Effect of Pressure on Fluid Damping in MEMS Torsional Resonators with Flow Ranging from Continuum to Molecular Regime

High quality factor of dynamic structures at micro and nano scale is exploited in various applications of micro electro-mechanical systems (MEMS) and nano electro-mechanical system. The quality factor of such devices can be very high in vacuum. However, when vacuum is not desirable or not possible, the tiny structures must vibrate in air or some other gas at pressure levels that may vary from atmospheric to low vacuum. The interaction of the surrounding fluid with the vibrating structure leads to dissipation, thus bringing down the quality factor. Depending on the ambient fluid pressure or the gap between the vibrating and the fixed structure, the fluid motion can range from continuum flow to molecular flow giving a wide range of dissipation. The relevant fluid flow characteristics are determined by the Knudsen number which is the ratio of the mean free path of the gas molecule to the characteristic flow length of the device. This number is very small for continuum flow and reasonably big for molecular flow. In this paper, we study the effect of fluid pressure on the quality factor by carrying out experiments on a MEMS device that consists of a double gimbaled torsional resonator. Such devices are commonly used in optical cross-connects and switches. We only vary fluid pressure to make the Knudsen number go through the entire range of continuum flow, slip flow, transition flow, and molecular flow. We experimentally determine the quality factor of the torsional resonator at different air pressures ranging from 760 Torr to 0.001 Torr. The variation of this pressure over six orders of magnitude ensures required rarefaction to range over all flow conditions. Finally, we get the variation of quality factor with pressure. The result indicates that the quality factor, Q, follows a power law, Q ∝P ^–r , with different values of the exponent r in different flow regimes. In the second part of the paper, we propose the use of effective viscosity for considering velocity slip conditions in solving Navier–Stokes equation numerically. This concept is validated with analytical results for a simple case and then compared with the experimental results presented in this paper. The study shows that the effective viscosity concept can be used effectively even for the molecular regime if the air-gap to length ratio is sufficiently small (h ₀/L<0.01). As this ratio increases, the range of validity decreases.     

Temperature factor effect on the structure of the separated flow within a supersonic gas stream

The effect of the temperature factor, that is, the ratio of the body temperature to the freestream stagnation temperature, on the structure of the separated flow formed in the presence of a concave corner in a supersonic stream is studied. The strong influence of the temperature factor on the separation zone length and the flow-generated aerodynamic characteristics is established. It is shown that for fairly large deflection angles this flow cannot be described by free interaction, or triple deck, theory.     

Optimal volume-corrected laplace-metropolis method

The article provides a refinement for the volume-corrected Laplace-Metropolis estimator of the marginal likelihood of DiCiccioet al. The correction volume of probability α in DiCiccioet al. is fixed and suggested to take the value α=0.05. In this article α is selected based on an asymptotic analysis to minimize the mean square relative error (MSRE). This optimal choice of α is shown to be invariant under linear transformations. The invariance property leads to easy implementation for multivariate problems. An implementation procedure is provided for practical use. A simulation study and a real data example are presented.     

ICF靶丸中液氢层厚度的确定

 利用靶丸内部的质量守恒，提出了计算惯性约束聚变靶丸内液氢层厚度的函数关系式，从而得到了求解惯性约束聚变靶丸内液氢层厚度的一般模型，并在此基础上对影响液氢层厚度的各类因素如温度、充气密度等进行了讨论。该模型的分析方法同样适用于惯性约束聚变中的燃料气体氘、氚或者氘氚混合物。     

火力发电厂废水中悬浮物处理的影响因素探讨

对火力发电厂废水中悬浮物处理的影响因素如pH值、水温、接触介质、进水水质、混凝剂加入量等进行分析，对主要影响因素水温和混凝剂加入量进行了试验分析，制定出废水悬浮物处理的合理方案。     

A new VOF‐based numerical scheme for the simulation of fluid flow with free surface. Part II: application to the cavity filling and sloshing problems

Finite element analysis of fluid flow with moving free surface has been performed in 2‐D and 3‐D. The new VOF‐based numerical algorithm that has been proposed by the present authors (Int. J. Numer. Meth. Fluids, submitted) was applied to several 2‐D and 3‐D free surface flow problems. The proposed free surface tracking scheme is based on two numerical tools; the orientation vector to represent the free surface orientation in each cell and the baby‐cell to determine the fluid volume flux at each cell boundary. The proposed numerical algorithm has been applied to 2‐D and 3‐D cavity filling and sloshing problems in order to demonstrate the versatility and effectiveness of the scheme. The proposed numerical algorithm resolved successfully the free surfaces interacting with each other. The simulated results demonstrated applicability of the proposed numerical algorithm to the practical problems of large free surface motion. It has been also demonstrated that the proposed free surface tracking scheme can be easily implemented in any irregular non‐uniform grid systems and can be extended to 3‐D free surface flow problems without additional efforts. Copyright © 2003 John Wiley & Sons, Ltd.     

Odd subgraphs and matchings

Let G be a graph and f : V(G)→{1,3,5,…}. Then a subgraph H of G is called a (1,f)-odd subgraph if deg_H(x){1,3,…,f(x)} for all xV(H). If f(x)=1 for all xV(G), then a (1,f)-odd subgraph is nothing but a matching. A (1,f)-odd subgraph H of G is said to be maximum if G has no (1,f)-odd subgraph K such that |K|>|H|. We show that (1,f)-odd subgraphs have some properties similar to those of matchings, in particular, we give a formula for the order of a maximum (1,f)-odd subgraph, which is similar to that for the order of a maximum matching.     

Invariants for normal completely positive maps on the hyperfinite II1 factor

We investigate certain classes of normal completely positive (CP) maps on the hyperfinite II₁ factorA. Using the representation theory of a suitable irrational rotation algebra, we propose some computable invariants for such CP maps. Dedicated to Professor K B Sinha     

The evolution of the cluster size distribution in a coagulation system

The coagulation equation with kernelK _ij =A+B(i+j)+C _ij and arbitrary initial conditions is studied analytically and a simple expression for the solution is found. For monodisperse initial conditions, we recover the known size distribution expressed in terms of a degeneracy factorN _k, which is determined by a recursion relation. For polydisperse initial conditions, a similar solution form is found, which includes a degeneracy factorN _kl, also determined by a recursion relation. The physical meaning ofN _kl and the recursion relation is given. A method to get explicit expressions forN _k andN _kl is illustrated. Finally, the pre-gel solution is given explicitly and a general method to get the post-gel solution is proposed.     

Rescaled mean spherical approximation structure factor for an aqueous suspension of polystyrene spheres

Rescaled mean spherical approximation (RMSA) has been used to calculate the structure factor for the aqueous suspension of polystyrene macroions with the interaction potential taken according to Derjaguin and Landau (1941) and Verwey and Overbeek (1948) (DLVO) model. The effects of charge over the macroion and size on the surface potential and therefore, the structure factor have been studied. The breakdown of the DLVO potential with an excess charge over the macroion (⩾800e) has been reported. The oscillation in the first peak height of structure factor versus wave vector curve with size has been correlated with the Debye length.