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.     

Analytical and numerical description for isothermal gas flows in microchannels

Analytical solutions for the pressure and the velocity profiles in a microchannel are derived from the quasi gasdynamic equations (QGD). An expansion method according to a small geometric parameter ɛ is undertaken to obtain the isothermal flow parameters. The deduced expression of the mass flow rate is similar to the analytical expression obtained from the Navier-Stokes equations with a second order slip boundary condition and gives results in agreement with the measurements. The analytical expression of the pressure predicts accurately the measured pressure distribution. The effects of the rarefaction and of the compressibility on pressure distributions are discussed. The numerical calculations based on the full system of the QGD equations were carried out for different sizes of the microchannels and for different gases. The numerical results confirm the validity of the analytical approach.     

Slip model for the ultra-thin gas-lubricated slider bearings of an electrostatic micromotor in MEMS

A new slip model derived by molecular dynamics has been used to investigate the ultra-thin gas-lubricated slider bearings beneath the three bushings of an electrostatic micromotor in micro-electro-mechanical systems (MEMS). Modified Reynolds equation is proposed based on the modified slip model. Analytical solutions for flow rate, pressure distribution, load carrying capacity and streamwise location using the modified Reynolds equation are obtained and compared with the results gained from those in the literature. It demonstrates that the new second-order slip model is of greater accuracy than that predicted by the first-order, second-order slip models and MMGL model and produces a good approximation to variable hard sphere (VHS) and variable soft sphere (VSS) models, which agree well with the solution obtained from the linearized Boltzmann equation. It is indicated that the slip effect reduces the pressure distribution and load carrying capacity, and shifts the streamwise location of the load carrying capacity, which should not be ignored to study the step-shaped slider bearings in micromotors for MEMS devices.     

CMOS-MEMS resonant pressure sensors: optimization and validation through comparative analysis

An optimized CMOS-MEMS resonant pressure sensor with enhanced sensitivity at atmospheric pressure has been reported in this paper. The presented work reports modeling and characterization of a resonant pressure sensor, based on the variation of the quality factor with pressure. The relevant regimes of air flow have been determined by the Knudsen number, which is the ratio of the mean free path of the gas molecule to the characteristic length of the device. The sensitivity has been monitored for the resonator design from low vacuum to atmospheric levels of air pressure. This has been accomplished by reducing the characteristic length and optimization of other parameters for the device. While the existing analytical model has been adapted to simulate the squeeze film damping effectively and it is validated at higher values of air pressure, it fails to compute the structural damping mechanisms dominant in the molecular flow regime, i.e. at lower levels of air pressure. This discrepancy has been solved by finite element modeling that has incorporated both structural and film damping effects. The sensor has been designed with an optimal geometry of 140 × 140 × 8 µm having 6 × 6 perforations along the row and column of the plate, respectively, for maximum Q, with an effective mass of 0.4 µg. An enhanced quality factor of 60 and reduced damping coefficient of 4.34 µNs/m have been obtained for the reported device at atmospheric pressure. The sensitivity of the manufactured device is approximately −0.09 at atmospheric pressure and increases to −0.3 at 40 kPa i.e. in the lower pressures of slip flow regime. The experimental measurements of the manufactured resonant pressure sensor have been compared with that of the analytical and finite element modeling to validate the optimization procedure. The device has been manufactured using standard 250 nm CMOS technology followed by an in-house BEOL metal-layer release through wet etching.

     

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

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 m.Thermally driven flows in microchannels are also described. They are particularly interesting for vacuum generation using microsystems without moving parts.     

A lattice Boltzmann study of gas flows in a long micro-channel

In this paper, the pressure-driven flow in a long micro-channel is studied via a lattice Boltzmann equation (LBE) method. With the inclusion of the gas–wall collision effects, the LBE is able to capture the flow behaviors in the transition regime. The numerical results are compared with available data of other methods. Furthermore, the effects of rarefaction and compressibility on the deviation of the pressure distribution from the linear one are also investigated.     

Analysis of Stokes flow in microchannels with superhydrophobic surfaces containing a periodic array of micro-grooves

Superhydrophobic surfaces have been demonstrated to be capable of reducing fluid resistance in micro- and nanofluidic applications. The objective of this paper is to present analytical solutions for the Stokes flow through microchannels employing superhydrophobic surfaces with alternating micro-grooves and ribs. Results are presented for both cases where the micro-grooves are aligned parallel and perpendicular to the flow direction. The effects of patterning the grooves on one or both channel walls are also analyzed. The reduction in fluid resistance has been quantified in terms of a dimensionless effective slip length, which is found to increase monotonically with the shear-free fraction and the periodic extent of each groove–rib combination normalized by the channel half-height. Asymptotic relationships have been derived for the normalized effective slip length corresponding to large and small limiting values of the shear-free fraction and the normalized groove–rib period. A detailed comparison has been made between transverse and longitudinal grooves, patterned on one or both channel walls, to assess their effectiveness in terms of enhancing the effective slip length. These comparisons have been carried out for small and large limiting values, as well as finite values of the shear-free fraction and normalized groove–rib period. Results for the normalized effective slip length corresponding to transverse and longitudinal grooves are further applied to model the Stokes flow through microchannels employing superhydrophobic surfaces containing a periodic array of micro-grooves inclined at an angle to the direction of the applied pressure gradient. Results are presented for the normalized effective slip lengths parallel to the direction of the applied pressure gradient and the normalized cross flow rate perpendicular to the direction of the applied pressure gradient.     

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.     

Experiments on rarefied gas flows through tubes

Experiments were carried out in order to evaluate the conductance of tubes of circular cross section as a function of the Knudsen number. The range of the rarefaction level spans from the free molecular flow to the continuum regime. A different experimental approach was followed with respect to previous researches in that the mass flow rate was assigned and the corresponding pressure drop was measured. Single tubes and a bundle of capillaries were adopted. The results are compared with the existing experimental data and analytical models. An analysis of the sensitivity of the conductance to the error in measurements, in free molecular flow, was also dealt with.     

A two-port model for wave propagation along a long circular microchannel

A compact model for oscillatory flow in a long microchannel with a circular cross-section is derived from the linearised Navier–Stokes equations. The resulting two-port model includes the effects of viscosity due to rarefied gas in the slip flow regime, inertia, compressibility and losses due to heat exchange. Both an acoustic impedance T network and an acoustic admittance Π network are presented for implementation in system level and circuit simulation tools. Also, reduced T and Π networks with constant component values are given to be used in the low frequency region. They are useful in time domain simulations, too. To verify the analytical model, simulations with a harmonic finite element solver for acoustic viscous flow are performed for microchannels exploiting the axisymmetry. The simulation results with both open and closed outlet conditions are compared with the two-port model with excellent agreement. Contribution of the slip conditions and the accuracy of the simple model are demonstrated.     

Experimental measurement on tangential momentum accommodation coefficient in a single microtube

The tangential momentum accommodation coefficient (TMAC) was investigated experimentally from the mass flow rate through a single microtube under the slip flow and the early part of the transition regime. The measurements were carried out by the constant-volume method under the mean Knudsen number smaller than 0.3, which is based on the mean pressure of the inlet and the outlet of the microtube, to apply the second-order slip boundary condition. To measure TMACs on various materials, quite large microtube was employed, which require the reduction in leakage. TMAC was obtained from the slip coefficient determined by the relation of the mass flow rate to the mean Knudsen number. The obtained mass flow rate was well explained by the theoretical equation. TMACs of deactivated-fused silica with argon, nitrogen, and oxygen were measured, showing the tangential momentum was not accommodated completely to the surface, and the values showed good agreement with previous studies. From the comparison between microtubes with different inner diameter, it is showed that TMAC is determined mainly by gas species and surface material.     

Determining the intrinsic permeability of tight porous media based on bivelocity hydrodynetics

For tight porous media, the permeability measured from the Darcy test is only an apparent permeability because the Klinkenberg effect occurs when the Knudsen number is high. To determine the intrinsic permeability, we need a simple and rigorous permeability correction that is valid in the entire flow regime. Thus, introducing a non-Maxwellian slip boundary condition, we develop the bivelocity hydrodynetics in this paper. The bivelocity hydrodynetics is defined by combining bivelocity hydrodynamics with kinetic theories. In the framework of the bivelocity hydrodynetics, we derive a simple and rigorous permeability correction based on the analytical solutions of rarefied gas flows in micro- and nano-tubes. After compared with conventional solutions and experiments, the present solutions are validated in the entire flow regime. Moreover, the validation is also a clear proof for the bivelocity theories.     

Subdynamic asymptotic behavior of microfluidic valves

Decreasing the Reynolds number of microfluidic no-moving-part flow control valves considerably below the usual operating range leads to a distinct "subdynamic" regime of viscosity-dominated flow, usually entered through a clearly defined transition. In this regime, the dynamic effects on which the operation of large-scale no-moving-part fluidic valves is based, cease to be useful, but fluid may be driven through the valve (and any connected load) by an applied pressure difference, maintained by an external pressure regulator. Reynolds number ceases to characterize the valve operation, but the driving pressure effect is usefully characterized by a newly introduced dimensionless number and it is this parameter which determines the valve behavior. This summary paper presents information about the subdynamic regime using data (otherwise difficult to access) obtained for several recently developed flow control valves. The purely subdynamic regime is an extreme. Most present-day microfluidic valves are operated at higher Re, but the paper shows that the laws governing subdynamic flows provide relations useful as an asymptotic reference.     

Lattice Boltzmann modeling of microchannel flows in the transition flow regime

Owing to its kinetic nature and distinctive computational features, the lattice Boltzmann method for simulating rarefied gas flows has attracted significant research interest in recent years. In this article, a lattice Boltzmann (LB) model is presented to study microchannel flows in the transition flow regime, which have gained much attention because of fundamental scientific issues and technological applications in various micro-electro-mechanical system (MEMS) devices. In the model, a Bosanquet-type effective viscosity is used to account for the rarefaction effect on gas viscosity. To match the introduced effective viscosity and to gain an accurate simulation, a modified second-order slip boundary condition with a new set of slip coefficients is proposed. Numerical investigations demonstrate that the results, including the velocity profile, the non-linear pressure distribution along the channel, and the mass flow rate, are in good agreement with the solution of the linearized Boltzmann equation, the direct simulation Monte Carlo (DSMC) results, and the experimental results over a broad range of Knudsen numbers. It is shown that taking the rarefaction effect on gas viscosity into consideration and employing an appropriate slip boundary condition can lead to a significant improvement in the modeling of rarefied gas flows with moderate Knudsen numbers in the transition flow regime.     

Numerical calculation of the bubbly two-phase flow around an airfoil

A model and the governing equations of two-phase bubbly flow are proposed; and the bubbly two-phase flow around an airfoil was calculated using a method based on the MacCormack method combined with the boundary-conforming curvilinear coordinates system. The compressibility and slip motion of the bubbles are considered in the calculation. The result revealed a wide ranging distribution of pressure and Mach number compared with single-phase flow. The effect of the slip velocity is also observed to be important in calculating the bubbly flow.     

Liquid–liquid microflows in micro-sieve dispersion devices with dual pore size

Here we present the liquid–liquid microflows and dispersion rules in micro-sieve devices with two different sized pores. The flow pattern, flux distribution and droplet size were investigated to discuss the effect of pore size deviation. Three flow patterns including dripping flow from a single active pore, dripping–dripping flow and dripping–jetting flow from two active pores were identified. A modified active pore model based on a pressure drop balance has been established. The model can predict the transition from a single active pore flow regime to a two active pore flow regime very well. In the latter regime, interactions between the small and large pores can result in dripping–dripping flow at low trans-pore flux and dripping–jetting flow at high trans-pore flux. Controlling the flow pattern in dripping–dripping flow is favorable to decreasing droplet polydispersity.     

Effects of interface deformation on flow through microtubes containing superhydrophobic surfaces with longitudinal ribs and grooves

This paper presents analytical and numerical results for Poiseuille flow through microtubes patterned with superhydrophobic surfaces which consist of alternative ribs and grooves aligned longitudinally with the flow direction. The superhydrophobic surface prevents the flowing liquid from penetrating the grooves and the liquid–gas interface experiences deformation as a consequence of a pressure difference across the interface. Employing a domain perturbation technique, the effects of a small interface deformation on the effective slip behavior are analytically quantified. For large interface deformations, numerical studies are performed to predict the effects of interface protrusion on the effective slip behavior of the superhydrophobic microtube. Comparisons are made between the effective slip behavior for tube and channel flows patterned with superhydrophobic surfaces containing alternating longitudinal ribs and grooves.     

Experiment and simulation of mixed flows in a trapezoidal microchannel

This paper presents experimental and numerical results of mixed electroosmotic and pressure driven flows in a trapezoidal shaped microchannel. A micro particle image velocimetry (μPIV) technique is utilized to acquire velocity profiles across the microchannel for pressure, electroosmotic and mixed electroosmotic-pressure driven flows. In mixed flow studies, both favorable and adverse pressure gradient cases are considered. Flow results obtained from the μPIV technique are compared with 3D numerical predictions, and an excellent agreement is obtained between them. In the numerical technique, the electric double layer is not resolved to avoid expensive computation, rather a slip velocity is assigned at the channel surface based on the electric field and electroosmotic mobility. This study shows that a trapezoidal microchannel provides a tapered-cosine velocity profile if there is any pressure gradient in the flow direction. This result is significantly different from that observed in rectangular microchannels. Our experimental results verify that velocity distribution in mixed flow can be decomposed into pressure and electroosmotic driven components.     

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.