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.     

3D phase field modeling of electrohydrodynamic multiphase flows

A 3D phase field model is developed to investigate the electrohydrodynamic (EHD) two phase flows. The explicit finite difference method, enhanced by parallel computing, is employed to solve the coupled nonlinear governing equations for the electric field, the fluid flow field and free surface deformation. Numerical tests indicate that an appropriate interpolation of densities within the interface is critical in ensuring numerical stability for highly stratified flows. The 3D phase field model compares well with the Taylor theory for the deformation of a single dielectric droplet in an electric field. Computed results show that the deformation of a leaky dielectric droplet in an electric field undergoes various stages before it reaches the final oblate shape. This is caused by the free charge relaxation near the fluid–fluid interface. The coalescence of four droplets in an electric field illustrates a truly 3D deformation behavior and a complex evolving fluid flow field associated with the participating droplets. The coalescence is a result of combined actions produced by the global electric force, the circulatory flows generated by the local electrohydrodynamic stress and the electrically-induced deformation. The 3D phase field model is also applied in modeling of an electrohydrodynamic patterning process for manufacturing nanoscaled structures, in which complex 3D flow structures develop as the electrically-induced deformation evolves.     

Interfacial instability of ferrofluid flow under the influence of a vacuum magnetic field

This study is to numerically test the interfacial instability of ferrofluid flow under the presence of a vacuum magnetic field. The ferrofluid parabolized stability equations(PSEs) are derived from the ferrofluid stability equations and the Rosensweig equations, and the characteristic values of the ferrofluid PSEs are given to describe the ellipticity of ferrofluid flow. Three numerical models representing specific cases considering with/without a vacuum magnetic field or viscosity are created to mathematically examine the interfacial instability by the computation of characteristic values. Numerical investigation shows strong dependence of the basic characteristic of ferrofluid Rayleigh-Taylor instability(RTI) on viscosity of ferrofluid and independence of the vacuum magnetic field.For the shock wave striking helium bubble, the magnetic field is not able to trigger the symmetry breaking of bubble but change the speed of the bubble movement. In the process of droplet formation from a submerged orifice, the collision between the droplet and the liquid surface causes symmetry breaking. Both the viscosity and the magnetic field exacerbate symmetry breaking. The computational results agree with the published experimental results.     

Lattice Boltzmann simulation of multiple droplets impingement and coalescence in an inkjet-printed line patterning process

Three-dimensional computations on the basis of the index-function lattice Boltzmann method are performed to simulate the process of multiple droplets impinging and coalescing into a line pattern on a solid substrate. The employed calculation model is validated by theoretical calculated values and experimental data from the literature. The influences of the equilibrium contact angle, droplet spacing and impinging velocity on the droplets impingement and coalescence behaviours are investigated. Numerical results demonstrate the width of the formed line depends significantly on the equilibrium contact angle and droplet spacing. The droplet spacing plays a significant role in controlling the coalescence moment of multiple droplets. The resolution of the printed pattern can be slightly increased with increase in impinging velocity.     

The influence of matrix viscoelasticity on the rheology of polymer blends

We examine the effects of matrix phase viscoelasticity on the rheological modeling of polymer blends with a droplet morphology. Two contravariant, second-rank tensor variables are adopted along with the translational momentum density of the fluid to account for viscoelasticity of the matrix phase and the ellipsoidal droplet shapes. The first microstructural variable is a conformation tensor describing the average extension and orientation of the molecules in the matrix phase. The other microstructural variable is a configuration tensor to account for the average shape and orientation of constant-volume droplets. A Hamiltonian framework of non-equilibrium thermodynamics is then adopted to derive a set of continuum equations for the system variables. This set of equations accounts for local conformational changes of the matrix molecules due to droplet deformation and vice versa. The model is intended for dilute blends of both oblate and prolate droplets, and droplet breakup and coalescence are not taken into account. Only the matrix phase is considered as viscoelastic; i.e., the droplets are assumed to be Newtonian. The model equations are solved for various types of homogeneous deformations, and microstructure/rheology relationships are discussed for transient and steady-state conditions. A comparison with other constrained-volume rheological models and experimental data is made as well.     

Investigation the effects of shear rate on stationary droplets coalescence by lattice Boltzmann

Coalescence is the process by which two or more droplets merge to form a single droplet. It can take place in many processes, ranging from meteorology to astrophysics. When two stationary droplets are suspended in a bulk vapor, if the gap between the droplets has been smaller than a critical value, the two droplets will coalesce. In this paper, single component, two phase flow is modeled under shear flow using a free energy lattice Boltzmann approach and the coalescence of stationary droplets are investigated for different cases of radius and shear rate. The results show that there is a critical gap between droplets and for the values larger than that they will not coalesce. Also in the case of constant thermophysical properties, this critical gap is a function of droplet radius and shear rate.     

Coalescence of two droplets impacting a solid surface

Coalescence of a falling droplet with a stationary sessile droplet is studied experimentally. High-speed video images are presented to show coalescence dynamics, shape evolution and contact line movement. Emphasis is put on spread length, which is the length of two coalesced droplets along their original centers. Experimental results have shown that the spread length can be larger or smaller than the ideal spread length, which is the spread diameter of individual droplet plus the center-to-center distance between the two droplets. Three different coalescence mechanisms based on comparing the maximum and the minimum spread lengths to the ideal spread length are identified. Correlations for the maximum and the minimum spread lengths are developed, which can be combined with the coalescence domains to determine the deposition conditions for forming continuous or discontinuous lines.     

Oldroyd-B黏弹性液滴碰撞过程的数值模拟

复杂的流变特性使凝胶推进剂的雾化过程存在一定困难,这制约了它的发展.聚合物胶凝剂的加入使凝胶推进剂具有黏弹性,从而在雾化时会产生黏弹性液滴,因此为了进一步认识凝胶推进剂的雾化机理、提高凝胶推进剂的雾化性能,对黏弹性液滴的碰撞行为进行数值模拟研究.针对凝胶推进剂雾化过程中出现的液滴撞击现象,考虑流体具有的黏弹性效应,采用...     

梯度润湿表面上液滴定向迁移及合并行为的格子Boltzmann模拟

采用改进的格子Boltzmann方法，对梯度润湿性表面上液滴的定向迁移及合并行为进行了数值模拟，该模型在精度和稳定性上都有很大改善，同时，研究了梯度润湿性表面上液滴定向迁移和合并的动力学特性，并对液滴尺寸及润湿梯度对液滴动力学特性的影响规律进行了分析。数值结果表明，液滴在梯度润湿性表面运动时会发生形变，且动态接触角逐渐减小。润湿梯度对液滴定向迁移行为有显著影响，润湿梯度越大，液滴左右侧接触线位移越大，润湿长度增加越快。但是液滴尺寸对接触线位移影响较小。润湿梯度对液桥宽度基本无影响，但对液滴初始合并时间有显著影响。     

Self-propelling,coalescing droplets

This work proposes and explores a new propulsion mechanism for sessile droplets which could be of interest for microfluidic applications. This mechanism relies on the Marangoni stresses resulting from the surface tension gradient arising when two droplets of different surface tensions coalesce. We argue that the tendency of the fluid to flow towards regions of higher surface tension is sufficient to displace the droplet. The coalescence of two miscible, partially wetting droplets with different surface tensions is investigated theoretically in this paper and modeled in the lubrication approximation framework. The problem is described by a set of three highly non-linear, coupled partial differential equations which is solved with a commercial finite element code. The analysis reveals two important dimensionless numbers which govern the flow characteristics, one related to the strength of the surface tension gradient and the other to the diffusion time scale. The numerical results confirm the occurrence of the self-propulsion behavior and a parametric study is performed to explore the role of the two dimensionless numbers on the propulsion speed and the total displacement. Unsurprisingly, self-propulsion is enhanced for larger values of the surface tension contrast between the two droplets and smaller values of the diffusion time scale which results in more time for the surface tension gradient to act.     

A modified SPH method to model the coalescence of colliding non-Newtonian liquid droplets

This paper develops a modified smoothed particle hydrodynamics (SPH) method to model the coalescence of colliding non-Newtonian liquid droplets. In the present SPH, a van der Waals (vdW) equation of state is particularly used to represent the gas-to-liquid phase transition similar to that of a real fluid. To remove the unphysical behavior of the particle clustering, also known as tensile instability, an optimized particle shifting technique is implemented in the simulations. To validate the numerical method, the formation of a Newtonian vdW droplet is first tested, and it clearly demonstrates that the tensile instability can be effectively removed. The method is then extended to simulate the head-on binary collision of vdW liquid droplets. Both Newtonian and non-Newtonian fluid flows are considered. The effect of Reynolds number on the coalescence process of droplets is analyzed. It is observed that the time up to the completion of the first oscillation period does not always increase as the Reynolds number increases. Results for the off-center binary collision of non-Newtonian vdW liquid droplets are lastly presented. All the results enrich the simulations of the droplet dynamics and deepen understandings of flow physics. Also, the present SPH is able to model the coalescence of colliding non-Newtonian liquid droplets without tensile instability.     

气相中双液滴正碰过程数值模拟研究

液滴碰撞现象普遍存在于动力装置燃烧室喷嘴的下游区域,影响燃料的雾化性能。为了揭示相同直径的双液滴中心碰撞机理,求解了轴对称坐标系下的N-S方程,采用VOF(Volume of Fluid)方法捕捉液滴碰撞过程中气液自由表面的演化规律。利用Qian等提供的实验结果对计算模型进行数值校验,验证了模型的准确性。在此基础上,研究了环境压强对液滴碰撞反弹后不同结果(分离和融合)的影响,分析了环境压强和Weber数对液滴碰撞分离的影响。结果表明,液滴在碰撞反弹后的状态(分离或融合)是由液滴间气膜压强与环境气动阻力共同作用的结果,环境压强对液滴碰撞分离过程基本没有影响;Weber数越大,碰撞过程中变形的幅度越大。     

流场中聚合物共混体系液滴形变的理论模型

讨论了两相聚合物共混体系中，悬浮于另一种牛顿（或粘弹）液体中的牛顿（或粘弹）液滴的形变理论模型．影响液滴形变的主要因素有两相的组成、粘度比和弹性比、动态界面张力、临界界面张力系数，外流场形式及其强度．对于两相均为牛顿流体的体系，理论预测能够与实验相符；对于两相（或其中一相）为粘弹流体的体系，由于弹性的影响而使液滴形变的研究变得复杂，理论模型尚需完善．建立完整的液滴形变理论模型还需深入研究界面层、微观分子形变、液滴之间及液滴和连续相介质之间的相互作用对液滴形变的影响     

A Hybrid Boundary/Finite Element Method for Simulating Viscous Flows and Shapes of Droplets in Electric Fields

A mixed boundary element and finite element numerical algorithm for the simultaneous prediction of the electric fields, viscous flow fields, thermal fields and surface deformation of electrically conducting droplets in an electrostatic field is described in this paper. The boundary element method is used for the computation of the electric potential distribution. This allows the boundary conditions at infinity to be directly incorporated into the boundary integral formulation, thereby obviating the need for discretization at infinity. The surface deformation is determined by solving the normal stress balance equation using the weighted residuals method. The fluid flow and thermal fields are calculated using the mixed finite element method. The computational algorithm for the simultaneous prediction of surface deformation and fluid flow involves two iterative loops, one for the electric field and surface deformation and the other for the surface tension driven viscous flows. The two loops are coupled through the droplet surface shapes for viscous fluid flow calculations and viscous stresses for updating the droplet shapes. Computing the surface deformation in a separate loop permits the freedom of applying different types of elements without complicating procedures for the internal flow and thermal calculations. Tests indicate that the quadratic, cubic spline and spectral boundary elements all give approximately the same accuracy for free surface calculations; however, the quadratic elements are preferred as they are easier to implement and also require less computing time. Linear elements, however, are less accurate. Numerical simulations are carried out for the simultaneous solution of free surface shapes and internal fluid flow and temperature distributions in droplets in electric fields under both microgravity and earthbound conditions. Results show that laser heating may induce a non-uniform temperature distribution in the droplets. This non-uniform thermal field results in a variation of surface tension along the surface of the droplet, which in turn produces a recirculating fluid flow in the droplet. The viscous stresses cause additional surface deformation by squeezing the surface areas above and below the equator plane.     

Role of droplet bridging on the stability of particle-containing immiscible polymer blends

The effect of micron-sized hydrophobic calcium carbonate particles on the stabilization of polydimethylsiloxane (PDMS)/polyisobutylene (PIB) immiscible model blends is investigated in this study. The analytical splitting of bulk and liquid–liquid interface contributions from the droplet bridging one is successfully performed due to the negligible contribution of hydrophobic microparticles to the bulk rheology of phases. The presence of particles at the fluid–fluid interface is supported by wetting parameter calculation and verified by optical microscopy observations. Moreover, direct visualizations shows that particles are able to form clusters of droplets by simultaneously adsorbing at two fluid–fluid interfaces and glue-dispersed droplets together, probably due to the patchy interactions induced by heterogeneous distribution of particles along the interface. Rheological studies show that the flow-induced coalescence is slowed down upon addition of particles and almost suppressed with the addition of 4 wt% particles. The linear viscoelastic response is modeled to estimate interfacial tension by considering the contribution of particle-induced droplet aggregation in addition to bulk and droplet deformation ones. From linear and nonlinear viscoelastic responses, the improved stability of filled polymer blends is attributed to the interfacial rheology and/or the bridged structure of droplets, even though the interfacial area is not fully covered by particles. Furthermore, Doi–Ohta scaling relations are investigated by employing stress growth response upon step-up of shear flow.     

Behavior of a ferrofluid layer with stable surface rupture subjected to a tangential magnetic field

The conditions of existence of a stable rupture in a horizontal ferrofluid layer on a liquid base are determined experimentally. The characteristics of ruptures of ferrofluid, kerosene, and water layers are compared to estimate the effect of different physicochemical properties of the liquid pairs used. It is found that switching-on a magnetic field parallel to the ferrofluid layer surface initiates deformation of the rupture and its simultaneous motion along the field. The dependences of the geometric parameters of the rupture on the magnetic field strength and the magnetic susceptibility of ferrofluid are established. The possibility of closing the stable rupture under the action of a magnetic field is demonstrated.     

含气泡油滴撞击油膜壁面时气泡的变形与破裂

油--气润滑过程中润滑油液滴受高速气流扰动易形成含气泡油滴,微气泡将对油滴撞击壁面时的运动过程以及壁面油膜 层的形成质量产生重要影响. 基于耦合的水平集--体积分数 方法,对含气泡油滴撞击油膜壁面行为进行数值模拟研究, 考察含气泡油滴撞击油膜壁面时气泡的变形运动过程,探讨气泡破裂的动力学机制,分析气泡大小、碰撞速度和液体黏度等因素对含气 泡油滴撞壁过程中气泡变形特征参数的影响规律. 研究表明:含气泡油滴撞击油膜壁面后气泡会发生变形,并破裂形成膜液滴;气泡随同 液滴运动过程中,气泡内外压力和速度梯度变化是使气泡发生破裂的主要诱因. 气泡大小对气泡破裂方式影响较大,气泡较小时发生单 点破裂,而气泡较大时更容易发生多处破裂. 不同大小气泡受力差异较大,气泡大小与破裂发生时刻没有明显相关性. 碰撞速度和液体 黏度对气泡的变形、破裂和破裂发生时刻都具有一定的影响. 碰撞速度越大,油滴动能越大,更容易产生气泡变形和破裂现象. 液体黏 度增大,在油滴撞壁运动前期促进气泡变形,而在运动后期可以阻延气泡破裂行为发生.      

An immersed‐boundary finite‐volume method for direct simulation of flows with suspended paramagnetic particles

In the present study, we have proposed an immersed‐boundary finite‐volume method for the direct numerical simulation of flows with inertialess paramagnetic particles suspended in a nonmagnetic fluid under an external magnetic field without the need for any model such as the dipole–dipole interaction. In the proposed method, the magnetic field (or force) is described by the numerical solution of the Maxwell equation without current, where the smoothed representation technique is employed to tackle the discontinuity of magnetic permeability across the particle–fluid interface. The flow field, on the other hand, is described by the solution of the continuity and momentum equations, where the discrete‐forcing‐based immersed‐boundary method is employed to satisfy the no‐slip condition at the interface. To validate the method, we performed numerical simulations on the two‐dimensional motion of two and three paramagnetic particles in a nonmagnetic fluid subjected to an external uniform magnetic field and then compared the results with the existing finite‐element and semi‐analytical solutions. Comparison shows that the proposed method is robust in the direct simulation of such magnetic particulate flows. This method can be extended to more general flows without difficulty: three‐dimensional particulate flows, flows with a great number of particles, or flows under an arbitrary external magnetic field. Copyright © 2010 John Wiley & Sons, Ltd.     

水击谐波流场中乳化油液滴粒径的PIV测量

为了掌握乳化油液滴在水击谐波流场中的碰撞、破裂、聚集和变形等微观形态的变化规律,采用粒子图像测速(PIV)技术对水平方管中乳化油液滴的水击谐波流场进行了测量,分析了在水击谐波流场中,不同激振力作用下乳化油液滴的粒径变化。测量结果表明,在水击谐波流场作用下,乳化油液滴平均粒径的增长率随着激振力的减小而减小,随着作用时间的增大呈增加趋势,直至粒径处于一种动态平衡;乳化液滴随着激振力增大到达波节聚集位置的时间减少,可见增大水击谐波激振力有利于乳化液滴的聚集并合并为大尺度的液滴,从而有效地提高了水击谐波流场作用下的油水分离效果。     

Motion of a deformable droplet of magnetic fluid in a rotating magnetic field

The problem of the magnetic field-driven rotation of a magnetic fluid droplet in a viscous nonmagnetic fluid is solved analytically and experimentally. The shape of the droplet and the magnetic fields and velocities of both fluids are calculated in the weak-field approximation. The droplet is flattened for any relations between the parameters of the system. The instability of the axisymmetric shape of the droplet is established experimentally. A result of the instability is a sudden change in the droplet shape from a flattened ellipsoid of revolution to a triaxial ellipsoid elongated in the equatorial plane. The critical magnetic Bond number is determined. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 22–30, January–February, 2000. The work was carried out with the support of the Russian Foundation for Basic Research (project No. 98-01-00182).