Tangential momemtum accommodation in microtube

Experimental investigations of isothermal steady flows for various gases have been carried out in a silica micro tube. This study is focused on the mass flow rate measurements of these flows in slip regime using a suitable powerful platform. First we analyse, for each gas, the pertinence of a first or second order continuum treatment; then we deduce from experiments, using the appropriate treatment, the tangential momentum accommodation coefficient (TMAC) of each gas. The TMAC obtained for the various pairs of gas (nitrogen, argon, helium)/surface (fused silica) exclude a full diffuse reflection.     

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

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.     

Molecular dynamics simulations of shear-driven gas flows in nano-channels

Using the recently developed smart wall molecular dynamics algorithm, shear-driven gas flows in nano-scale channels are investigated to reveal the surface–gas interaction effects for flows in the transition and free molecular flow regimes. For the specified surface properties and gas–surface pair interactions, density and stress profiles exhibit a universal behavior inside the wall force penetration region at different flow conditions. Shear stress results are utilized to calculate the tangential momentum accommodation coefficient (TMAC) between argon gas and FCC walls. The TMAC value is shown to be independent of the flow properties and Knudsen number in all simulations. Velocity profiles show distinct deviations from the kinetic theory based solutions inside the wall force penetration depth, while they match the linearized Boltzmann equation solution outside these zones. Results indicate emergence of the wall force field penetration depth as an additional length scale for gas flows in nano-channels, breaking the dynamic similarity between rarefied and nano-scale gas flows solely based on the Knudsen and Mach numbers.     

Two-dimensional molecular dynamic simulations on accommodation coefficients in nanochannels with various wall configurations

The accommodation coefficients are often utilized in slip boundary conditions to characterize gas-wall interactions. Due to the insufficient transport of momentum and energy in nanochannels, the accommodation coefficients are always less than unity and greatly influenced by temperature and surface structures. In the present paper, a statistical algorithm of the accommodation coefficients was described in molecular dynamic method. The accommodation coefficients were calculated for various wall configurations in two-dimensional nanochannels. The channels were constituted by several layers of platinum atoms, which vibrated and attached to face centered cubic (FCC) lattice sites. The results revealed that the NMAC and EAC are sensitive with the spring constant and wall atom layers. Subtle distinctions in FCC lattice and nanoscale roughness had strong effects. On FCC (1 1 1) lattice plane, the TMAC in isothermal flows was larger, while the NMAC and EAC in thermal conductions are smaller, than those on FCC (1 1 0) lattice plane. Moreover, larger roughness induced more normal momentum transferred into tangential momentum so that the NMAC decreases while the TMAC and EAC increases for larger roughness. In addition, the accommodation coefficients are also affected by rarefaction that the TMAC and EAC decrease as the Knudsen number increases.     

A unified approach for nonslip and slip boundary conditions in the lattice Boltzmann method

This work proposed a unified approach to impose both nonslip and slip boundary conditions for the lattice Boltzmann method (LBM). By introducing the tangential momentum accommodation coefficient (TMAC), the present implementation can determine the change of the tangential momentum on the wall and then impose the correct boundary conditions for LBM. The simulation results demonstrate that this implementation is equivalent to the first-order slip model.     

Surface–gas interaction effects on nanoscale gas flows

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 (ε_wf/ε_ff). 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 ε_wf/ε_ff 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 ε_wf/ε_ff. Using MD predicted shear stress values and kinetic theory, tangential momentum accommodation coefficients (TMAC) are calculated as a function of ε_wf/ε_ff, 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.     

Smoluchowski thermostat: a numerical demonstration of the tangential momentum accommodation coefficient

In this research, numerical simulations are used to study the flow of a monoatomic rarefied gas inside a model cylinder with a nanopore diameter. In the rarefied gas regime, gas–gas collisions seldom occur and molecular transport can be fully attributed to gas–wall collisions. The most important parameter in describing the gas–surface interaction is the tangential momentum accommodation coefficient (TMAC). This parameter represents the fraction of gas–wall collisions that are reflected diffusely. Present work shows that the Smoluchowski thermostat, a simulation technique that mimics a diffusive gas–wall collision satisfies Maxwell’s boundary condition by calculating this coefficient numerically. The method involves placing an ideal gas in a weak constant gravitational field. The TMAC is determined by computing the correlation between the impinging and outgoing drift velocities at the cylinder surface. While most simulation studies compare their computed TMAC’s with other relevant studies, the method discussed here enables one to compare the computed TMAC with an input parameter.     

Incident energy dependence of the scattering dynamics of water molecules on silicon and graphite surfaces: the effect on tangential momentum accommodation

The scattering dynamics of water molecules on solid surfaces was investigated using the molecular beam technique. In contrast to the experiments previously reported in the literature, the range of incident energy was chosen to cover the typical kinetic energies of gas molecules in equilibrium at room temperature (35–130 meV). Even in such a narrow energy range, the angular distribution of scattered molecules is heavily affected by the incident energy, exhibiting both a nearly cosine distribution and a lobular distribution, which has a clear peak close to the specular direction. Interestingly, the tangential momentum accommodation coefficients (TMACs) estimated from the scattering experiments show opposite energy dependences on graphite (0001) and silicon (100) surfaces. As the incident energy increases, the TMAC decreases on the graphite surface, whereas it increases on the silicon surface. These trends can be attributed to the relatively large adsorption energy of water molecules on these surfaces and the atomic-scale surface corrugation, although a rigorous understanding requires further analysis by molecular dynamics simulations. Our findings suggest the need for an elaborate slip-flow model that takes account of the incident energy effect to accurately analyze water vapor flow in micro/nanostructures, which is ubiquitous in nature and engineering applications.     

Double-disk rotating viscous micro-pump with slip flow

In this paper numerical solution was provided for the 2D, axisymmetric Navier-Stokes equations coupled with energy equation for gaseous slip flow between two micro rotating disks pump. A first-order slip boundary condition was applied to all internal solid walls. The objective is to study the effect of Knudsen number, rotational Reynolds number and gap height on pump head, flow rate, coefficient of moments and overall micro-pump efficiency. Pump head, flow rate, coefficient of moments and pump efficiency were calculated for various pump operating conditions when the mass flow rate is applied at the pump inlet port. Detailed investigations were performed for rotational Reynolds number equals to 10. Effect of gap height between the two disks was studied. Effect of rotational Reynolds number on maximum flow rate and maximum pressure rise was simulated. The present numerical results for no-slip were compared with previously published experimental and theoretical data and found to be in a very good agreement. Knudsen number Kn values were found to be major parameters that affect the performance of pump. Pump performance decreases with increasing Kn. Optimal pump performance occurs around middle point of pump operating range. Pump operating range decreases with increasing Kn numbers. Pump performance is found to experience a steep degradation for Kn approaching 0.1. Maximum flow rate increases with rotational speed almost linearly. Maximum pressure rise also increases with rotational speed. Reducing gap height results in increasing maximum pressure rise, while increasing gap height results in larger maximum flow rate.     

The influence of slip conditions, wall properties and heat transfer on MHD peristaltic transport

The effects of both wall slip conditions and heat transfer on peristaltic flow of MHD Newtonian fluid in a porous channel with elastic wall properties have been studied under the assumptions of long-wavelength and low-Reynolds number. The analytical solution has been derived for the stream function and temperature. The results for velocity, temperature, stream function and heat transfer coefficient obtained in the analysis have been evaluated numerically and discussed briefly. The numerical result shows that more trapped bolus appears with increasing Knudsen number.     

Improved modified Reynolds equation for thin-film gas lubrication from an extended slip velocity boundary condition

An extended slip velocity boundary condition is derived from the regularized 13 moment equations firstly. Different from the existing slip velocity boundary condition, the slip coefficients of the extended one are not fixed, which will change with the wall accommodation coefficient and the Knudsen number of the gas flow. Using the extended slip velocity condition, an improved modified Reynolds equation for thin-film gas lubrication is established. From solving the improved modified Reynolds equation, the pressure distribution of the slider gas bearing is obtained and has a better agreement with that from the direct simulation Monte Carlo method under different pitch angles and wall velocities. It is found that the improved modified Reynolds equation can predict a more accurate pressure distribution of the slider gas bearing than the Fukui and Kaneko’s lubrication model from the linearized Boltzmann equation in the near transition regime.     

静电悬浮转子微陀螺高努森数下的空气阻尼力矩

针对工作于高真空密封腔中的静电悬浮MEMS陀螺微转子,为建立其旋转动力学模型,实现高精度恒速旋转控制,需要得到高努森数条件下的空气阻尼力矩特性。结合复杂结构的轮状扁平微转子,根据分子动理论和Christian模型,推导出压膜阻尼力矩特性;根据稀薄气体动力学,在考虑腔室内壁和转子表面具有不同切向动量适应系数的条件下,推导出滑膜阻尼力矩解析公式。以根据腔内空气流动特征进行的分区直接蒙特卡罗模拟(DSMC)结果为参考,总阻尼力矩系数的解析误差约为16%,表明解析模型具有较好的精度,对于高努森数条件下类似微器件的建模和系统设计具有较高的实用价值。     

Geometry effects on rarefied nanochannel flows

A three dimensional molecular dynamics method was used to study the effect of different geometries for rarefied gas flows in nanochannels. Argon molecules have been used. The velocity profiles in the channel were obtained and analyzed with three different channel geometries: a circular, a rectangular (square), and a slit channel. A channel width of 50 nm was used for the simulation. It was found that when using the same driving force, the maximum velocity of the flow increases when the geometry changes in the order from circular to rectangular to slit geometry, where the latter becomes 2–2.5 times as large compared with either the rectangular or circular channel. For Kn larger than 1.0, the rectangular channel showed a similar maximum and slip velocity as the circular channel while the velocity profile was qualitatively similar to the slit channel. The effect of different Knudsen numbers on the velocity profiles was also investigated. We found that for Kn larger than 2–3, the Knudsen number has a relatively small influence on the slip velocity for circular channels and rectangular channels. The effect of the accommodation coefficient on the average flow velocity for all three geometries was studied and expressed as an allometric equation model.     

An improved model of carbon nanotube conveying flow by considering comprehensive effects of Knudsen number

Although the theoretical model of carbon nanotube conveying flow has been evolving from under macroscale theory framework to under nanoscale theory framework, for now, the small-scale effects have yet to be considered thoroughly. Herein, after extending the compatibility condition, we propose an improved model. Compared with the previous models, the improved model is not only dependent on the nonlocal parameter, but also comprehensively takes all the factors related to Knudsen number, namely effective viscosity, slip boundary condition and non-uniform flow profile, into account. Based on this model, a formula of critical flow velocity is derived in addition to numerical results and our model gives a considerably decreased critical flow velocity. Besides, when Knudsen number and nonlocal parameter increase, the critical flow velocity goes down dramatically, which indicates that the effects of Knudsen number cannot be neglected, and we demonstrate that the dispute over nonlocal parameter may impair the reliability of theoretical prediction of critical flow velocity. We also find that the effects of nonlocal parameter and Knudsen number on critical flow velocity are probably uncoupled.     

Non-isothermal flow of rarefied gas through a long pipe with elliptic cross section

A non-isothermal rarefied gas flow trough a long tube with an elliptical cross section due to pressure and temperature gradients is studied on the basis of the S-model kinetic equation in the whole range of the Knudsen number covering both free molecular regime and hydrodynamic one. A wide range of the pipe section aspect ratio is considered. The mass flow rate is calculated as a function of the pressures and temperatures on the tube ends. The thermomolecular pressure effect has been modeled and the coefficient of the thermomolecular pressure difference has been calculated in whole range of the Knudsen number and in wide range of the pipe section aspect ratio.     

Analytical solution of mixed electromagnetic/pressure driven gaseous flows in microchannels

The influences of wall-slip/jump conditions on the fluid flow and heat transfer for hydrodynamically and thermally fully developed electrically conducting gaseous flow subject to an electromagnetic field inside a parallel plate microchannel with constant heat flux at walls are studied under the assumptions of a low-magnetic Reynolds number. The governing equations are non-dimensionalized and then analytical solutions are derived for the friction and the heat transfer coefficients. The fluid flow and the heat transfer characteristics obtained in the analytical solutions are discussed in detail for different parameters such as the Knudsen, Hartmann, and Brinkman numbers. The velocity profiles verify that even with a constant Knudsen number, applying a stronger electromagnetic field gives rise to an increase in the slip velocity. The results also reveal that on increasing the Hartmann number, the heat transfer rate as well as the friction factor is enhanced, whereas it tends to suppress the movement of the fluid. Further, it is found that the Nusselt and the Poiseuille numbers are less sensitive to the electromagnetic field effects with increase in rarefaction.     

Analytical solution of gas lubricated slider microbearing

The analytical solution of a two-dimensional, isothermal, compressible gas flow in a slider microbearing is presented. A higher order accuracy of the solution is achieved by applying the boundary condition of Kn ² order for the velocity slip on the wall, together with the momentum equation of the same order (known as the Burnett equation). The analytical solution is obtained by the perturbation analysis. The order of all terms in continuum and momentum equations and in boundary conditions is evaluated by incorporating the exact relation between the Mach, Reynolds and Knudsen numbers in the modelling procedure. Low Mach number flows in microbearing with slowly varying cross-sections are considered, and it is shown that under these conditions the Burnett equation has the same form as the Navier–Stokes equation. Obtained analytical results for pressure distribution, load capacity and velocity field are compared with numerical solutions of the Boltzmann equation and some semi-analytical results, and excellent agreement is achieved. The model presented in this paper is a useful tool for the prediction of flow conditions in the microbearings. Also, its results are the benchmark test for the verifications of various numerical procedures.     

Nonlocal vibration and instability analysis of carbon nanotubes conveying fluid considering the influences of nanoflow and non-uniform velocity profile

In this article, the influences of non-uniform velocity profile attributable to slip boundary condition and viscosity of fluid on the dynamic instability of carbon nanotubes (CNTs) conveying fluid are investigated. The nonlocal elasticity theory and the Euler–Bernoulli beam theory are employed to derive partial differential equation of nanotubes conveying fluid. Furthermore, a dimensionless momentum correction factor (MCF) is obtained as a function of Knudsen number (Kn) so as to insert the effects of non-uniform velocity profile into the equation of motion. In continuation, complex eigen-frequencies of the system are attained with respect to different boundary conditions, the momentum correction factor, slip boundary condition and nonlocal parameter. The results delineate that considering the effects of non-uniform velocity profile could diminish predicted critical velocity of flow. Therefore, the divergence instability occurs in the lower values of flow velocity. In addition, the MCF decreases through enhancement of Kn; hence, the effects of non-uniform velocity profile are more noticeable for liquid fluid than gas fluid.     

Numerical simulation of gas flow in rough microchannels: hybrid kinetic–continuum approach versus Navier–Stokes

The flow field in a rough microchannel is numerically analyzed using a hybrid solver, dynamically coupling kinetic and Navier–Stokes solutions computed in rarefied and continuum subareas of the flow field, respectively, and a full Navier–Stokes solver. The rough surface is configured with triangular roughness elements, with a maximum relative roughness of 5 % of the channel height. The effects of Mach number, Knudsen number (or Reynolds number) and roughness height are investigated and discussed in terms of Poiseuille number and mass flow rate. Discrepancies between full Navier–Stokes and hybrid solutions are analyzed, assessing the range of validity of Navier–Stokes equations provided with first-order slip boundary conditions for modeling gas flow along a rough surface. Results indicate that the roughness increases Poiseuille number and decreases mass flux in comparison with those for the smooth microchannel. Increasing rarefaction results in further enhancement of roughness effect. At the same time, the compressibility effect is more noticeable than the roughness one, although the compressibility effect is alleviated by increase in the rarefaction. It was found that, although the Navier–Stokes solution of the flow in a smooth channel is accurate up to Kn = 0.1, when relative roughness height is higher than 1.25 % significant errors already appear at Kn = 0.02.