Inertial effects on the flow of capsules in cylindrical channels

A front-tracking method was used to study moderate to large-sized capsules flowing in cylindrical channels at Reynolds numbers ranging from 0.1 to 225. Two different constitutive equations, the Neo-Hookean and Skalak laws were considered to describe the mechanics of the thin membrane. The effect of capsule size, elastic capillary number, and Reynolds number on the shape, migration velocity, and extra pressure loss were determined. The deformation of the capsules was strongly tied to the size of the capsule compared with the channel diameter with larger capsules deforming more due to the confining effect of the wall. As the Reynolds number was increased, capsules were more elongated in the direction of flow. The effect of Reynolds number was more apparent as the elastic capillary number was increased. Both the migration velocity and extra pressure loss were seen to depend primarily on the size of the capsule with deformation playing a secondary role. The Neo-Hookean membrane showed a larger deformation than the Skalak law due to its strain softening nature. The Neo-Hookean membrane also displayed a failure phenomenon of continuous deformation at large enough elastic capillary numbers not seen in the Skalak law membranes. This limiting elastic capillary number was shown to decrease as Reynolds number became larger. The membrane strain was largest at the front of the capsule indicating the most likely region where the capsule would fail.     

Lateral migration of a capsule in a plane Poiseuille flow in a channel

Three-dimensional numerical simulation is presented on the motion of a deformable capsule undergoing large deformation in a plane Poiseuille flow in a channel at small inertia. The capsule is modeled as a liquid drop surrounded by an elastic membrane which follows neo-Hookean law. The numerical methodology is based on a mixed finite-difference/Fourier transform method for the flow solver and a front-tracking method for the deformable interface. The methodology can address large deformation of a capsule over a wide range of capsule-to-medium viscosity ratio. An extensive validation of the methodology is presented on capsule deformation in linear shear flow and compared with the boundary-element/integral simulations. Motion of a capsule in wall-bounded parabolic flow is simulated over an extended period of time to consider both transient and steady-state motion. Lateral migration of the capsule towards the centerline of the channel is observed. Results are presented over a range of capillary number, viscosity ratio, capsule-to-channel size ratio, and lateral location. After an initial transient phase during which the capsule deforms very quickly, the flow of the capsule is observed to be a quasi-steady process irrespective of capillary number (Ca)$(\mathit{Ca})$, capsule-to-channel size ratio (a/H)$(a/H)$, and viscosity ratio (λ)$(\lambda )$. Migration velocity and capsule deformation are observed to increase with increasing Ca$\mathit{Ca}$ and a/H$a/H$, but decrease with increasing λ$\lambda$, and increasing distance from the wall. Numerical results on the capsule migration are compared with the analytical results for liquid drops, and capsules with Hookean membrane which are valid in the limit of small deformation. Unlike the prediction for liquid drops, capsules are observed to migrate toward the centerline for 0.2?λ?5$0.2?\lambda ?5$ range considered here. The migration velocity is observed to depend linearly on (a/H)³$(a/H{)}^3$, in agreement with the small-deformation theory, but non-linearly on Ca$\mathit{Ca}$ and the distance from the wall, in violation of the theory. Using the present numerical results and the analytical results, we present a correlation that can reasonably predict migration velocity of a capsule for moderate values of a/H$a/H$ and Ca$\mathit{Ca}$.     

An approach for prediction of the elasto-plastic behavior of particulate reinforced composites

A numerical approach is presented in this paper for the calculation of the elasto-plastic deformation behavior of particulate reinforced composites. The effect of shape and arrangement of particulate on the elastic modulus and tensile deformation behavior were estimated. The approach presented can consider the shape and arrangement effect of reinforcement particulate via a simple parameter called the geometrical factor (Gf). Elastic moduli and tensile deformation estimations for the particulate reinforced composites were studied. The results of proposed approach were in very good agreement with the results of finite element analysis.     

Dynamic fluid–structure interaction of an elastic capsule in a viscous shear flow at moderate Reynolds number

The unsteady behavior of a 2-D circular elastic capsule was investigated in three viscous shear flows. An immersed boundary method (IBM) has been used to solve the dynamic fluid-structure interaction of the capsule. Computations were carried out in finite parameter ranges where the Reynolds number is Re=1-40 and the capillary number is Ca=0.0005-0.05, which is the ratio of the external viscous shear stress to the resistant elastic tensions of the membrane. For the simple shear flow, the effect of inertia on the transient behavior of the capsule was studied. For the pulsatile shear flow, two values of the peak fluid strain, T_f=1 and 5, were considered for the quasi-steady capsule mechanics. The capsule shows a cyclic structural response that includes subharmonics as the Reynolds number is elevated to 10 and 40. The capsule dynamic response includes a phase lag, which is a function of the capillary number, the Reynolds number, and the peak fluid strain. Finally, the capsule flowing in the Couette flow shows lateral migration due to the transient lift force, which is higher for lower Ca and higher Re. When capsules with diverse elasticity are dispersed along the velocity gradient, the capsule with a hard membrane experienced greater lift than the one with a soft membrane.     

Dynamics of an elastic capsule in moderate Reynolds number Poiseuille flow

The dynamic motions and lateral equilibrium positions of a two-dimensional elastic capsule in a Poiseuille flow were explored at moderate Reynolds number (10 ⩽ Re ⩽ 100) as a function of the initial lateral position (y₀), Re, aspect ratio (ɛ), size ratio ($\lambda$), membrane stretching coefficient (φ) and bending coefficient (γ). The transition between tank-treading (TT) and swinging (SW) to tumbling (TU) motions was observed and the lateral equilibrium positions of the capsules varied according to the conditions. The initial behavior of the elastic capsule was influenced by variation in the initial lateral position (y₀), but the equilibrium position and dynamic motion of the capsule were not affected by such variation. The capsules had a stronger tendency toward TU motion at higher values of Re, φ and γ, whereas the capsules underwent TT or SW motion as the values of ɛ and $\lambda$ increased. Under moderate Re Poiseuille flows, capsules tended to migrate across streamlines to a specific equilibrium position. The lateral equilibrium position shifted toward the centerline at larger $\lambda$ and migrated toward the wall at larger $\epsilon \text{,}\varphi \text{and}\gamma$. As Re increased, the equilibrium position first shifted toward the bottom wall, then toward the channel center. However, different equilibrium position trends were obtained around the SW–TU transition. The capsule undergoing TU motion tended to migrate downward toward the bottom wall more than the capsule undergoing SW motion, all other conditions being similar.     

Mechanical Properties of Microcapsules Used in a Self-Healing Polymer

The elastic modulus and failure behavior of poly(urea-formaldehyde) shelled microcapsules were determined through single-capsule compression tests. Capsules were tested both dry and immersed in a fluid isotonic with the encapsulent. The testing of capsules immersed in a fluid had little influence on mechanical behavior in the elastic regime. Elastic modulus of the capsule shell wall was extracted by comparison with a shell theory model for the compression of a fluid filled microcapsule. Average capsule shell wall modulus was 3.7 GPa, regardless of whether the capsule was tested immersed or dry. Microcapsule diameter was found to have a significant effect on failure strength, with smaller capsules sustaining higher loads before failure. Capsule size had no effect on the modulus value determined from comparison with theory.     

Spark-generated bubble near an elastic sphere

The interaction between a spark-generated bubble and an elastic sphere is investigated. A spark-generated bubble is created at various distances horizontally away from a suspended elastic sphere made of silicone rubber or super absorbent polymer (of shear modulus of elasticity G of between 5 and 312 kPa), using a low-voltage spark discharge method. We observe pronounced deformation and elongation of the elastic sphere when the spark-bubble is generated very close to a sphere. This happens when the elastic sphere has a small modulus of elasticity and a small size ratio R’ between the bubble and the elastic sphere (i.e. the bubble and the sphere have similar radii). Numerical simulations are also conducted using a Boundary Element Method (BEM) model coupled with a Finite Element Method (FEM) solver. The simulation results compare well with the experimental data. The numerical model is then extended to study the effects of elasticity and experimental parameters, such as the dimensionless stand-off distance H’, and size ratio R’, on the degree of deformation of the elastic cell and the dynamics of the bubble.     

Investigation of Near-Surface Mechanical Properties of Materials Using Atomic Force Microscopy

As ultra-thin films or small-scale structures become widely used in electronics and biology, knowledge concerning their near-surface mechanical properties of the materials is increasingly important. Atomic force microscopy (AFM) is employed to determine near-surface elastic modulus via force-penetration curves acquired during indentation. Samples include polydimethylsiloxane (PDMS), parylene, mica, and single-crystal silicon, and indentations are performed with single-crystal silicon and silicon nitride AFM tips. An analysis algorithm based on the secant modulus method is proposed to extract the true penetration curves from the experimental displacement curves. The penetration data is then analyzed in terms of Hertzian model to estimate the elastic modulus. Three concerns in applying nanoscale AFM indentation to the measurement of the elastic modulus of an ultra-thin material are addressed. First, the effect of the lateral force caused by the inclined angle of the cantilevered probe is investigated theoretically and by numerical simulation. A second concern is local plastic deformation induced by a sharp probe tip. In this case, numerical results show a relatively small effect on the force-penetration curves if the plastic deformation is limited to the central area below the probe tip. The deviation of the elastic-plastic simulation from the elastic estimation depends on the yield strength of the material. Finally, the effect of stiffness matching between the AFM probe and the sample is a key issue that is studied numerically, and appropriate stiffness matching criteria are suggested.     

Effective equations describing the flow of a viscous incompressible fluid through a long elastic tubeEquations efficaces décrivant l'écoulement d'un fluide visqueux incompressible dans un tuyau long et élastique

We study the flow of a viscous incompressible fluid through a long and narrow elastic tube whose walls are modeled by the Navier equations for a curved, linearly elastic membrane. The flow is governed by a given small time dependent pressure drop between the inlet and the outlet boundary, giving rise to creeping flow modeled by the Stokes equations. By employing asymptotic analysis in thin, elastic, domains we obtain the reduced equations which correspond to a Biot type viscoelastic equation for the effective pressure and the effective displacement. The approximation is rigorously justified by obtaining the error estimates for the velocity, pressure and displacement. Applications of the model problem include blood flow in small arteries. We recover the well-known Law of Laplace and provide a new, improved model when shear modulus of the vessel wall is not negligible. To cite this article: S. ?ani?, A. Mikeli?, C. R. Mecanique 330 (2002) 661–666.     

Elastic-plastic finite element polycrystal model

A finite element polycrystal model is formulated with the initial strain method where the stiffness matrix in FEM is based on the elastic modulus. For the determination of time-independent slips, a new numerical scheme, “successive integration method,” is proposed, which uses only Schmid's law. The numerical result for a simple tension of nonhardening FCC metal is compared with other theories. Crystal lattice rotations are presented for some examples of loading. A numerical experiment is done to show the evolution of anisotropy due to plastic deformation. The numerical code of the present model is quite simple and can be applied to arbitrary loading paths.     

Simulation of flow around a thin,flexible obstruction by means of a deforming grid overlapping a fixed grid

This paper presents a numerical method for simulation of coupled flows, in which the fluid interacts with a thin deformable solid, such as flows in cardiovascular valves. The proposed method employs an arbitrary Lagrangian–Eulerian (ALE) method for flow near the solid, embodied in the outflow domain in which the mesh is fixed. The method was tested by modelling a two‐dimensional channel flow with a neo‐Hookean obstacle, an idealization of the coupled flow near a cardiovascular valve. The effects of the Reynolds number and the dimensionless elastic modulus of the material on the wall shear stress, the size of the downstream reverse flows, and the velocity and pressure profiles were investigated. The deformation of the obstacle, the pressure drop across the obstacle, and the size of the top reverse flow increased as the Reynolds number increased. Conversely, increasing the elastic modulus of the obstacle decreased the deformation of the obstacle and the size of the top reverse flows, but did not affect the pressure drop across the obstacle over the range studied. Copyright © 2007 John Wiley & Sons, Ltd.     

FRP筋粘结式锚具的界面径向弹性模量分析

建立了FRP(Fibre Reinforced Polymer)筋粘结式锚具粘结界面的Rib-scale模型和Bar-scale模型,然后利用Fourier-Bessel级数推导了FRP筋、混凝土以及钢套筒等在径向应力作用下的解析解.解析解与数值解吻合较好,验证了用Fourier-Bessel级数表达的解析解的有效性....     

An IB-LBM study of continuous cell sorting in deterministic lateral displacement arrays

The deterministic lateral displacement(DLD) is an important method used to sort particles and cells of different sizes. In this paper, the flexible cell sorting with the DLD method is studied by using a numerical model based on the immersed boundary-lattice Boltzmann method(IB-LBM). In this model, the fluid motion is solved by the LBM, and the cell membrane–fluid interaction is modeled with the LBM.The proposed model is validated by simulating the rigid particle sorted with the DLD method, and the results are found in good agreement with those measured in experiments. We first study the effect of flexibility on a single cell and multiple cells continuously going through a DLD device. It is found that the cell flexibility can significantly affect the cell path,which means the flexibility could have significant effects on the continuous cell sorting by the DLD method. The sorting characteristics of white blood cells and red blood cells are further studied by varying the spatial distribution of cylinder arrays and the initial cell–cell distance. The numerical results indicate that a well concentrated cell sorting can be obtained under a proper arrangement of cylinder arrays and a large enough initial cell–cell distance.     

Simultaneous solution of the Navier-Stokes and elastic membrane equations by a finite element method

Fluid flow through a significantly compressed elastic tube occurs in a variety of physiological situations. Laboratory experiments investigating such flows through finite lengths of tube mounted between rigid supports have demonstrated that the system is one of great dynamical complexity, displaying a rich variety of self-excited oscillations. The physical mechanisms responsible for the onset of such oscillations are not yet fully understood, but simplified models indicate that energy loss by flow separation, variation in longitudinal wall tension and propagation of fluid elastic pressure waves may all be important. Direct numerical solution of the highly non-linear equations governing even the most simplified two-dimensional models aimed at capturing these basic features requires that both the flow field and the domain shape be determined as part of the solution, since neither is known a priori. To accomplish this, previous algorithms have decoupled the solid and fluid mechanics, solving for each separately and converging iteratively on a solution which satisfies both. This paper describes a finite element technique which solves the incompressible Navier-Stokes equatikons simultaneously with the elastic membrane equations on the flexible boundary. The elastic boundary position is parametized in terms of distances along spines in a manner similar to that which has been used successfully in studies of viscous free surface flows, but here the membrane curvature equation rather than the kinematic boundary condition of vanishing normal velocity is used to determine these diatances and the membrane tension varies with the shear stresses exerted on it by the fluid motions. Bothy the grid and the spine positions adjust in response to membrane deformation, and the coupled fluid and elastic equations are solved by a Newton-Raphson scheme which displays quadratic convergence down to low membrane tensions and extreme states of collapse. Solutions to the steady problem are discussed, along with an indication of how the time-dependent problem might be approached.     

考虑初始缺陷与弹性模量的岩石变形破坏过程模拟方法

针对扰动作用下岩石存在初始缺陷的特点,通过探讨含初始缺陷岩石变形力学特征,根据含初始缺陷岩石弹性模量的变化情况,建立了初始损伤的确定方法。然后,引入几何损伤理论,通过分析三轴压缩条件下岩石损伤演化规律,建立了含初始缺陷岩石损伤模型,进而建立了考虑初始缺陷与弹性模量的岩石统计损伤本构模型,并给出了参数的确定方法。最后,通过砂岩三轴压缩试验资料分析得出:本文理论模型能够反映不同围压作用下含初始缺陷岩石的变形破坏全过程,与试验曲线较为接近;且常见损伤模型是本文理论模型的特例,表明本文模型和方法具有一定的合理性与可行性。     

An experimental/computational approach to identify moduli and residual stress in MEMS radio-frequency switches

In this paper, we identify the Young's modulus and residual stress state of a free-standing thin aluminum membrane, used in MEMS radio-frequency (rf) switches. We have developed a new methodology that combines a membrane deflection experiment (MDE) and three-dimensional numerical simulations. Wafer-level MDE tests were conducted with a commercially available nanoindenter. The accuracy and usefulness of the MDE is confirmed by the repeatability and uniformity of measured load-deflection curves on a number of switches with both wedge and Berkovich tips. It was found that the load-deflection behavior is a function of membrane elastic properties, initial residual stress state and corresponding membrane shape. Furthermore, it was assessed that initial membrane shape has a strong effect on load-deflection curves; hence, its accurate characterization is critical. Through an iterative process and comparison between MDE data and numerical simulations, the Young's modulus and residual stress state, consistent with measured membrane shape, were identified. One important finding from this investigation is that variations in membrane elastic properties and residual stress state affect the load-deflection curve in different regimes. Changes in residual stress state significantly affect the load-deflection slope at small values of deflection. By contrast, variations in Young's modulus result in changes in load-deflection slope at large deflections. These features are helpful to decouple both effects in the identification process.     

Nanoindentation modeling of a nanodot-patterned surface on a deformable substrate

A numerical model was developed to simulate the nanoindentation of a Ni nanodot-patterned surface (NDPS) on a deformable Si substrate. Each contacting nanodot on the Si substrate was treated individually in this model and the interaction among the nanodots was considered through the elastic deformation of the Si substrate. The load–deformation relationship for the single-asperity contact between the indenter tip and a nanodot was determined using finite element analysis. A nanoindentation experiment on a Ni NDPS was performed to test the developed model. The simulation and experimental results were found to be in good agreement. The experimentally verified model was used to explore the effects of substrate deformation and surface roughness caused by the Ni nanodots on the nanoindentation behavior. It was found that the effect of the substrate and the effect of roughness must be considered. A detailed study of the substrate deformation shows that the interaction among nanodots, through the substrate, can contribute a considerable portion of the total deformation under a nanodot. The yield strength of the nanodot was found to have a significant effect on the contact deformation, while the elastic modulus was found to have little effect.     

Phenomenological constitutive model for a CNT turf

Carbon nanotubes (CNT), grown on a substrate, form a turf – a complex structure of intertwined, mostly nominally vertical tubes, cross-linked by adhesive contact and few bracing tubes. The turfs are compliant and good thermal and electrical conductors. In this paper, we consider the micromechanical analysis of the turf deformation reported earlier, and develop a phenomenological constitutive model of the turf. We benchmark the developed model using a finite element implementation and compare the model predictions to the results two different nanoindentation tests.The model includes: nonlinear elastic deformation, small Kelvin–Voigt type relaxation, caused by the thermally activated sliding of contacts, and adhesive contact between the turf and the indenter. The pre-existing (locked-in) strain energy of bent nanotubes produces a high initial tangent modulus, followed by an order of magnitude decrease in the tangent modulus with increasing deformation. The strong adhesion between the turf and indenter tip is due to the van der Waals interactions.The finite element simulations capture the results from the nanoindentation experiments, including the loading, unloading, viscoelastic relaxation during hold, and adhesive pull-off.     

基于近场动力学的单晶冰弹性各向异性数值模拟方法

天然冰材料在变形与破坏行为上的各向异性特征是冰与结构相互作用中产生复杂载荷过程的关键诱因, 而天然冰各向异性的根源则在于单晶冰的各向异性. 目前, 学术界针对单晶冰各向异性的数值模拟方法研究仍较为缺乏. 为了准确再现天然冰材料的特殊力学性质, 本文基于近场动力学理论, 提出了一种单晶冰弹性各向异性的数值模拟方法. 该方法的核心思想是将单晶冰杨氏模量沿不同加载方向的变化规律引入到近场动力学力密度向量的影响函数中. 以前人实验测试得到的杨氏模量值为参考, 通过开展与C轴呈0°, 45°和90°三个加载方向的单晶冰单轴压缩数值模拟实验, 提出了针对该影响函数的修正和辅助参数标定方法, 最终在15°, 30°, 60°和75°等其他四个加载方向进行了验证. 结果表明: 本文提出的针对影响函数的修正与参数标定方法, 能够较为便捷地找到数值模型杨氏模量与参考杨氏模量相一致的影响函数最优解, 即本文提出的基于影响函数的近场动力学数值模拟方法, 能够合理、准确地模拟单晶冰的弹性各向异性行为. 本文研究成果可为后续多晶冰各向异性数值模拟方法的建立提供基础性参考.      

Downstream pressure and elastic wall reflection of droplet flow in a T-junction microchannel

This paper discusses pressure variation on a wall during the process of liquid flow and droplet formation in a T-junction microchannel. Relevant pressure in the chan-nel, deformation of the elastic wall, and responses of the droplet generation are analyzed using a numerical method. The pressure difference between the continuous and dis-persed phases can indicate the droplet-generation period. The pressure along the channel of the droplet flow is affected by the position of droplets, droplet-generation period, and droplet escape from the outlet. The varying pressures along the channel cause a nonuniform deformation of the wall when they are elastic. The deformation is a vibration and has the same period as the droplet generation arising from the process of droplet formation.