A comparative study of analytical squeeze film damping models in rigid rectangular perforated MEMS structures with experimental results

Several analytical models exist for evaluating squeeze film damping in rigid rectangular perforated MEMS structures. These models vary in their treatment of losses through perforations and squeezed film, in their assumptions of compressibility, rarefaction and inertia, and their treatment of various second order corrections. We present a model that improves upon our previously reported work by incorporating more accurate losses through holes proposed by Veijola and treating boundary cells and interior cell differently as proposed by Mohite et al. We benchmark all these models against experimental results obtained for a typical perforated MEMS structure with geometric parameters (e.g., perforation geometry, air gap, plate thickness) that fall well within the acceptable range of parameters for these models (with the sole exception of Blech’s model that does not include perforations but is included for historical reasons). We compare the results and discuss the sources of errors. We show that the proposed model gives the best result by predicting the damping constant within 10% of the experimental value. We study the validity of the proposed model over the entire range of perforation ratios (PR) by comparing its results with numerically computed results from 3D Navier-Stokes equation. These results are also compared with other analytical models. The proposed model shows considerably better results than other models, especially for large values of PR.     

Squeeze-film damping of circular microplates vibrating in a tilting motion

Accurate determination of squeeze-film damping (SFD) plays an important role in the design of high-Q microresonators. Many analytical models for predicting SFD on the microplate vibrating in a tilting motion have been well established in the past. However, most of the previous works focused on the rectangular torsion microplates. There are few analytical models for the SFD on the circular microplate vibrating in the tilting motion. Only one model was developed by Xia et al. (Microfluid Nanofluid 19:585–593, 2015). However, the gas in the air gap was treated as an incompressible gas in their model, and the perforation effect was not considered. This paper first studies the SFD on a non-perforated circular microplate vibrating in the tilting motion. The effects of both gas compressibility and rarefaction are considered in a modified Reynolds equation. The air pressure under the circular microplate is approximated by using Bessel series. A more accurate analytical expression for the damping and spring constants has been developed. Then, the model for the non-perforated microplates is extended to include the perforation effect. The present models are validated by comparison of the numerical results obtained by finite element method over a wide range of frequency and perforation ratios.     

Comparative numerical study of FEM methods solving gas damping in perforated MEMS devices

Three different numerical strategies are presented for the estimation of the damping force acting on perforated movable MEMS dampers. Results from the 2D Perforated Profile Reynolds (PPR) method and the simplified 2D ANSYS method are compared with accurate full 3D flow simulations. Altogether, 32 different topologies are compared varying, e.g., the dimensions of the square damper and the square holes, and the number of holes. The case of uniform perforation and perpendicular motion is studied. Oscillation in the low frequency regime is assumed, that is, the compressibility and inertia of the gas are ignored in the study. While the PPR method is in good agreement with the 3D simulations, the forces given by the ANSYS method were considerably smaller. The reasons for this are studied, and a compact expression to explain the small forces is derived.     

A silicon rectangular micro-orifice for gas flow measurement at moderate Reynolds numbers: design,fabrication and flow analyses

This work describes a micro-flowmeter for moderate flow rates of gases based on a differential pressure measurement. The micro-flowmeters consist of a microfabricated silicon–glass rectangular micro-orifice plate, with external pressure measurement. We experimentally evaluate the effects of geometrics parameters, Reynolds number and compressibility on the discharge coefficient. The paper examines a series of 13 rectangular micro-orifice sizes, with orifice hydraulic diameters ranging from 115 to 362 µm. The behavior of the discharge coefficient is presented for orifice Reynolds numbers ranging from 200 to 18000. Agreement is shown between the experimental and numerical results of the discharge coefficient. The micro-flowmeters measure moderate flow of air ranging from 1 to 106 mg/s. This demonstration implements a design method of micro-flowmeters that can be used in a broad range of microfluidic applications, such as microreactors and power MEMS.     

Analysis of laminar-to-turbulent transition for isothermal gas flows in microchannels

In this work the laminar-to-turbulent transition in microchannels of circular cross-section is studied experimentally. In order to single out the effects of relative roughness, compressibility and channel length-to-diameter ratio on the Reynolds number at which transition occurs, experimental runs have been carried out on circular microchannels in fused silica—smooth for all purposes—and in stainless steel (which possess a high surface roughness), with a diameter between 125 and 180 μm and a length of 5–50 cm through which nitrogen flows. For each tube the friction factor has been computed. The values of the critical Reynolds number have been determined plotting the Poiseuille number (i.e., the product of the friction factor, f, times the Reynols number, Re) as a function of the average Mach number between inlet and outlet. The transitional regime was found to start no earlier than at values of the Reynolds number around 1,800–2,000. It has been observed that surface roughness has no effect on the hydraulic resistance in the laminar region for a relative roughness lower than 4.4%, and that friction factor obeys the Poiseuille law, if it is correctly computed taking compressibility into account. It is found that recent correlations for the prediction of the critical Reynolds number in microchannels that link the relative roughness of the microtubes to the critical Reynolds number do not agree with the present results.     

A numerical molecular dynamics approach for squeeze-film damping of perforated MEMS structures in the free molecular regime

Accurate determination of the squeeze-film damping in rare air is crucial for the design of high-Q MEMS devices. In the past, for the MEMS structures with no perforations, there have been two approaches to treating the squeeze-film damping in rare air: the approach based on the continuum assumption and the approach using molecular dynamics (MD) method. The amount of squeeze-film damping can be controlled by providing perforations in microstructures. To model perforation effects on squeeze-film damping, many methods have been proposed. However, almost all the previous methods are based on the continuum assumption. Only one paper focuses on analytical modeling of squeeze-film damping of a perforated microplate using the MD method. Hutcherson and Ye (J Micromech Microeng 14:1726–1733, 2004) developed a novel MD method to model the squeeze-film damping in free molecular regime. The method possesses high computational efficiency. However, their work is valid only for non-perforated rectangular microplate. This paper presents a numerical MD approach for calculating the squeeze-film damping of a perforated rectangular plate and a perforated circular plate in free molecular regime. In Hutcherson and Ye’s work, the microplate is non-perforated. After each collision with the non-perforated plate, all the molecules are reflected to the substrate. In this paper, the plate is perforated. For the molecules in the air gap striking the surface of the perforated microplate, some of the molecules are reflected to the substrate. The rest leave the air gap through the perforations. This paper is an extension of the work done by Hutcherson and Ye (J Micromech Microeng 14:1726–1733, 2004). The accuracy of the present numerical MD approach is verified by comparing its results with the experimental results available in the literature and the finite element method results.     

Numerical simulation of the fluid–structure interaction between air blast waves and free-standing plates

A numerical method is used to compute the flow field corresponding to blast waves of different incident profiles propagating in air and impinging on free-standing plates. The method is suitable for the consideration of compressibility effects in the fluid and their influence on the plate dynamics. The history of the pressure experienced by the plate is extracted from numerical simulations for arbitrary blast strengths and plate masses and used to infer the impulse per unit area transmitted to the plate. The numerical results complement some recent analytical solutions in the intermediate range of plate masses and arbitrary blast intensities where exact solutions are not available. The resulting beneficial effect of the fluid–structure interaction (FSI) in reducing transmitted impulse in the presence of compressibility effects is discussed. In particular, it is shown that in order to take advantage of the impulse reduction provided by the FSI effect, large plate displacements are required which, in effect, may limit the practical applicability of exploiting FSI effects in the design of blast-mitigating systems.     

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.     

Experimental validation of compact damping models of perforated MEMS devices

Measured damping coefficients of six different perforated micromechanical test structures are compared with damping coefficients given by published compact models. The motion of the perforated plates is almost translational, the surface shape is rectangular, and the perforation is uniform validating the assumptions made for compact models. In the structures, the perforation ratio varies from 24 to 59%. The study of the structure shows that the compressibility and inertia do not contribute to the damping at the frequencies used (130–220 kHz). The damping coefficients given by all four compact models underestimate the measured damping coefficient by approximately 20%. The reasons for this underestimation are discussed by studying the various flow components in the models.     

A multiphase compressible model for the simulation of multiphase flows

A compressible model able to manage incompressible two-phase flows as well as compressible motions is proposed. After a presentation of the multiphase compressible concept, the new model and related numerical methods are detailed on fixed structured grids. The presented model is a 1-fluid model with a reformulated mass conservation equation which takes into account the effects of compressibility. The coupling between pressure and flow velocity is ensured by introducing mass conservation terms in the momentum and energy equations. The numerical model is then validated with four test cases involving the compression of an air bubble by water, the liquid injection in a closed cavity filled with air, a bubble subjected to an ultrasound field and finally the oscillations of a deformed air bubble in melted steel. The numerical results are compared with analytical results and convergence orders in space are provided.     

On the modified Reynolds equation model for the prediction of squeeze-film gas damping in a low vacuum

The Reynolds equation coupled with an effective viscosity model is often employed to predict squeeze-film damping of plate resonators in a low vacuum. Due to the lack of a sound theoretical foundation, a study is carried out to evaluate the performance of such an approach in the free-molecule regime and results are presented in this paper. An experimentally validated Monte Carlo simulation approach for the simulation of air damping is developed and employed for this study. First, effective viscosity models are developed for a parallel-plate resonator and a rotational resonator based on experimental measurements. These models are then coupled with Reynolds equation and employed to simulate air damping of resonators of the same type but with differing dimensions. The results are compared with Monte Carlo simulation results. It has been found that the modified Reynolds equation approach cannot accurately compute air damping for a general class of resonators and hence cannot serve as a predictive tool. The deficiency lies in the effective viscosity model that is assumed to be a function of Knudsen number only. Possible extensions of the modified Reynolds equation approach in the highly rarefied regime are also discussed.     

Convective–diffusive transport in parallel lamination micromixers

Effective mixing and a controllable concentration gradient are important in microfluidic applications. From the scaling law, decreasing the mixing length can shorten the mixing time and enhance the mixing quality. The small sizes lead to small Reynolds numbers and a laminar flow in microfluidic devices. Under these conditions, molecular diffusion is the main transport effect during the mixing process. In this paper, we present complete 2D analytical models of convective–diffusive transport in parallel lamination micromixers for a binary system. An arbitrary mixing ratio between solute and solvent is considered. The analytical solution indicates the two important parameters for convective–diffusive transport in microchannels: the Peclet number and the dimensionless mixing length. Furthermore, the model can also be extended to the mixing of multiple streams—a common and effective concept of parallel mixing in microchannels. Using laser machining and adhesive bonding, polymeric micromixers were fabricated and tested to verify the analytical results. The experimental results agree well with the analytical models.This revised version was published online in March 2005 with corrections to Eq. 12.     

A fine grid finite element computation of two-dimensional high Reynolds number flows

A velocity—pressure integrated, mixed interpolation, Galerkin finite element computation of the Navier-Stokes equations using fine grids, is presented. In the method, the velocity variables were interpolated using complete quadratic shape functions: and the pressure was interpolated using linear shape functions defined on a triangular element, which is contained inside the quadratic element for velocity variables. Comprehensive computational results for a cavity flow for Reynolds number of 400 through 10,000 and a laminar backward-facing step flow for Reynolds number of 100 through 900 are presented in this paper. Many high Reynolds number flows involve convection dominated motion as well as diffusion dominated motion (such as the fluid motion inside the subtle pressure driven recirculation zones where the local Reynolds number may become vanishingly small) in the flow domain. The computational results for both of the fluid motions compared favorably with the high accuracy finite difference computational results and/or experimental data available.     

Static bending of perforated nanobeams including surface energy and microstructure effects

This article aims to present comprehensive model and analytical solution to investigate the static bending behavior of regularly squared cutout perforated thin/thick nanobeams incorporating the coupled effect of the microstructure and surface energy for the first time. The perforation influence is considered to be deriving equivalent geometrical and material characteristics. The modified couple stress theory is adopted to incorporate the microstructure effect while the modified Gurtin–Murdoch surface elasticity model is employed to incorporate the surface stress effect in perforated nanobeams. A variational formulation based on minimization of the total potential energy principle is employed to derive the equilibrium equations of perforated nanobeams based on both Euler–Bernoulli and Timoshenko beams theories are developed to investigate the associated effect of the shear deformation due to perforation process. Additionally, Poisson’s effect is also incorporated. Analytical closed-form for the non-classical bending profiles as well as the rotational displacement are developed for both beam theories considering the simultaneous effect of both couple stress and surface stress for both uniformly distributed and concentrated loading patterns. The verification of the developed model is verified and compared with previous works, and an excellent agreement is obtained. The applicability of the developed model is demonstrated and applied to study and analyze the nonclassical bending behavior of regularly squared perforated simply supported beams under different loading conditions. Additionally, effects of the perforation configuration parameters, beam size as well as beam aspect ratio on the bending behavior of perforated beams in the presence of microstructure and surface stress effects are also investigated and analyzed. The obtained results reveal that both couple stress and surface stress significantly affect the bending behavior of regularly squared cutout perforated beam structures. Results obtained are supportive for the design, analysis and manufacturing of perforated NEMS applications.

     

Analytic damping model for an MEM perforation cell

The concept of the perforation cell is specified for compact modelling of perforated gas dampers with micromechanical dimensions. Both, analytic expressions and FEM simulations, are used to derive its flow resistance. An extensive set of FEM simulations is performed to characterize the flow resistance of the cell, and to derive approximations for different flow regions by fitting simple functions to them. Sinusoidal small-amplitude velocities are assumed, and micromechanical dimensions are considered with rare gas effects in the slip flow regime (Knudsen number <0.1). The model is capable of modelling all practical combinations of the perforation cell dimensions in a wide range of perforation ratios (1,...,90%). Its validity is verified with a Navier–Stokes solver, and it is shown to be accurate (relative error <4.5%) in the continuum and slip flow regimes. Estimates for cut-off frequencies due to inertial and compressibility effects are specified in a way that the maximum operation frequency of the model can be easily tested. Using a harmonic FEM solver, these estimates are verified. The perforation cell model is also applied to estimate the damping in a perforated rectangular damper (4,...,64 square holes). The damping predicted by the simple model is in moderate agreement with that obtained with 3D FEM simulations.     

Prediction of elastic compressibility of rock material with soft computing techniques

Mechanical and physical properties of sandstone are interesting scientifically and have great practical significance as well as their relations to the mineralogy and pore features. These relations are however highly nonlinear and cannot be easily formulated by conventional methods. This paper investigates the potential of the technique named as the relevance vector machine (RVM) for prediction of the elastic compressibility of sandstone based on its characteristics of physical properties. Based on the fact that the hyper-parameters may have effects on the RVM performance, an iteration method is proposed in this paper to search for optimal hyper-parameter value so that it can produce best predictions. Also, the qualitative sensitivity of the physical properties is investigated by the backward regression analysis. Meanwhile, the hyper-parameter effect of the RVM approach is discussed in the prediction of the elastic compressibility of sandstone. The predicted results of the RVM demonstrate that hyper-parameter values have evident effects on the RVM performance. Comparisons on the results of the RVM, the artificial neural network and the support vector machine prove that the proposed strategy is feasible and reliable for prediction of the elastic compressibility of sandstone based on its physical properties.     

Effects of disk surface roughness on static flying characteristics of air bearing slider by using a combined method of Reynolds equation and rough disk surface

Reynolds equation was modified with adding the surface roughness parameters to analyze the effects of disk surface roughness on the static flying characteristic of an air bearing slider. However, the modification demands the complicated mathematical expressions and related knowledge of physics and mathematics. In this paper, a combined method of Reynolds equation without introducing the roughness parameters and rough disk surface is proposed to investigate the effects of disk surface roughness on the static flying characteristics of an air bearing slider, it is different from those models of modified Reynolds equation introducing the disk surface roughness used by many researchers. More importantly, this method avoids the complicated numerical calculation resulted from the mathematical expressions including the Peklenik parameter \(\gamma\) and roughness R_a. By using an Ω air bearing slider, we investigated the effects of disk surface roughness on the static flying characteristics of this slider, the results show that the Peklenik parameter \(\gamma\) and roughness R_a have a significant influence on the pressure distribution, the load carrying capacity and the location of the pressure centre.     

涡激旋转下方柱小幅振荡模态的自由流线-边界层理论模型

针对刚体在流体中涡激旋转的数值结果表明,方柱在低雷诺数涡激旋转中有六种模态.本文针对其中的小幅振荡模态,通过综合无黏绕流情况下的自由流线理论与有黏平板的边界层理论,提出了一种理论模型;通过与浸没边界法所得数值结果进行对比,验证了该理论模型的有效性;并分析了小幅振荡模态的主要驱动力,解释了出现周期性振荡的原因.     

Compact model for a MEM perforation cell with viscous, spring, and inertial forces

A compact model for calculating damping, inertial, and spring forces in a perforated squeeze-film damper is reported. The repetitive pressure patterns around each perforation are utilized by analyzing the visco-acoustic wave transmission around the hole in a cylindrical volume, called perforation cell. The model is needed in applications where the acoustic wavelength of the oscillation is comparable with the dimensions of the perforation cell. The model is constructed of acoustic impedance two-ports. A novel model is derived for the air gap region, and a published two-port model is used for the hole. The impedances for these two-ports are derived from the low reduced frequency model that is equivalent with linearized, harmonic Navier–Stokes equations for acoustic wave propagation in thin channels. This model considers also the transition from the isothermal conditions at low frequencies to the adiabatic ones at high frequencies. The dimensions of MEMS structures are considered using slip conditions for velocities and temperatures. Also, an easy-to-use simplified model for frequencies where the squeeze number and the Reynolds numbers are below unity is derived. The analytical compact model is verified with FEM simulations using a harmonic solver for linearized Navier–Stokes equations with slip boundary conditions in a wide range of perforation ratios. The maximum relative error in the damping coefficient in the simulated cases was 20% upto the first resonant frequency.     

Lattice Boltzmann simulation of gas flow over micro-scale airfoils

In order to understand aerodynamic issues related to design and performance of micro air aircraft, in this paper isothermal low Reynolds number flows over micro-scale airfoils are simulated using the kinetic lattice Boltzmann method [Niu et al., Phys. Rev. E 76 (2007) 036711 ]. The sample of the micro-scale airfoil investigated is a flat plate with a 5% thickness ratio. Investigation shows that low Reynolds number flows over the micro-scale airfoils are viscous and compressible, and that rarefied effects in these kinds of flows are dominant. It is also found that the lift coefficients of the micro-scale plate airfoils are always smaller than the drag coefficients of them at Reynolds numbers less than 100, and this observation is consistent with the previous studies.