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.     

High Order Computation of the History Term in the Equation of Motion for a Spherical Particle in a Fluid

The historical evolution of the equation of motion for a spherical particle in a fluid and the search for its general solution are recalled. The presence of an integral term that is nonzero under unsteady motion and viscous conditions allowed simple analytical or numerical solutions for the particle dynamics to be found only in a few particular cases. A general solution to the equation of motion seems to require the use of computational methods. Numerical schemes to handle the integral term of the equation of motion have already been developed. We present here adaptations of a first order method for the implementation at high order, which may employ either fixed or variable computation time steps. Some examples are shown to establish comparisons between diverse numerical methods.     

Support loss for beam undergoing coupled vibration of bending and torsion in rocking mass resonator

Rocking mass resonator is widely used to design various sensors and actuators, which is a dual-axial symmetry resonator with high sensitivity. Q_support is the dominant energy loss mechanism influencing its high sensitivity. The anchor types and support loads applied to attachment points of rocking mass resonator are analyzed. Then support loss is simplified as a model with a beam attached to a finite thickness plate at its end. The general formulations for power radiated into support structure are given. An accurate analytical model of support loss for rocking mass resonator has been developed and verified by experiments. When the thickness of resonator is 240 μm, the measured Q can achieve a value of 589.1; while the thickness of resonator is reduced to 60 μm, the measured Q can achieve a value more than 8500. The derived model is general and might be applicable to various micro beam resonators and anchor types, providing significant insight to design of high-Q rocking mass devices.     

微悬臂梁DNA生物传感器纳米力学分析的重力模型及其应用

基于微悬臂梁技术的DNA生物传感器的纳米挠度与核苷长度、嫁接密度、杂交率、重力、盐溶液浓度、时间、温度变化、缓冲液流场特性等诸多因素有关.利用Zhang两变量方法,建立了静态模式中自重作用下微梁正应力、切应力、层间挤压应力及其挠度预测的解析模型.作为应用,采用McKendry实验参数,研究了重力因素对静态纳米挠度的影响,发现参考梁技术中生物层自重对纳米挠度的影响可以忽略,但单梁技术中非生物层自重对纳米挠度的影响不可以忽略;同时,利用DNA生物传感器表面应力的Zhang解析模型,考虑自重和表面应力因素,给出了微梁的最优尺寸,从而可使单梁技术中自重因素对静态挠度的影响降到最低.     

Numerical Simulation of Oldroyd-B Fluid in a Contraction Channel

A spectral element method coupled with the elastic viscous split stress technique for computing viscoelastic flows is presented. The rate of deformation tensor is introduced as an additional variable in the momentum equation, but not in the constitutive equation. The nonlinear rheological model, Oldroyd-B, is chosen to simulate the flow of a viscoelastic fluid. Numerical solutions are investigated based on a planar 4:1 abrupt contraction channel flow benchmark problem with different Weissenberg numbers. The results show a good agreement with other numerical predictions.     

Dynamics of a single guy cable

A numerical method has been used for selected dynamic analyses of single guy cables (both windward and leeward). This method has been validated by comparing its results to those of analytical formulae. Natural frequencies and mode shapes have been determined for both windward and leeward guys, vibrating about their deflected positions under increasing average wind speeds. A simple wind approach has been used to investigate the feasibility of large amplitude oscillations.     

Simulation of n-R planar manipulators

Modeling and simulation of n-DOF planar manipulators with revolute joints are presented in the paper. First, numerical effective equations to solve the direct kinematics of n-degree-of-freedom manipulators are derived. Following this, the dynamic model based on Lagrangian equation is developed. In the model the characteristics of planar manipulators are utilized and therefore the complexity of the model is increasing slower with the number of degrees of freedom compared with the general type of manipulators. Next, some cost functions for n-R manipulators and their gradients are presented. The derived models represent the basis of the software package “Planar Manipulators Toolbox” implemented in MATLAB/SIMULINK. The package allows the user to create and simulate different systems considering planar manipulators. MATLAB functions and SIMULINK blocks provided, include forward kinematics, dynamics and several utility functions.     

Formation of inverse Chladni patterns in liquids at microscale: roles of acoustic radiation and streaming-induced drag forces

While Chladni patterns in air over vibrating plates at macroscale have been well studied, inverse Chladni patterns in water at microscale have recently been reported. The underlying physics for the focusing of microparticles on the vibrating interface, however, is still unclear. In this paper, we present a quantitative three-dimensional study on the acoustophoretic motion of microparticles on a clamped vibrating circular plate in contact with water with emphasis on the roles of acoustic radiation and streaming-induced drag forces. The numerical simulations show good comparisons with experimental observations and basic theory. While we provide clear demonstrations of three-dimensional particle size-dependent microparticle trajectories in vibrating plate systems, we show that acoustic radiation forces are crucial for the formation of inverse Chladni patterns in liquids on both out-of-plane and in-plane microparticle movements. For out-of-plane microparticle acoustophoresis, out-of-plane acoustic radiation forces are the main driving force in the near-field, which prevent out-of-plane acoustic streaming vortices from dragging particles away from the vibrating interface. For in-plane acoustophoresis on the vibrating interface, acoustic streaming is not the only mechanism that carries microparticles to the vibrating antinodes forming inverse Chladni patterns: In-plane acoustic radiation forces could have a greater contribution. To facilitate the design of lab-on-a-chip devices for a wide range of applications, the effects of many key parameters, including the plate radius R and thickness h and the fluid viscosity μ, on the microparticle acoustophoresis are discussed, which show that the threshold in-plane and out-of-plane particle sizes balanced from the acoustic radiation and streaming-induced drag forces scale linearly with R and \(\sqrt \mu\), but inversely with \(\sqrt h\).     

Influence of viscous dissipation in a fluid between concentric rotating spheres

The steady motion of a viscous fluid contained between two concentric spheres which rotate about a common axis with different angular velocities is considered. A second-order method which was introduced previously by the authors to obtain numerical solutions of a class of Navier-Stokes problems coupled with a superimposed thermal field is successfully extended here to investigate the influence of the internal generation of heat by viscous dissipation on the thermal field. The resulting thermal field solutions are presented for various values of the Eckert number and the rotation ratio. It is shown that the inclusion of heat generation by viscous dissipation significantly alters the thermal field behavior, while the relative rates of rotation of the two spheres do not change the general character of the thermal field.     

Subspace identification of circulant systems

This article concerns the identification of a class of large scale systems called “circulant systems”. Circulant systems have a special property that allows them to be decomposed into simpler subsystems through a state transformation. This property has been used in literature for control design, and here we show how it can be used for system identification. The approach that is proposed here will both reduce the complexity of the problem as well as provide models which have a circulant structure that can be exploited for control design. A novel identification algorithm for circulant systems based on subspace identification is presented. The algorithm is then tested in simulation on an academic example of circulant system and on a realistic finite element model of a vibrating plate.     

Dissipative particle dynamics simulations for fibre suspensions in newtonian and viscoelastic fluids

A versatile model of fibre suspensions in Newtonian and viscoelastic fluids has been developed using dissipative particle dynamics method. The viscoelastic fluid is modelled by linear chains with linear connector spring force (the Oldroyd-B model), which is known to be a reasonable model for the so-called Boger fluid (a dilute suspension of polymer in a highly viscous solvent). The numerical results are in excellent agreement with the analytical results of the Oldroyd-B model in simple shear flow. An effective meso-scale model of fibre in DPD is proposed and then incorporated with simple Newtonian fluid and our Boger fluid to enable entirely study rheological properties of fibre suspensions in both Newtonian and viscoelastic solvents. The numerical results are well compared with available experimental data and other numerical models.     

On analytical and finite element modelling of piezoelectric extension and shear bimorphs

A number of well-established analytical and numerical modelling techniques of classical extension bimorphs are assessed and evaluated using general purpose finite element codes. This allows the clarification of several confusing common validation practices, using well-known experimental data and analytical formulas. New analytical formulas are presented and evaluated for the relatively new shear bimorph concept and a more general extension–shear one, resulting from the combination of the extension- and shear-piezoelectric modes via an off-axes polarisation. The corresponding new simple analytical formulas accurately reproduced the three-dimensional piezoelectric finite element results, confirming their usefulness for preliminary designs.     

Flow past an array of cells that are adherent to the bottom plate of a flow channel

Parallel-plate flow channels are used extensively in cell-biological research to investigate cell-substrate adhesion. However, an analytical relationship between the fluid force acting on a cell that is adherent to the bottom plate of a channel and the flow rate into the channel is yet to be established. A finite-difference scheme was used to evaluate the three-dimensional laminar flow past an array of uniformly distributed cells that are adherent to the bottom plate of a parallel-plate flow channel. Computational results indicated that the fluid force acting on a spherical cell can be computed within 10% accuracy by using the solution given by Goldman et al. [Goldman, A. J., Cox, R. G. and Brenner, H., Slow viscous motion of a sphere parallel to a plane wall. I. Motion through quiescent fluid. Chem. Engng Sci., 1967, 22, 637–651. Goldman, A. J., Cox, R. G. and Brenner, H., Slow viscous motion of a sphere parallel to a plane wall. II. Couette flow. Chem. Engng Sci., 1967, 22, 653–660.] — for a single sphere in contact with a planar wall in infinite shear flow — when the ratio of the cell radius (R_S) to the gap thickness between parallel plates (h) is less than (1/15). Goldman et al.'s solution begins to significantly overestimate the actual fluid force as the (R_S/h) ratio becomes larger than 1/15. When R_S/h) = 1/5, the fluid force computed by Goldman et al. is greater than the actual force by 30%. As an originally spherical cell aligns and elongates in the direction of flow, the fluid force acting on it decreases by 25%. In all cases, cell spreading leads to a more uniform distribution of fluid shear stress on the cell surface. Further computations indicate that fluid force on a spherical cell with surface projections (rough cell) is slightly smaller than that for a smooth spherical cell whose radius is equal to the maximum radial dimension of the rough cell.     

Numerical study of stress-transport turbulence models: Implementation and validation issues

The primary goal of this work is to implement, validate and compare in shear-free and simple wall-bounded turbulent flows the performance of five stress-transport turbulence models that have recently appeared in the open literature. A secondary goal of this work is to analyze and study the effort and difficulties encountered by programmers when implementing turbulence models developed by other researchers. The need for standardized procedures and for the development of efficient numerical techniques is advocated as a means to reduce the model-variance and code dependency of turbulent models. The second-order models chosen for this study are the Launder-Shima, the Jakirlic-Hanjalic, the elliptic-blending model of Manceau, the Turbulent Potential Model proposed by Perot and an unidentified model. For comparison reasons, Wilcox k-ω eddy-viscosity model was included in the study. The validation and the study of the performance of the models were performed through the comparison of the numerical solutions with experimental data and analytical solutions. The five benchmark flowfields considered in this study encompass the shear-free and wall-bounded regimes and are the flat plate without pressure gradient, the flow over a plate with a moderately adverse pressure gradient, and the self-similar flows of the mixing layer, the plane jet and the axi-symmetric jet. The tested stress-transport models produced results in general agreement with the experiments. However, no clear advantage of the stress-transport model over Wilcox k-ω model was noticed in these simple flowfields. The Launder-Shima model could not predict accurately the skin friction on a flat plate but it performed well in all the other cases. Although the test cases used were simple, a major difficulty encountered in this effort is the unreliability of the open literature as a resource for turbulence model implementation. A general lack of consistency was observed between model versions published in different journals or at different times. The detrimental effect that such a lack of structure and consistency has on the CFD community is discussed.     

Decoupled Energy Stable Schemes for a Phase Field Model of Three-Phase Incompressible Viscous Fluid Flow

We develop a numerical approximation for a hydrodynamic phase field model of three immiscible, incompressible viscous fluid phases. The model is derived from a generalized Onsager principle following an energetic variational formulation and is consisted of the momentum transport equation and coupled phase transport equations. It conserves the volume of each phase and warrants the total energy dissipation in time. Its numerical approximation is given by a set of easy-to-implement, semi-discrete, linear, decoupled elliptic equations at each time step, which can be solved efficiently using fast solvers. We prove that the scheme is energy stable. Mesh refinement tests and three numerical examples of three-phase viscous fluid flows in 3D are presented to benchmark the effectiveness of the model as well as the efficiency of the numerical scheme.     

The simpler GMRES method combined with finite volume method for simulating viscoelastic flows on triangular grid

An efficient solver integrating the restarted simpler generalized minimal residual method (SGMRES(m)) with finite volume method (FVM) on triangular grid is developed to simulate the viscoelastic fluid flows. In particular, the SGMRES(m) solver is used to solve the large-scale sparse linear systems, which arise from the course of FVM on triangular grid for modeling the Newtonian and the viscoelastic fluid flows. To examine the performance of the solver for the nonlinear flow equations of viscoelastic fluids, we consider two types of numerical tests: the Newtonian flow past a circular cylinder, and the Oldroyd-B fluid flow in a planar channel and past a circular cylinder. It is shown that the numerical results obtained by the SGMRES(m) are consistent with the analytical solutions or empirical values. By comparing CPU time of different solvers, we find our solver is a highly efficient one for solving the flow equations of viscoelastic fluids.     

A correct derivation of acceleration parameter for hopscotch and checkerboard (P,Q)-cyclic relaxation schemes

The correct method for applying the von Neumann stability analysis to composite finite difference schemes for numerical solution of partial differential equations is investigated. Our results provide justification of the hopscotch method and give correction to earlier analyses [1,7]. The methods employed here to analyze checkerboard and hopscotch iterative processes are also applicable to the study of more general composite (P, Q)-cyclic finite difference schemes.     

Viscous damping model for laterally oscillating microstructures

Viscous energy loss in oscillating fluid-film dampers that provide frictional shear for laterally-driven planar microstructures is investigated. It is found that Stokes-type fluid motion models viscous damping more accurately than Couette-type flow field. This paper characterizes the damping property of a fluid layer in terms of viscous energy dissipation, then derives analytic damping formulae for practical Q estimation. Theoretical Q-factors are compared to the experimental values, measured from surface-micromachined polysilicon resonators. Data reported by previous investigators are also analyzed and compared. The experimental results indicate that the Stokes-type damping model presents a more general damping treatment with better Q estimation, although discrepancies of 10 to 20% still remain between the estimated and measured Q