Lumped Mass Modelling and Chaotic Behavior of an Elastic Levitated Droplet

In this paper, a nonlinear dynamicsanalysis of simulated data was considered to study the timeevolution of an electromagnetically levitated (ELM) flexibledroplet. The flexible droplet is modelled as a system with lumpedmasses, elastic springs, and rigid links. The behavior of thelevitated droplet was investigated. Quantities characterizing timeseries data such as attractor dimension or largest Lyapunovexponent were computed. Calculations were performed for dropletsof aluminum and copper. Thus, its chaotic behavior was analyzed.     

索杆体系的机构运动及其与弹性变形的混合问题

基于有限单元平衡方程推导了索杆体系机构运动的通解、特解及其路径跟踪的迭代求解列式，以及机构运动与弹性变形混合问题的迭代求解列式。从机构运动和弹性变形问题的比较角度出发，解释了机构运动中的若干基本定义，对索杆体系的机构运动和弹性变形混合问题作了分类。数值算例表明：采用本文方法可以非常有效地跟踪求解索杆体系机构运动、弹性变形及其混合问题的全过程路径，得到机构运动的静定状态和混合问题的平衡状态。     

The Motion Behavior of a Group of Charged Droplets in Liquid-Liquid Electrostatic Spray

The droplet sizes and electrical charges under different applied electrical voltages are experimentally measured for a liquid-liquid electrostatic spray system. Considering droplet size and charge distributions, the two-dimensional motion for a group of charged droplets in a liquid-liquid electrostatic atomization system is simulated. From measured droplet size and charge distributions, the simulation can obtain the velocities and positions in a two-dimensional domain for all simulated droplets at different times. The various forces acting on droplet as well as their effects on droplet velocity and trajectory are analyzed and the liquid-liquid electrostatic atomization characteristics are revealed. In addition, for one-dimensional motion trajectory of larger droplet, the comparison between simulation and experiment is also conducted and a general agreement can be obtained.     

Efficient Evaluation of the Elastic Forces and the Jacobian in the Absolute Nodal Coordinate Formulation

This paper develops a new procedure for evaluating the elastic forces, the elastic energy and the jacobian of the elastic forces in the absolute nodal coordinate formulation. For this procedure, it is fundamental to use some invariant sparse matrices that are integrated in advance and have the property of transforming the evaluation of the elastic forces in a matrix multiplication process. The use of the invariant matrices avoids the integration over the volume of the element for every evaluation of the elastic forces. Great advantages can be achieved from these invariant matrices when evaluating the elastic energy and calculating the jacobian of the elastic forces as well. The exact expression of the jacobian of the differential system of equations of motion is obtained, and some advantages of using the absolute nodal coordinate formulation are pointed out. Numerical results show that there is important time saving as a result of the use of the invariant matrices.     

Unsteady Motion of a Maxwellian Fluid Droplet in a Maxwellian Medium under the Action of Monotonic and Periodic Forces

Unsteady motion of a Maxwellian fluid droplet, which arises in a quiescent Maxwellian medium under the action of monotonic and periodic forces, is considered. In the initial period of time smaller than the relaxation time, the droplet is affected by elastic forces on the part of the fluid; moreover, the droplet itself is a viscoelastic material. A solution of the problem in the first approximation is found. The dependence of the amplitude of droplet velocity and the shift of the phase of oscillations on the relaxation time of the external and internal media and also on the frequency of oscillations of the driving force is analyzed. The passage to the limit in terms of density and viscosity of the internal medium is performed.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 55–65, July–August, 2005.     

Dynamics of a Two-Pendulum Model with Impact Interaction and an Elastic Support

This paper examines the dynamic behavior of a double pendulummodel with impact interaction. One of the masses of the two pendulumsmay experience impacts against absolutely rigid container wallssupported by an elastic system forming an inverted pendulum restrainedby a torsional elastic spring. The system equations of motion arewritten in terms of a non-smooth set of coordinates proposed originallyby Zhuravlev. The advantage of non-smooth coordinates is that theyeliminate impact constraints. In terms of the new coordinates, thepotential energy field takes a cell-wise non-local structure, and theimpact events are treated geometrically as a crossing of boundariesbetween the cells. Based on a geometrical treatment of the problem,essential physical system parameters are established. It is found thatunder resonance parametric conditions of the linear normal modes thesystem's response can be either bounded or unbounded, depending on thesystem's parameters. The ability of the system to absorb energy from anexternal source essentially depends on the modal inclination angle,which is related to the principal coordinates.     

航空发动机轴承腔中油滴运动与沉积的特性分析

本文在获得轴承腔中气相介质流场的基础上,采用Lagrangian方法建立油滴在气相介质流场中运动的分析模型,通过瞬时步进法数值模拟油滴的运动过程,获得了油滴直径和旋转轴转速对油滴运动过程中的速度和轨迹影响的规律.基于获得的油滴与腔壁碰撞前的运动状态,以及结合油滴与腔壁的碰撞模型,实现了油滴直径和旋转轴转速对碰撞后油滴沉积率和动量转移率影响规律的分析.结果表明:油滴直径和旋转轴转速对油滴速度及轨迹,以及油滴沉积率及动量转移率都有很大影响,而且前者的影响更为明显.与国外同等条件下的试验结果对比表明,本文提出的油滴运动与沉积特性分析方法具有较好的可靠性和精度.碰撞前后油滴运动状态和沉积率及动量转移率的计算,为下一步油膜厚度和速度的计算,继而为轴承腔润滑设计和换热分析提供了初始条件.     

A numerical study of droplet motion/departure on condensation of mixture vapor using lattice Boltzmann method

Droplet motion/departure, which is governed by external force acceleration coefficient, droplet radius and surface wettability on solid surfaces under external forces such as gravitational force, play a significant role in characterizing condensation heat transfer, especially when high fractional non-condensable gases (NCG) present. However, due to the challenge in visualizing the vapor/steam velocity field imposed by droplet motion/departure, the detailed mechanism of droplet motion/departure on condensing surfaces has not been completely investigated experimentally. In this study, droplet motion/departures on solid surfaces under external forces and their interactions with steam flow are simulated using two dimensional (2D) multiphase lattice Boltzmann method (LBM). Large external force acceleration coefficient, droplet radius and contact angle, lead to large droplet deformation and high motion/departure velocity, which significantly shortens the droplet residual time on the solid surface. Our simulation shows that steam vortices (lateral velocity) induced by droplet motion/departure can greatly disturb the vapor flow and would be intensified by increasing external force acceleration coefficient, droplet radius, and contact angle. In addition, the location of vortex center shifts in the ascending direction with increase of these factors. The average lateral velocities induced by droplet motion/departure at various conditions are obtained. The mass transfer resistance is substantially reduced owing to the droplet motion/departure, leading to an enhanced heat flux. The experimental results are compared to validate the influence of droplet motion/departure on condensation heat transfer performance, especially for steam–air mixture with the presence of high fractional NCG.     

基于表面力模型的液滴冲击弹性基底SPH模拟

采用光滑粒子动力学SPH方法建立液滴冲击弹性基底的流固耦合数值模型,给出描述粘性流体和弹性固体运动的SPH离散方程和数值处理格式,引入人工耗散项来抑制标准SPH方法的数值震荡。为模拟液滴的表面张力效应,通过精确检测边界粒子,采用拉格朗日插值方法计算表面法向量和曲率,结合界面理论中的连续表面力CSF方法,建立了适用于自由表面液滴的表面力模型,方形液滴变形的模拟结果与拉普拉斯理论解吻合较好。随后,采用SPH流固耦合模型模拟1.0 mm直径水滴以不同速度(0.2 m/s～3.0 m/s)冲击两种薄板型基底,分析了基底弹性变形对液滴铺展、收缩以及回弹行为的影响。     

Forces acting on water droplets falling in oil under the influence of an electric field: numerical predictions versus experimental observations

The combination of an electric field and a moderate turbulent flow is a promising technique for separating stable water–oil emulsions. Field-induced charges on the water droplets will cause adjacent droplets to align with the field and attract each other. The present work describes the forces that influence the kinematics of droplets falling in oil when exposed to an electric field. Mathematical models for these forces are presented and discussed with respect to a possible implementation in a multi-droplet Lagrangian framework. The droplet motion is mainly due to buoyancy, drag, film-drainage, and dipole–dipole forces. Attention is paid to internal circulations, non-ideal dipoles, and the effects of surface tension gradients.Experiments are performed to observe the behavior of a droplet falling onto a stationary one. The droplet is exposed to an electric field parallel to the direction of the droplet motion. The behavior of two falling water droplets exposed to an electric field perpendicular to the direction of their motion is also investigated until droplet coalescence. The droplet motion is recorded with a high-speed CMOS camera. The optical observations are compared with the results from numerical simulations where the governing equations for the droplet motion are solved by the RK45 (Runge Kutta) Fehlberg method with step-size control and low tolerances. Results, using different models, are compared and discussed in detail. A framework is otlined to describe the kinematics of both a falling rigid spherical particle and a fluid droplet under the influence of an electric field.     

Spreading dynamics of droplet on an inclined surface

A three-dimensional unsteady theoretical model of droplet spreading process on an inclined surface is developed and numerically analyzed to investigate the droplet spreading dynamics via the lattice Boltzmann simulation. The contact line motion and morphology evolution for the droplet spreading on an inclined surface, which are, respectively, represented by the advancing/receding spreading factor and droplet wetted length, are evaluated and analyzed. The effects of surface wettability and inclination on the droplet spreading behaviors are examined. The results indicate that, dominated by gravity and capillarity, the droplet experiences a complex asymmetric deformation and sliding motion after the droplet comes into contact with the inclined surfaces. The droplet firstly deforms near the solid surface and mainly exhibits a radial expansion flow in the start-up stage. An evident sliding-down motion along the inclination is observed in the middle stage. And the surface-tension-driven retraction occurs during the retract stage. Increases in inclination angle and equilibrium contact angle lead to a faster droplet motion and a smaller wetted area. In addition, increases in equilibrium contact angle lead to a shorter duration time of the middle stage and an earlier entry into the retract stage.     

Droplet spindown in a high-temperature gas environment

The spindown and heating of a spherical droplet in an initially undisturbed infinite fluid is investigated by means of a numerical model based on finite-difference discretization techniques. The nonevaporating droplet enters the hot gas while rotating about a diameter and has no translational motion with respect to the suspending medium. Special attention is given to the transient secondary (nonrotational) motion developed as a result of shear interaction between the two phases. The results indicate that for droplet sizes and rotation frequencies representative of droplet combustion applications; i.e., Reynolds ∼ O(0.1), the secondary motion in both phases remains weak and heat transport is conduction-dominated. On the other hand, the secondary motion is strengthened with increased values of the rotational Reynolds number. The characteristic time for droplet spindown is found to be proportional to the square of the droplet radius. The results also show that the rotational deceleration time is of the same order of magnitude with the translational response time of the droplet. Finally, the thermocapillary stress effects on fluid dynamics and heat transfer are investigated in this flow configuration.     

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.     

Axisymmetric oscillations of a cylindrical droplet with a moving contact line

Forced oscillations of a cylindrical droplet of an inviscid liquid surrounded by another liquid and bounded in the axial direction by rigid planes are investigated. The system is affected by vibrations whose force is directed parallel to the axis of symmetry of the droplet. The velocity of motion of the contact line is proportional to the deviation of the contact angle from the value at which the droplet is in equilibrium. Linear and nonlinear oscillations are considered. The conditions of the occurrence of resonance are determined.     

Computational study of microparticle effect on self-propelled jumping of droplets from superhydrophobic substrates

We present three-dimensional numerical simulations, employing a lattice Boltzmann method for three-phase system of liquid, gas, and solid, and investigate the influence of a solid particle on the dynamic and departure of a droplet after coalescence on superhydrophobic substrates. A particle can be removed autonomously by the jumping motion of the droplet, which partially or fully covers the particle. This spontaneous removal from superhydrophobic substrates is achieved by converting surface energy to kinetic energy, independent of gravity. We discussed the effect of size, wettability and initial placement of particle on the evolution of lateral and vertical motion of the droplet. The results indicate that the droplet with a fully immersed particle, as in the floating mechanism, reaches to the same equilibrium height as a particle-free droplet. However, the droplet with a partially immersed particle, as in the lifting mechanism, can have a substantial jumping velocity compared to a particle-free droplet. As the size of the partially immersed particle approaches its critical limit, which is equal to the size of the droplet, the droplet jumping and transport from the substrate is enhanced. Besides the particle size, the particle wettability can result in a considerable droplet jumping velocity. A particle with a neutrally wetting contact angle (i.e. 90°) is found to elevate the transport of the droplet to a higher distance from the substrate relative to a partially wetting case (i.e. 60°). In the lifting removal mechanism, unlike the floating removal mechanism, the particle initial placement is highly critical for the detachment of the merged droplet from the substrate, as well as the elevation of the detached droplet to a longer distance from the substrate. For a partially immersed particle, the critical particle initial position from the substrate above which the droplet-particle system does not jump away from the substrate is independent of particle size and wettability and is about 1.5r_d where r_d is the initial size of the droplet.     

微液滴在PDMS软基体表面的动态摩擦行为研究

通过固液界面摩擦力测试装置研究了微液滴在PDMS软基体表面运动时的动态摩擦学行为,并对微液滴体积、滑动速度及软基体力学性能对固液界面动态摩擦行为的影响进行了分析. 结果表明:微液滴在软基体表面运动时表现出最大静摩擦力和动态摩擦力. 最大静摩擦力与微液滴黏度和速度梯度呈正比,动态摩擦力与微液滴体积、滑动速度和基体力学性能有关. 随着微液滴体积的增加,三相接触线长度增加,动态摩擦力增加;随着相对滑动速度增加,三相接触线长度及接触角滞后增加,动态摩擦力增加;随着软基体弹性模量降低,固液界面黏附力增加,固液界面运动能量耗散增加,动态摩擦力增加. 研究结果可为PDMS软基体表面微液滴的精确驱动和运动参数优化提供理论指导,也可进一步丰富固液界面摩擦理论.      

Experiments and analysis on self-motion behaviors of liquid droplets on gradient surfaces

A surface with surface energy gradient was fabricated by using chemical vapor deposition technology with dodecyltrichlorosilane (C₁₂H₂₅Cl₃Si), and its property was characterized by sessile drop method and Atomic Force Microscope scanning. Visualization experiments were carried out to investigate the motion behaviors of water and ethylene glycol droplets on horizontal and inclined gradient surfaces. And system free energy transition was analyzed to understand the mechanics of the droplet self-motion. The results show that the height and density of the silane molecules groups determined surface energy distribution on the surface. The liquid droplets were self-propelled to move horizontally or uphill from hydrophobic zone to hydrophilic zone on horizontal and inclined gradient surface. The motion process of the droplet experienced an accelerating stage and a creeping decelerating stage; the velocity and the displacement as well as the creeping frequency were proportional to the droplet size. The velocity of 2 ml water droplet reached 42 mm/s on the horizontal surface and 18 mm/s on the inclined surface, while that for ethylene glycol droplet reached 7 mm/s on the horizontal surface. The droplet motion was resulted from the energy transition among interfacial energy, kinetic energy, gravitational potential energy, and viscous dissipation energy. The interfacial energy released from deformation of the droplet is the main source for the motion.     

Geometrical nonlinear elasto-plastic analysis of tensegrity systems via the co-rotational method

An efficient finite element formulation is presented for geometrical nonlinear elasto-plastic analyses of tensegrity systems based on the co-rotational method. Large displacement of a space rod element is decomposed into a rigid body motion in the global coordinate system and a pure small deformation in the local coordinate system. A new form of tangent stiffness matrix, including elastic and elasto-plastic stages is derived based on the proposed approach. An incremental-iterative solution strategy in conjunction with the Newton-Raphson method is employed to obtain the geometrical nonlinear elasto-plastic behavior of tensegrities. Several numerical examples are given to illustrate the validity and efficiency of the proposed algorithm for geometrical nonlinear elasto-plastic analyses of tensegrity structures.     

DYNAMIC MODELING OF MULTI-BODY SYSTEM BASED ON GAUSS'S PRINCIPLE

基于高斯最小拘束原理,以释放中的绳系卫星为背景,建立地球引力场内变长度大变形柔索联系的多体系统动力学模型. 利用基尔霍夫动力学比拟方法将柔索的变形转化为刚性截面沿中心线的转动,使包含刚性分体与变形体的刚柔耦合系统转化为由柔索的刚性截面与刚性分体组成的广义多刚体系统. 由于刚性截面的局部小变形沿弧坐标的积累不受限制,适合描述柔索的超大变形. 文中对此刚柔耦合多体系统导出其在地球引力场中的拘束函数,考虑各分体在空间中相对位置的几何约束条件,利用拉格朗日乘子构成以条件极值问题为特征的数学模型. 将高斯原理用于多体系统动力学的建模,其特点是以寻求函数极值的变分方法代替微分方程,通过数值计算直接得出运动规律. 其形式统一,不随系统拓扑结构的变化而改变,也无需区分树系统或非树系统.对于带控制的多体系统,动力学分析还可根据技术需要与系统的优化结合进行.