8.27 kPa, enabling slip correction measurements for Knudsen numbers as large as 83. A condensation particle counter was modified for low pressure application. The slip correction factor obtained for the three particle sizes is fitted well by the equation: C = 1 + Kn (α + β exp(-γ/Kn)), with α = 1.165, β = 0.483, and γ = 0.997. The first quantitative uncertainty analysis for slip correction measurements was carried out. The expanded relative uncertainty (95 % confidence interval) in measuring slip correction factor was about 2 % for the 100.7 nm SRM particles, about 3 % for the 19.90 nm PSL particles, and about 2.5 % for the 269 nm SRM particles. The major sources of uncertainty are the diameter of particles, the geometric constant associated with NDMA, and the voltage.

Rarefaction and compressibility effects on steady and transient gas flows in microchannels

Abstract: The main theoretical and experimental results from the literature about steady pressure-driven gas microflows are summarized. Among the different gas flow regimes in microchannels, the slip flow regime is the most frequently encountered. For this reason, the slip flow regime is particularly detailed and the question of appropriate choice of boundary conditions is discussed. It is shown that using second-order boundary conditions allows us to extend the applicability of the slip flow regime to higher Knudsen numbers that are usually relevant to the transition regime. The review of pulsed flows is also presented, as this kind of flow is frequently encountered in micropumps. The influence of slip on the frequency behavior (pressure gain and phase) of microchannels is illustrated. When subjected to sinusoidal pressure fluctuations, microdiffusers reveal a diode effect which depends on the frequency. This diode effect may be reversed when the depth is shrunk from a few hundred to a few μm. Thermally driven flows in microchannels are also described. They are particularly interesting for vacuum generation using microsystems without moving parts.

Lattice Boltzmann simulation of rarefied gas flows in microchannels

Abstract: For gas flows in microchannels, slip motion at the solid surface can occur even if the Mach number is negligibly small. Since the Knudsen number of the gas flow in a long microchannel can vary widely and the Navier-Stokes equations are not valid for Knudsen numbers beyond 0.1, an alternative method that can be applicable to continuum, slip and transition flow regimes is highly desirable. The lattice Boltzmann equation (LBE) approach has recently been expected to have such potential. However, some hurdles need to be overcome before it can be applied to simulate rarefied gas flows. The first major hurdle is to accurately model the gas molecule and wall surface interactions. In addition, the Knudsen number needs to be clearly defined in terms of LBE properties to ensure that the LBE simulation results can be checked against experimental measurements and other simulation results. In this paper, the Maxwellian scattering kernel is adopted to address the gas molecule and surface interactions with an accommodation coefficient (in addition to the Knudsen number) controlling the amount of slip motion. The Knudsen number is derived consistently with the macroscopic property based definition. The simulation results of the present LBE model are in quantitative agreement with the established theory in the slip flow regime. In the transition flow regime, the model captures the Knudsen minimum phenomenon qualitatively. Therefore, the LBE can be a competitive method for simulation of rarefied gas flows in microdevices.

Predicting the thermal resistance of nanosized constrictions.

Abstract: Various devices and technologies using nanowires and nanoparticles are under intense investigation because of their promise. In these devices, nanowires or nanoparticles are typically in contact with another surface. The contact between a nanowire and a nanoparticle with a substrate forms a constriction of the order of a few nanometers. A continuum description of heat transport at these nanosized constrictions will break down. In this paper, an analytical model is presented in which the relevant length scales have been taken into consideration. The results show that the constriction resistance of nanoconstrictions is much higher than those predicted using macroscopic approaches. The Knudsen number is the key parameter for constriction formed between the same materials, whereas the microscopic Biot number based on phonon thermal boundary resistance is the key parameter for constriction formed between dissimilar materials. Finally, the model is applied to calculate the thermal resistance of the nanowire/planar interface.

Gas slippage effect on microscale porous flow using the lattice Boltzmann method

Abstract: A lattice Boltzmann method is developed for gaseous slip flow at the pore scale in microscale porous geometries. Flow characteristics through various porous structures are studied for different Knudsen numbers and inlet to outlet pressure ratios. It is found that the gas permeability is larger than the absolute permeability of porous media due to the gas slippage effect. Furthermore, the rarefaction influence on the gas permeability is more evident for porous structures with low porosity. The Klinkenberg equation is confirmed for the simulated porous structures. However, the second-order term of the Knudsen number $({\mathrm{Kn}}^{2})$ cannot be neglected for gaseous flow with relatively high Knudsen numbers. A model for predicting the pressure drop of the flow through microscale porous media is presented based on the Ergun equation and the Carman-Kozeny equation by taking into account the effects of gas rarefaction and compressibility.

Lattice Boltzmann method for gaseous microflows using kinetic theory boundary conditions

Abstract: The lattice Boltzmann method is developed to study gaseous slip flow in microchannels. An approach relating the Knudsen number with the relaxation time in the lattice Boltzmann evolution equation is proposed by using gas kinetic equation resulting from the Bhatnagar–Gross–Krook collision model. The slip velocity at the solid boundaries is obtained with kinetic theory boundary conditions. The two-dimensional micro-Couette flow, micro-Poiseuille flow, and micro-lid-driven cavity flow are simulated using the present model. It is found that the numerical results agree well with available analytical and benchmark solutions.

Analytical calculation of slip flow in lattice Boltzmann models with kinetic boundary conditions

Abstract: We present a mathematical formulation of kinetic boundary conditions for lattice Boltzmann schemes in terms of reflection, slip, and accommodation coefficients. It is analytically and numerically shown that, in the presence of a nonzero slip coefficient, the lattice Boltzmann develops a physical slip flow component at the wall. Moreover, it is shown that the slip coefficient can be tuned in such a way to recover quantitative agreement with the analytical and experimental results up to second order in the Knudsen number.

Lattice Boltzmann method at finite Knudsen numbers

Abstract: A modified lattice Boltzmann model with a stochastic relaxation mechanism mimicking "virtual" collisions between free-streaming particles and solid walls is introduced. This modified scheme permits to compute plane channel flows in satisfactory agreement with analytical results over a broad spectrum of Knudsen numbers, ranging from the hydrodynamic regime, all the way to quasi-free flow regimes up to Kn ~ 30.

Rarefaction and compressibility effects of the lattice-Boltzmann-equation method in a gas microchannel

Abstract: A wall equilibrium boundary condition for an implicit lattice-Boltzmann-equation method is proposed to simulate gas flows in a microchannel with rough surface on the characteristic length of gas molecules. The boundary condition is based on the assumption that impinging molecules reach equilibrium with the surface. The molecular mean free path used to define the Knudsen number is determined by the lattice speed and the relaxation time of the lattice-Boltzmann equation. With the wall equilibrium boundary condition and the appropriate relation defined for the Knudsen number and the relaxation time, the computed slip velocity and nonlinear pressure distribution along the microchannel are in excellent agreement with analytical solutions.

Flow of gaseous mixtures through rectangular microchannels driven by pressure, temperature, and concentration gradients

Abstract: The flow of binary gaseous mixtures through rectangular microchannels due to small pressure, temperature, and molar concentration gradients over the whole range of the Knudsen number is studied. The solution is based on a mesoscale approach, formally described by two coupled kinetic equations, subject to diffuse scattering boundary conditions. The model proposed by McCormack substitutes the complicated collision term and the resulting kinetic equations are solved by an accelerated version of the discrete velocity method. Typical results are presented for the flow rates and the heat fluxes of two different binary mixtures (Ne–Ar and He–Xe) with various molar concentrations, in two-dimensional microchannels of different aspect (height to width) ratios. The formulation is very efficient and can be used instead of the classical method of solving the Navier–Stokes equations with slip boundary conditions, which is restricted by the hydrodynamic regime. Moreover, the present formulation is a good alternative to the direct simulation Monte Carlo method, which often becomes computationally inefficient.

The driven cavity flow over the whole range of the Knudsen number

Abstract: The flow of a rarefied gas in a rectangular enclosure due to the motion of the upper wall is solved over the whole range of the Knudsen number. The formulation is based on the two–dimensional linearized Bhatnagar-Gross-Krook (BGK) kinetic equation with Maxwell diffuse-specular boundary conditions. The integro-differential equations are solved numerically implementing the discrete velocity method. The discontinuity at the boundaries between stationary and moving walls is treated accordingly. A detailed investigation of the rarefaction effects on the flow pattern and quantities is presented over the whole range of the Knudsen number and various aspect (height/width) ratios. Numerical results of flow characteristics, including the streamlines, the velocity profiles, the pressure and temperature contours, and the drag force of the moving wall, are presented for different aspect ratios and various degrees of gas rarefaction from the free molecular through the transition up to the continuum limit. On several occasions, depending upon the flow parameters, in addition to the main vortex, corner eddies are created. As the depth of the cavity is increased, these eddies grow and merge into additional vortices under the top one. The mesoscale kinetic-type approach proves to be efficient and suitable for problems that incorporate multiscale physics, such as the present nonequilibrium flow.

Developing Free-Convection Gas Flow in a Vertical Open-Ended Microchannel Filled with Porous Media

Abstract: The developing hydrodynamic and thermal behaviors of free convection gas flow in a vertical open-ended parallel-plate microchannel filled with porous media are investigated numerically. The extended Darcy-Brinkman-Forchheimer model is used to model the flow in porous medium and the solid and fluid media are not assumed in local thermal equilibrium. The microflow regime considered is the slip flow regime. The slip in velocity and jump in temperature are found to decrease in the axial direction of the flow. The friction factor is found to decrease as Knudsen number, Forchheimer number and Grashof number are increased. However, the friction factor is found to increase as Darcy number increased. On the other hand, Nusselt number is found to decrease as Knudsen number, Darcy number and thermal conductivity ratio are increased, whereas it increased as Forchheimer number, Grashof number and Biot number are increased.

Simulation of gas flow in microchannels with a sudden expansion or contraction

Abstract: Two-dimensional simulations based on the isothermal lattice-Boltzmann method have been undertaken on microchannels with a sudden expansion or contraction. The study provides insight into the analysis of flows in complicated microdevices. The flow is pressure driven, and computations are performed for several Knudsen numbers, and area and pressure ratios, allowing the effects of compressibility and rarefaction to be assessed. The pressure drop for both the converging and diverging channels shows a discontinuity in slope at the junction, and is accompanied by a jump in velocity. The pressure drop in each section can be predicted well by the theory for straight channels. The mass flow ratio between converging and diverging channels is close to unity, and the streamlines are attached in both cases. It is deduced that compressibility and rarefaction have opposite effects on the flow. These results suggest that complex channels of the type considered here can be understood in terms of their primary units, and they experience only small secondary losses.

Extending the validity of squeezed-film damper models with elongations of surface dimensions

Abstract: Compact models for micromechanical squeezed-film dampers with gap sizes comparable to the surface dimensions are presented. Two different models considering both the border flow and non-uniform pressure distribution effects are first derived for small squeeze numbers. In the first 'surface extension' model the border effects are considered simply by calculating the damping with extended surface dimensions, and in the second 'border flow channel' model an additional short fictitious flow channel is placed at the damper borders. Utilizing a large amount of two-dimensional (2D) FEM simulation results by varying the damper dimensions, mainly the ratio a/h between the surface length and the air gap height, surface elongations are extracted using both elongation models. Both linear and torsional modes of motion are considered at the continuum flow regime. These results show that the 'surface extension' model is superior, since the extracted elongation Δa is almost constant (Δa = 1.3h), leading to a very simple model. Next, the rare gas effects are included in the 'surface extension' model in the slip flow regime (Knudsen number 0 4 in the linear motion and for a/h > 10 in the torsional motion. The model assumes incompressible flow and thus the maximum frequency where the models are valid is limited. In typical MEMS topologies where the elongations must be considered, this means that the models are valid below frequencies of 500 kHz. To also model rectangular 2D squeezed-film dampers, these elongations are applied directly in the surface length and width used in the compact models. Comparison with three-dimensional (3D) FEM simulations shows that the new model gives excellent results, and it extends the validity range of existing compact models. The maximum relative error of the models is smaller than 10% for a/h > 16 in the linear motion and for a/h > 16 in the torsional motion. The new surface extension model is useful in simulating both the circuit level and the system level behavior of gas-damped microelectromechanical devices with aspect ratios greater than 2 in the time and frequency domains.

Direct Simulation Monte Carlo Simulations of Hypersonic Flows With Shock Interactions

Abstract: The capabilities of a relatively new direct simulation Monte Carlo (DSMC) code are examined for the problem of hypersonic laminar shock/shock and shock/boundary-layer interactions, where boundary-layer separation is an important feature of the flow Flow about two model configurations is considered, where both configurations (a biconic and a hollow cylinder-flare) have recent published experimental measurements The computations are made using the DS2V code of Bird, a general two-dimensional/axisymmetric time-accurate code The current focus is on flows produced in ground-based facilities at Mach 12 and 16 test conditions with nitrogen as the test gas and the test models at zero incidence The freestream Knudsen numbers, with the characteristic length equal to the test model diameter, range from 00008 to 00004, consequently demanding computations for DSMC simulations Results presented highlight the sensitivity of the calculations to grid resolution, sensitivity to physical modeling parameters, and comparison with experimental measurements Information is provided concerning the flow structure and surface results for the extent of separation, heating, pressure, and skin friction

A continuum approach to phoretic motions: Thermophoresis

Abstract: A purely continuum theory for the thermophoretic velocity of aerosol and hydrosol particles in the zero Knudsen number, near continuum limit, Kn = 0 + , valid for both gases and liquids, is proposed. This theoretical result is based upon a fundamentally modified version of the traditional equations governing continuum fluid motion, one which accounts for an intrinsic difference in a fluid's barycentric (mass-based) velocity and its kinematic velocity of volume, this difference arising during molecular transport processes in fluids within which a mass density gradient exists. Our continuum-scale approach contains no free parameters, nor does it rely upon any sub-continuum, molecular concepts, such as Maxwell's thermally-induced velocity-slip condition. The resulting expression for the thermophoretic velocity of a non-Brownian, spherical particle agrees both constitutively and phenomenologically with available correlations of such velocity data in gases, as well as with the more limited data for liquids. Furthermore, the effect of shape and orientation is discussed for the case of non-spherical particles, with specific results furnished for effectively non-conducting particles. Agreement of the theory with the data furnishes explicit experimental support of the non-traditional fluid-mechanical equations utilized herein.

Direct Numerical Solution Of The Boltzmann Equation

Abstract: Progress in computer hardware and improvement of numerical methods made solution of the Boltzmann equation for rather complex gas dynamic problems real. The method developed by the author is based on a projection technique for evaluation of the collision operator. The computed collision integral is conservative by density, impulse, and energy, and became equal to zero when the solution has a form of the Maxwellian distribution. The later feature sharply increases its efficiency, especially for the near equilibrium flows. The method is extended on a mixture of gases and the gases with internal degrees of freedom, where it can incorporate real physical parameters of molecular potential and of internal energy spectrum. Examples of computations for a range of Mach and Knudsen numbers are presented.

Acoustic properties of rarefied gases inside pores of simple geometries.

Abstract: Analytical solutions describing propagation of monochromatic acoustic waves inside long pores of simple geometries and narrow flat slits are obtained with accounting for gas rarefaction effects. It is assumed that molecular nature of gas is important in Knudsen layers near solid boundaries. Outside the Knudsen layers, the continuum approach is used. This model allows for extension of acoustic analysis to regions of low pressures and microscopic cross-sectional sizes of channels. The problem is solved using linearized Navier-Stokes equations with the boundary conditions that resulted from the first-order approximation with respect to small Knudsen number Kn. For slits and pores of circular and square cross sections, the theoretical dependencies of the dynamic density in the low-frequency range are compared with those that resulted from known experimental data on steady-state flows of rarefied gases in uniform channels. Despite the formal restriction Kn 0.01, especially at low frequencies. The obtained results may be used for analyses of acoustic properties of waveguides, perforated panels, micro-channels and pores in wide range of gas pressures as well as for stationary flows of rarefied gases through long uniform pipes etc.

Derivation of 13 Moment Equations for Rarefied Gas Flow to Second Order Accuracy for Arbitrary Interaction Potentials

Abstract: A recent approach to derive transport equations for rarefied gases from the Boltzmann equation within higher orders of the Knudsen number [H. Struchtrup, Phys. Fluids, 16 (2004), pp. 3921--3934] is used to derive a set of 13 moment equations for arbitrary molecular interaction potentials. It is shown that the new set of equations is accurate to second order, while Grad's original 13 moment equations are of second order accuracy only for Maxwell molecules and Bhatnagar--Gross--Krook models.

Gas flow in microchannels - a Lattice Boltzmann method approach

Abstract: Gas flow in microchannels can often encounter tangential slip motion at the solid surface even under creeping flow conditions. To simulate low speed gas flows with Knudsen numbers extending into the transition regime, alternative methods to both the Navier–Stokes and direct simulation Monte Carlo approaches are needed that balance computational efficiency and simulation accuracy. The lattice Boltzmann method offers an approach that is particularly suitable for mesoscopic simulation where details of the molecular motion are not required. In this paper, the lattice Boltzmann method has been applied to gas flows with finite Knudsen number and the tangential momentum accommodation coefficient has been implemented to describe the gas-surface interactions. For fully-developed channel flows, the results of the present method are in excellent agreement with the analytical slip-flow solution of the Navier–Stokes equations, which are valid for Knudsen numbers less than 0.1. The present paper demonstrates that the lattice Boltzmann approach is a promising alternative simulation tool for the design of microfluidic devices.

Diffuse-reflection boundary conditions for a thermal lattice Boltzmann model in two dimensions: evidence of temperature jump and slip velocity in microchannels.

Abstract: We discuss the implementation of diffuse reflection boundary conditions in a thermal lattice Boltzmann model for which the upwind finite difference scheme is used to solve the set of evolution equations recovered after discretization of the velocity space. Simulation of heat transport between two parallel walls at rest shows evidence of temperature jumps at the walls that increase with Knudsen number. When the walls move in opposite directions with speeds $\ifmmode\pm\else\textpm\fi{}{u}_{W}$, fluid velocity slip is observed at the walls, together with temperature jumps.

Oscillatory shear-driven gas flows in the transition and free-molecular-flow regimes

Abstract: We investigate oscillatory shear-driven gas flows in the transition and free-molecular-flow regimes. Analytical results valid through slip flow and the early transition regime are obtained using a recently proposed, rigorous second-order slip model with no adjustable coefficients. Analytical solution of the collisionless Boltzmann equation provides a description of the high Knudsen number limit (Kn⪢1) including the bounded shear layers present in the limit of high oscillation frequency. These layers are analogous to the Stokes layers observed in the Kn⪡1 limit, but contrary to the latter, they exhibit a nonconstant wave speed as demonstrated by Park, Bahukudumbi, and Beskok in Phys. Fluids. 16, 317 (2004). All theoretical results are validated by direct Monte Carlo simulations. We find that the second-order slip results are in good agreement with direct simulation Monte Carlo (DSMC) solutions up to Kn≈0.4; in some cases these results continue to provide useful approximations to quantities of engineering i...

Viscosity and slip velocity in gas flow in microchannels

Abstract: It was shown that as a result of the interaction of part of the molecules with a wall, the viscosity in the Knudsen layer is lower in comparison with the viscosity in the bulk fluid. The slip condition of Maxwell depends on the normal gradient of the tangential velocity on the wall. The reduction of the viscosity increases the tangential velocity and its normal gradient, and, therefore, increases the slip. The correction factors for the Maxwell expression for one-dimensional plane and tubular isothermal and incompressible flows as a function of the Knudsen number and of the kind of interaction of molecules with the wall were derived.

Transient Free Convection Fluid Flow in a Vertical Microchannel as Described by the Hyperbolic Heat Conduction Model

Abstract: The transient hydrodynamics and thermal behaviors of fluid flow in an open-ended vertical parallel-plate microchannel are investigated analytically under the effect of the hyperbolic heat conduction model. The model that combines both the continuum approach and the possibility of slip at the boundary is adopted in this study. The effects of Knudsen number Kn and thermal relaxation time τ on the microchannel hydrodynamics and thermal behaviors are investigated using the hyperbolic and the parabolic heat conduction models. It is found that as Kn increases, the slip in the hydrodynamic and thermal boundary condition increases. Also, this slip increases as τ decreases.