Velocity Slip Boundary Condition Research Articles

1306 1階・2階滑り境界条件を用いた中程度希薄流の壁面滑り効果の数値的考察(OS13-2.流体力学の先端的な数値解法と応用(2),OS・一般セッション講演)

The present paper introduces a numerical method to a simple but general form of a higher-order slip boundary condition on a simple planar wall. The present numerical method for the continuum model in combination with first and second-order velocity slip boundary conditions is applied to two-dimensional rid-driven cavity flow. Rarefaction effects by slip conditions can be clearly seen in stream lines and velocity distributions depending on slip models.

Analysis of a porous-inclined slider bearing lubricated with magnetic fluid considering thermal effects with slip velocity

A theoretical model of a porous-inclined slider bearing lubricated with magnetic fluid has been considered together with slip velocity boundary condition. Our aim is to study the influence of various dimensionless parameters arising out of the analysis of the model. By assuming the viscosity µ = µ0 exp [-β(tm - t0)] of magnetic fluid, the expressions for mean temperature and load capacity have been obtained. It has been observed that both mean temperature field and load capacity are the functions of slip parameter, magnetic parameter, thermal parameter and permeability parameter. The dependence of the mean temperature field as well as of load capacity on these parameters has been seen graphically.

Numerical modelling of a microreactor for thermocatalytic decomposition of toxic compounds

Numerical modelling of a microreactor for thermocatalytic decomposition of toxic compounds In this paper a three-dimensional model for determination of a microreactor's length is presented and discussed. The reaction of thermocatalytic decomposition has been implemented on the base of experimental data. Simplified Reynolds-Maxwell formula for the slip velocity boundary condition has been analysed and validated. The influence of the Knudsen diffusion on the microreactor's performance has also been verified. It was revealed that with a given operating conditions and a given geometry of the microreactor, there is no need for application of slip boundary conditions and the Knudsen diffusion in further analysis. It has also been shown that the microreactor's length could be practically estimated using standard models.

Mass flow measurement through rectangular microchannel from hydrodynamic to near free molecular regimes

The mass flow rate through microchannels with rectangular cross section is measured for a wide Knudsen number range (0.0025-26.2) in isothermal steady conditions. The experimental technique called ‘Constant Volume Method’ is used for the measurements. This method consists of measuring the small pressure variations in the tanks upstream and downstream of the microchannel. The measurements of the mass flow rate are carried out for three gases (Helium, Nitrogen and Argon). The microchannel internal surfaces are covered with a thin gold coat of mean roughness characterized by Ra = 0.87nm (RMS). The continuum approach (Navier-Stokes equations), associated to the first order velocity slip boundary condition, was used in the slip regime (Knudsen number varies from 0.0025 to 0.1). The experimental velocity slip and the accommodation coefficients based on the Maxwell kinetic boundary condition were deduced. In the transitio-nal and near free molecular regimes the linearized kinetic BGK model was used to calculate numerically the mass flow rate. From the comparison of the numerical and measured values of the mass flow rate the accommodation coefficient was also obtained.

Simulation of Liquid Flowing over Surfaces with Micro-Structure

The exist shear-free air-water interface at the fluid-solid houndary is one important reason for drag reduction of ultrahydrophobic surface. Rectangular microchannel flows with various periodic micro-structured wall by introducing velocity slip boundary condition at low Reynolds number is investigated through computational fluid dynamics simulations(CFD). The purpose of the current study is to numerically find out the effects of periodic micro-structured wall on the flow resistance in rectangular microchannel with the different spacings between microridges ranging from 15 to 60 μm. The simulative results indicate that pressure drop with different spacing between microridges increases linearly with flow velocity and decreases monotonically with slip velocity; pressure drop reduction also increases with the spacing between microridges at the same condition of slip velocity and flow velocity.

Stokes' Second Problem with Velocity Slip Boundary Condition

The damping effect of microfluidics has great effect on the dynamic characteristics of MEMS devices. Based on the separation of variables and the integral transform methods, the Stokes' second problem is solved at the microscale velocity slip boundary condition and the analytical solution for velocity distribution is obtained. Furthermore, the expression of the penetration depth is gotten for Stokes model in this article. Through analysis, it’s found that due to the effect of the microscale velocity slip boundary condition, the velocity oscillation amplitude and the penetration depth have been reduced. Then the shear stress, the damping force and elastic force on the plate have been investigated. It’s shown that both the elastic coefficient and the damping coefficient increases as the oscillation frequency rises.

Numerical Investigations of Rarefied Gas Flows Using the Continuum Description by the CIP Method

Rarefied gas flows in the slip regime are investigated by the CIP method. The two-dimensional Couette/Poiseuille flows and two/three-dimensional rid-driven cavity flows are calculated using the Navier-Stokes (N-S) equation with the Maxwell's velocity slip boundary condition for various Knudsen numbers and accommodation coefficients. Numerical solutions show that the N-S equation with the slip boundary can give comparable solutions to analytical solutions and kinetic-type approaches. The continuum model is sufficiently applicable to the slip flow regime on the range of Knudsen numbers of some typical MEMS as 10-3 < Kn < 10-1 .

Thermophoresis of a Micro-Particle in Gaseous Media with Effect of Thermal Stress Slip

The present article pertains to a theoretical analysis for thermophoresis of a micro-particle in rarefied gas environment. Different from the conventional analyses, thermal stress effect in velocity slip boundary condition is taken into account. This interfacial thermal characteristic is always ignored in previous theoretical studies. The present theory with consideration of the thermal stress slip provides an appropriate modification and thus agrees well with the measured data of thermophoretic mobility of various particles. Order-of-magnitude analysis also qualitatively demonstrates enhancement of the thermal stress slip with reducing particle size or increasing characteristic temperature gradient. Finally, a parametric analysis shows that the thermal stress slip effect becomes more pronounced at conditions of high particle thermal conductivity and high Knudsen number as well.

Mathematical modelling and experimental investigation of gas flow in minichannels and microchannels

The first part of this study is focused on the numerical modelling and experimental investigation of transonic flow through a 2D model of the male rotor-housing gap in a dry screw compressor. Numerical simulations of the clearance flow are performed with the help of the in-house compressible Navier-Stokes solver. Experimental measurements based on the Schlieren method in Toepler configuration are carried out. The objective of the second part of the study is to derive the analytical solution of gas microflow development in a gap between two parallel plates. The microflow is assumed to be laminar, incompressible and the velocity slip boundary conditions are considered at the walls. The constant velocity profile is prescribed at the inlet. For the mathematical description of the problem, the Oseen equation is used. The analytical results are compared with the numerical ones obtained using the developed incompressible Navier-Stokes solver including the slip flow boundary conditions.

Effect of Velocity-Slip Boundary Conditions on Jeffery–Hamel Flow Solutions

In the present work, the Jeffery–Hamel flow problem has been studied using both first- and second-order velocity-slip models, and then compared with the no-slip model. The objectives are to observe the behavior of the flow predicted by the two slip models and to establish criteria for using the two velocity-slip models. The study concentrates on examining the effect of the change in the Knudsen number (Kn) on the velocity profiles, magnitude of slip at the wall, and skin friction coefficient. Assuming that a difference between the two slip models of the order of 10% or less justifies the use of the simple first-order model, the transitional Kn numbers have been found. These Kn numbers depend on the flow direction, being either inflow or outflow. Also, there are three distinct regions that specify where to use each of the no-slip, first-order, and second-order slip models. Further, the reversal of the flow has been investigated as a function of the Kn number and for different Re⋅α, where Re is Reynolds number and α is the wall angle. Using the second-order slip models, it is found that as the Kn number increases, reversal occurs at Re⋅α smaller than the 10.31 value at which flow reversal happens in the no-slip model, and increasing the Kn number leads to a reduction in the skin friction coefficient in all cases except when reversal occurs.

An Extension to the Navier–Stokes Equations to Incorporate Gas Molecular Collisions With Boundaries

We investigate a model for microgas-flows consisting of the Navier–Stokes equations extended to include a description of molecular collisions with solid-boundaries together with first- and second-order velocity-slip boundary conditions. By considering molecular collisions affected by boundaries in gas flows, we capture some of the near-wall effects that the conventional Navier–Stokes equations with a linear stress-/strain-rate relationship are unable to describe. Our model is expressed through a geometry-dependent mean-free-path yielding a new viscosity expression, which makes the stress-/strain-rate constitutive relationship nonlinear. Test cases consisting of Couette and Poiseuille flows are solved using these extended Navier–Stokes equations and we compare the resulting velocity profiles with conventional Navier–Stokes solutions and those from the BGK kinetic model. The Poiseuille mass flow rate results are compared with results from the BGK-model and experimental data for various degrees of rarefaction. We assess the range of applicability of our model and show that it can extend the applicability of conventional fluid dynamic techniques into the early continuum-transition regime. We also discuss the limitations of our model due to its various physical assumptions and we outline ideas for further development.

Fluid dynamics simulation of the high shear mixing process

The Eulerian–Eulerian two-fluid approach for modelling multiphase flows is used to simulate the flow in a high shear mixer. The results are compared with experimental velocity profiles for the solids phase at the wall in the mixer obtained using a high speed camera (Darelius et al. Chem. Eng. Sci. 62 (2007) 2366). The governing equations are closed using relations from the Kinetic Theory of Granular Flow (KTGF) combined with a frictional stress model due to Johnson and Jackson and Schaeffer and inter-phase drag due to Wen and Yu. In addition, calculations are presented for a model with a constant particle phase viscosity (CPV). Free slip and partial slip boundary conditions for the solid phase velocity at the vessel wall and the impeller have been utilized. The results show that the bed height could be well predicted by the partial slip model, whereas the free slip model could not capture the experimentally found bed height satisfactorily. For the KTGF model, the swirling motion of the rotating torus that is formed by the moving powder bed was over-predicted and the tangential wall velocity was under-predicted, probably due to the fact that the frictional stress model needs to be further developed, e.g. to tackle cohesive particles in dense flow. The CPV model gave predictions in good agreement with the experiments for a solids viscosity of 0.1 Pa s and a wall slip parameter of 0.005 m/Pa s. However, for a very low or very high value of the particle phase viscosity and for a high value of the wall slip parameter the agreement with experiments was poor. Interestingly, values of the viscosity that are commonly employed for fluidized beds seem applicable also in the present case.

CVD in Weakly Rarefied Rotating Disk Flows

AbstractCVD in a weakly rarefied rotating disk flow is numerically investigated using the SPIN code modified by including slip boundary conditions for velocity, concentration, and temperature, and thereby extending the capability of the code to describe weakly rarefied flows. A model reaction mechanism for silicon deposition, including the gas‐phase decomposition of silane to silylene and the surface reactions of silane and silylene, is used in order to demonstrate rarefied gas effects. Results show that, by taking into account the temperature slip, the deposition rate decreases from its value based on the continuum model, mainly due to either the decrease of the concentration of silylene, which is the dominant deposition species at high temperatures, or the decrease of the sticking coefficient of silane, which is the dominant deposition species at low temperatures. Compared to that of the temperature slip, the influence on the deposition rate of the velocity and concentration slips is negligible.

Microchannel heat transfer and dispersion of nanoparticles in slip flow regime with constant heat flux

If the hydrodynamic diameter of a channel is comparable with the mean free path of the gas molecules moving inside the channel, the fluid can no longer be considered to be in thermodynamic equilibrium and a variety of non-continuum or rarefaction effects can occur. To avoid enormous complexity and extensive numerical cost encountered in modeling of nonlinear Boltzmann equations, the Navier–Stokes equations can be solved considering the concepts of slip flow regime and applying slip velocity boundary conditions at the solid walls. In this study, the Navier–Stokes and energy equations for fluid flow in a microchannel in slip flow regime were solved analytically and temperature and velocity profiles were evaluated. The effect of different parameters on heat transfer and dispersion of nanoparticles was discussed. Dispersion of particles due to drag force, Saffman lift force, Brownian forces and gravity force was studied using a Lagrangian approach. The presented results could provide guidelines for heat transfer modeling in nanofluids.

Mathematical modeling of AC electroosmosis in microfluidic and nanofluidic chips using equilibrium and non-equilibrium approaches

AC electroosmotic micropumps are suggested to be powerful tools for electrolyte dosing in various micro- and nanofluidic systems. In this paper, we compare two modeling approaches for studying the AC electroosmosis in the following micro and nanochannel systems: (i) a traveling-wave AC pump with a spatially continuous wave of electric potential applied on a planar boundary, (ii) a traveling-wave AC pump with a wave of electric potential applied on a set of discrete planar electrodes, and (iii) an AC pump with a set of non-planar electrodes. The equilibrium approach is based on the use of capacitor–resistor boundary conditions for electric potential and the slip boundary conditions for velocity at electrode surfaces. The non-equilibrium approach uses the mathematical model based on the Poisson equation and the non-slip boundary conditions. We have observed discrepancies between the predictions given by the both models and then we have identified their possible reasons. The comparison of the equilibrium and non-equilibrium results further showed three important actualities: (a) how the equilibrium model overestimates or underestimates the net velocity, (b) how the velocity maxima in the frequency characteristics can be shifted, if the equilibrium model assumptions are not satisfied, (c) the parametric region where the equilibrium model is applicable. Because the data are obtained in a dimensionless form, they can be exploited for AC electroosmotic studies. We discuss the limitations of the equilibrium and non-equilibrium models and compare selected predictions with available experimental data.

Long time existence of solutions to 2d Navier–Stokes equations with heat convection

Global existence of regular solutions to the Navier&#8211;Stokes equations for $(v,p)$ coupled with the heat convection equation for $\theta $ is proved in the two-dimensional case in a bounded domain. We assume the slip boundary conditions for velocity a

A Deviation from Darcy's Law due to Unsaturated Flow in Dual-Scale Porous Media

The unsaturated flow encountered in the fibrous dual-scale porous media found in the liquid composite molding processes during the manufacture of polymer composites is the object of study here. The delayed absorption of fiber bundles in such media leads to the formation of a sink term in the continuity equation. A rapidly varying sink term near a moving macroscopic front in the dual-scale porous medium gives rise to two extra force terms in the volume-averaged momentum balance equation: a Brinkman term and an interfacial kinetic effect term. In this numerical work, the relative importance of these two terms in the gap-averaged momentum balance equation is studied. A finite element-based numerical method is used to model the 2-D isothermal flow between two parallel liquid-absorbing surfaces representing resin flow in the interbundle regions of a dual-scale porous medium during the unsaturated flow. It is observed that the Brinkman and the interfacial kinetic effect force terms become significant in a small region with rapidly varying sink term behind the moving flow front. The Beavers-Joseph slip-velocity boundary condition imposed at the gap-tow interfacial surfaces does not have any notable impact on the results. However, the correction factors tend to become less significant with an increase in the local Reynolds number. But the overall results suggest that a modified form of Darcy's law with the Brinkman and interfacial kinetic effect correction terms needs to be used near the flow front during liquid composite modeling mold-filling simulation in a dual-scale fiber mat.

A slip model for rarefied gas flows at arbitrary Knudsen number

A slip model for wall bounded rarefied gas flows is derived from kinetic theory. A corresponding modified Reynolds lubrication equation is obtained from the slip velocity boundary conditions at walls for high Knudsen number gas flows. The slip model in a simplest form has predictions very close to the numerical solutions of linearized Boltzmann equation in the whole Knudsen number range, and is preferable to the widely applied 1st order (Maxwell slip model), 2nd order, and 1.5 order slip models.

DSMC Simulation: Validation and Application to Low Speed Gas Flows in Microchannels

A direct simulation Monte Carlo method (DSMC) solver, adapted to the subsonic microflow, is developed under the object-conception language (C++). Some technical details critical in these DSMC computations are provided. The numerical simulations of gas flow in a microchannel are carried out using the developed DSMC solver. Streamwise velocity distributions in the slip flow regime are compared with the analytical solution based on the Navier–Stokes equations with the velocity slip boundary condition. Satisfactory agreements have been achieved. Furthermore, the domain of the validity of this continuum approach is discussed. Simulations are then extended to the transitional flow regime. Streamwise velocity distributions are also compared with the results of the numerical solutions of the linearized Boltzmann equation. We emphasize the influence of the accommodation coefficient on the velocity profiles and on the mass flow rate. The simulation results on the mass flow rate are compared with the experimental data, which allow us to validate the “experimental” technique of the determination of the accommodation coefficient.

Pressure Drop Predictions in Microfibrous Materials Using Computational Fluid Dynamics

Three-dimensional computational fluid dynamics simulations are performed for the flow of air through microfibrous materials for void fractions of 0.41 and 0.47 and face velocities ranging between 0.04ms and 1.29m∕s. The microfibrous materials consist of activated carbon powder with diameters of 137×10−6m entrapped in a matrix of cylindrical fibers with diameters of 8×10−6m. These sintered microfibrous materials are a new class of patented materials with properties that are advantageous compared to traditional packed beds or monoliths. Microfibrous materials have demonstrated enhanced heat and mass transfer compared to packed beds of particles of similar dimensions. In this paper, the simulations are used to predict the pressure drop per unit length through the materials and to analyze the details of the flow that are difficult to interrogate experimentally. Various geometric approximations are employed in order to allow the simulations to be performed in an efficient manner. The Knudsen number, defined as the ratio of the mean free path between molecular collisions to the fiber diameter, is 0.011; thus, velocity-slip boundary conditions are employed and shown to have only a minor effect on the pressure drop predictions. Significant effort is made to estimate numerical errors associated with the discretization process, and these errors are shown to be negligible (less than 3%). The computational predictions for pressure drop are compared to available experimental data as well as to two theory-based correlations: Ergun’s equation and the porous media permeability equation. The agreement between the simulations and the experiments is within 30% and is reasonable considering the significant geometric approximations employed. The errors in the simulations and correlations with respect to experimental data exhibit the same trend with face velocity for both void fractions. This consistent trend suggests the presence of experimental bias errors that correlate with the face velocity. The simulations generally underpredict the experimental pressure drop for the low void fraction case and overpredict the experimental pressure drop for the high void fraction case.