A phenomenological model for wall effects on the deformation of an ellipsoidal drop in viscous flow

The simple phenomenological model developed by Maffettone and Minale (J Non-Newt Fluid Mech 78:227–241, 1998) for the deformation of a single ellipsoidal drop in a viscous flow is extended here to predict the drop deformation in confined viscous flows. The model is capable of describing the transient evolution of an ellipsoidal drop subjected to a generic flow field. The steady-state predictions are analytical and recover the theoretical limits of Shapira and Haber (Int J Multiph Flow 16:305–321, 1990) for steady small deformation of a drop in a confined simple shear flow. Model predictions are compared with data available in the literature that cover a wide range of parameter values, and the agreement is good.     

Effect of interfacial modifier on single drop deformation and breakup in step increasing shear flow

This work deals with in situ visualisation of deformation and breakup of a copolymer modified single Newtonian drop immersed in a Newtonian homogenous matrix. The experiments were carried out on a model system made of poly-isobutylene as the suspending fluid and two poly-dimethylsiloxanes with different molecular weights as the drop phase with viscosity ratios 0.036 and 1.13, below and above but close to unity. Three weight concentrations 0.5%, 2% and 10% of the block copolymer laying below, close to and above the critical concentration of the total drop surface coverage were examined. Single drop deformation experiments were carried out in a home-designed Couette quartz cell connected to a home-modified Paar Physica Rheometer. The variation in the length-to-diameter ratio (L/D) versus shear rate and capillary number was measured both in steady and in transient regimes till breakup. The results indicated a weaker resistance of copolymer modified drops against hydrodynamic stresses at both viscosity ratios as compared to the clean drop. However, the drop deformation was found to be complex and depends on the copolymer concentration and the viscosity ratio.     

Deceleration and deformation of a liquid drop in a gas stream

An analytic method is developed for calculating the nonstationary motion and spreading of two-dimensional and axisymmetric liquid drops in a gas stream. The method is based on an expansion of the Navier-Stokes equations in a small parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 58–69, March–April, 1981.     

Dynamics of separation of a single drop in air

The processes of formation of alcohol and water drops, as well as formation of small fragments—satellites, are traced using the high-speed filming. The trajectory of a water drop satellite is nonmonotonic, at first the satellite moves upward against the gravity force, reaches the oscillating residual fluid at the nozzle exit, and then starts to move down. From the satellite, a microdroplet is ejected, which bounces off the residual fluid at the nozzle, returns back to the satellite and merges. In the case of an alcohol drop, no accompanying microdroplet is formed, and the satellite follows a nearballistic trajectory.     

The impact of a single drop on a wetted solid surface

 The impact of single drops on a thin liquid film was studied to understand the mechanism of secondary atomisation of sprays colliding on a wetted, cold, solid surface. To span a wide range of conditions various mixtures of water and glycerol were used. The use of Weber number, Ohnesorge number and non-dimensional film thickness to describe the peculiarities of the phenomenon allowed to carry out the experiments under appropriate similarity conditions. The impact of millimetric drops was analysed in detail by photographic means, using both still photography to study impact morphology, and laser sheet visualisation to investigate secondary droplet formation. Two mechanisms of splash were identified, depending essentially on the liquid viscosity (Ohnesorge number), a parameter which appears to play an important role also in defining the splash morphology. A photographic documentation is annexed. The characteristic times of the crown formation, the non-linear evolution of cusps (jet formation) and the surface roughness influence are further discussed. The experimental results allow to propose an empirical correlation for the splashing/deposition limit, for a wide range of conditions, and a comparison to available previous works is presented. The influence of the film thickness and liquid viscosity on the splash is confirmed and quantified. Received: 1 March 1996/Accepted: 12 November 1996     

Two-dimensional study of drop deformation under simple shear for Oldroyd-B liquids

The effect of viscoelasticity on the deformation of a circular drop suspended in a second liquid in shear is investigated with direct numerical simulations. A numerical algorithm based on the volume-of-fluid method for interface tracking is implemented in two dimensions with the Oldroyd-B constitutive model for viscoelastic liquids. The code is verified against a normal mode analysis for the stability of two-layer flow in a channel; theoretical growth rates are reproduced for the interface height, velocity and stress components. Drop simulations are performed for drop and matrix liquids of different viscosities and elasticities. A new feature is found for the case of equal viscosity, when the matrix liquid is highly elastic and surface tension is low; hook-like structures form at the drop tips. This is due to the growth of first normal stress differences that occur slightly above the front tip and below the back tip as the matrix elasticity increases above a threshold value.     

Models for the formation of gas bubbles at a single submerged orifice in a non-Newtonian fluid

Nitrogen bubbles were emitted by a single horizontal orifice submerged in an aqueous solution of sodium carboxymethylcellulose (C.M.C.). Their behaviour follows the Ostwald Dé Waele rheological model: τ = mE_oⁿ; τ is the shear stress and E_o the shear rate.     

Modelling of a single drop impact onto liquid film using particle method

The process of single liquid drop impact on thin liquid surface is numerically simulated with moving particle semi‐implicit method. The mathematical model involves gravity, viscosity and surface tension. The model is validated by the simulation of the experimental cases. It is found that the dynamic processes after impact are sensitive to the liquid pool depth and the initial drop velocity. In the cases that the initial drop velocity is low, the drop will be merged with the liquid pool and no big splash is seen. If the initial drop velocity is high enough, the dynamic process depends on the liquid depth. If the liquid film is very thin, a bowl‐shaped thin crown is formed immediately after the impact. The total crown subsequently expands outward and breaks into many tiny droplets. When the thickness of the liquid film increases, the direction of the liquid crown becomes normal to the surface and the crown propagates outward. It is also found that the radius of the crown is described by a square function of time: r_C = [c(t ? t₀)]^0.5. When the liquid film is thick enough, a crown and a deep cavity inside it are formed shortly after the impact. The bottom of the cavity is initially oblate and then the base grows downward to form a sharp corner and subsequently the corner moves downward. Copyright © 2004 John Wiley & Sons, Ltd.     

The optimal drop shape for vortices generated by drop impacts: the effect of surfactants on the drop surface

Subsurface vortices are frequently created when a falling drop strikes a flat water surface. Prior work has demonstrated that the shape of the drop at the point of impact is critical in determining how deep or how fast the resulting vortex will penetrate into the water bulk. In the present study, the details of this phenomena are explored by using surfactants to vary surface tension. Specifically, Triton X-100 monolayers are created on the surface of the drop, and on the flat water surface. The results of these experiments suggest that there is no single optimal drop shape resulting in best vortex penetration. Rather, the data suggest that the optimal shape depends on the surface tension of the falling drop. An attempt is made to reconcile contradictory results in the literature using this result.

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激波诱导高速气流中液滴的初期变形

基于激波管平台和高速摄影方法对平面激波诱导高速气流中液滴的早期变形现象进行实验研究。研究发现在相近的We数或Re数下,实验参数的改变可导致液滴形态发展出现显著差异。这种差异主要体现在背风面的脊状环形突起、褶皱区以及后驻点区的凹凸形态。对刚性圆球外流的数值模拟显示,液滴变形早期形态与外流场结构和表面气动力分布之间存在鲜明的对应关系。最后采用简化理论推导出一组估测液滴早期变形的表达式。将数值模拟所得气动力数据代入计算发现:导致液滴变形的主要驱动力是液滴表面不均匀压力的挤压效应,而不是界面剪切摩擦所引起的切向流动堆积效应,前者高出后者约2个数量级;此外,采用压力作用理论计算所得液滴外形在主要变形特征和变形量级上均可与实验图像很好地吻合。     

Production of secondary drops during the single water drop impact onto a plane water surface

The normal impact of single water drops onto a plane water surface was studied experimentally to reveal the amount of secondary drops produced from the rim of crown-like interfacial structure. Within the experimental ranges tested, the ratio of the total mass of secondary drops to the mass of primary drop was approximately within 0–1 and correlated well as the function of dimensionless parameter K that consisted of the impact Weber number We and the Ohnesorge number Oh (K = We Oh ^−0.4). The dependences of the number and the mean diameter of secondary drops on K and dimensionless film thickness were also investigated.     

Effects of viscosity ratio on deformation of a viscoelastic drop in a Newtonian matrix under steady shear

Deformation of an Oldroyd B drop in a Newtonian matrix under steady shear is simulated using a front tracking finite difference method for varying viscosity ratio. For drop viscosity lower than that of the matrix, the long-time steady deformation behavior is similar to that of the viscosity matched system—the drop shows reduced deformation with increasing Deborah number due to the increased inhibiting viscoelastic normal stress inside the drop. However for higher viscosity ratio systems, the drop response is non-monotonic—the steady drop deformation first decreases with increasing Deborah number but above a critical Deborah number, it increases with further increase in Deborah number, reaching higher than the viscous case value for some viscosity ratios. We explain the increase in deformation with Deborah number by noting that at higher viscosity ratios, strain rate inside the drop is reduced, thereby reducing the inhibiting viscoelastic stress. Furthermore, similar to the viscosity matched system, the drop inclination angle increases with increasing Deborah number. A drop aligned more with the maximum stretching axis at 45 degree of the imposed shear, experiences increased viscous stretching. With increased ratio of polymeric viscosity to total drop viscosity, the drop deformation decreases and the inclination angle increases. Our simulation results compare favorably with a number of experimental and computational results from other researchers.     

Numerical study of a single drop impact onto a liquid film up to the consequent formation of a crown

In this paper, the whole dynamic process of a single drop impact onto a thin liquid surface up to the consequent formation of a thin crown is numerically studied using the smoothed particle hydrodynamics (SPH) method. Especially, the gravity, artificial viscosity, and surface tension are introduced into the model. The obtained SPH numerical results are compared with experimental results. The numerical model of the SPH method is valid for simulating the dynamic process of a single drop impact onto a liquid surface. Meanwhile, it is found that the whole dynamic process mainly depends on the depth of the liquid pool and the initial velocity of the droplet.     

单晶有限变形的热力耦合计算模型

以弹性变形梯度作为基本变量,结合热力学理论构造了单晶有限变形的热、力耦合计算模型。该模型考虑了温度、变温速率以及塑性耗散等条件对单晶有限变形的影响,相对于传统的以弹性变形梯度为基本变量的晶体塑性模型,算法能够体现温度效应的影响。采用隐式的积分方法对建立的控制方程进行计算以保证求解过程的稳定。以1100Al单晶为例计算了不同升温、降温速率,以及不同应变率影响下的材料应力-应变的响应。结果表明,模型能较好地反映变温过程中,单晶各向异性性质的演化以及应力、应变之间关系的变化。     

A continuum model for intermittent deformation of single crystal micropillars

Strain bursts are often observed during compression tests of single crystal micropillars. In this work, we formulate a new continuum model that accounts for the strain bursts within the framework of crystal plasticity. The strain bursts are separated from the loading stage (nearly elastic loading) by introducing a dimensionless constant in the continuum model, and are detected by load serrations. The boundary conditions in the context of micropillar compression are studied and they are shown to be changing and unpredictable as plastic deformation proceeds. To evaluate the validity of our model, finite element simulations of the uniaxial compression tests on nickel micropillars are performed. Our simulations produce clearly visible strain bursts during the plastic flow and the produced intermittent flows are comparable with the experimental observations. For the bulk crystal, a series of strain bursts is identified in the course of plastic flow, despite an apparently smooth stress–strain response. We also show that the intermittent flow is intensified in the micrometer-scale due to both increasing numbers of the successive strain bursts and increasing amplitude of the strain burst, when the specimen size decreases. Finally, we show that the occurrences of the strain bursts are always associated with negative values of the second-order work.     

A new approach for the ellipsoidal statistical model

In this paper we aim to introduce a systematic way to derive relaxation terms for the Boltzmann equation based on the minimization problem for the entropy under moments constraints (Levermore in J. Stat. Phys. 83:1021–1065, 1996; Schneider in M2AN 38:541–561, 2004). In particular the moment constraints and corresponding coefficients are linked with the eigenfunctions and eigenvalues of the linearized collision operator through the Chapman–Enskog expansion. Then we deduce from this expansion a single relaxation term of BGK type. Here we stop the moments constraints at order two in the velocity v and recover the ellipsoidal statistical model (Holway in Rarefied Gas Dynamics, vol I, pp 193–215, 1966).      

A comparison of boundary element and finite element methods for modeling axisymmetric polymeric drop deformation

A modified boundary element method (BEM) and the DEVSS‐G finite element method (FEM) are applied to model the deformation of a polymeric drop suspended in another fluid subjected to start‐up uniaxial extensional flow. The effects of viscoelasticity, via the Oldroyd‐B differential model, are considered for the drop phase using both FEM and BEM and for both the drop and matrix phases using FEM. Where possible, results are compared with the linear deformation theory. Consistent predictions are obtained among the BEM, FEM, and linear theory for purely Newtonian systems and between FEM and linear theory for fully viscoelastic systems. FEM and BEM predictions for viscoelastic drops in a Newtonian matrix agree very well at short times but differ at longer times, with worst agreement occurring as critical flow strength is approached. This suggests that the dominant computational advantages held by the BEM over the FEM for this and similar problems may diminish or even disappear when the issue of accuracy is appropriately considered. Fully viscoelastic problems, which are only feasible using the FEM formulation, shed new insight on the role of viscoelasticity of the matrix fluid in drop deformation. Copyright © 2001 John Wiley & Sons, Ltd.     

Effect of impingement angle on the outcome of single water drop impact onto a plane water surface

Experiments of single water drop impact onto a plane water surface were carried out to investigate the effect of impingement angle on the total mass of secondary drops produced during the collision. When the impingement angle (the angle between the velocity vector of primary drop and the normal vector to water surface) was less than 50°, an increase in the impingement angle led to a remarkable increase in the total mass of secondary drops; this could be attributed to a significant increase in the secondary drop size. However, no secondary drop was observed within the experimental ranges tested when the impingement angle exceeded 70°.