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.     

Effect of the contact-line dynamics on the oscillations of a compressed droplet

This paper studies the natural and forced oscillations of a deformed droplet of an inviscid liquid surrounded by a different liquid and bounded in the axial direction by solid planes. In equilibrium, the droplet is a figure of revolution and the ratio of its radius to height is significant. The equilibrium contact angle between the side surface of the droplet and the solid surface is different from a right angle. The motion of the contact line is taken into account by setting an effective boundary condition. It is shown that three characteristic ranges of natural frequencies exist.     

Simulating drop formation at an aperture by means of a Multi-Component Pseudo-Potential Lattice Boltzmann model

The growth and release of a pendent liquid droplet with the complex motion of the phase interface as a result of a liquid flow from an aperture has been studied by means of a Multi-Component Pseudo-Potential Lattice Boltzmann method. In this method, automatic component separation is attained by means of the Shan and Chen (1993) interaction strength G. We demonstrate that droplet formation can satisfactorily be described by combining three elements: using two components α and β (of the same density and viscosity), the G-driven separation of α and β, and gravity working on just α such that it becomes heavy and behaves as a liquid, while the gravity-free β mimics a gas. We present several time sequences of the growth and release of a pendent liquid droplet. Although the simulations were just 2–D, the dynamics of the necking, the tear shape of the droplet, and the motion of the apex after pinch-off all qualitatively agree with literature. The results are interpreted in terms of non–dimensional Bond, Ohnesorge and Archimedes numbers. We find convincing agreement between the relationships derived by dimensional analysis and the numerical simulations.     

PTV investigation of phase interaction in dispersed liquid–liquid two-phase turbulent swirling flow

An investigation of dispersed liquid–liquid two-phase turbulent swirling flow in a horizontal pipe is conducted using a particle tracking velocimetry (PTV) technique and a shadow image technique (SIT). Silicone oil with a low specific gravity is used as immiscible droplets. A swirling motion is given to the main flow by an impeller installed in the pipe. Fluorescent tracer particles are applied to flow visualization. Red/green/blue components extracted from color images taken with a digital color CCD camera are used to simultaneously estimate the liquid and droplet velocity vectors. Under a relatively low swirl motion, a large number of droplets with low specific gravity tend to accumulate in the central region of the pipe. With increasing droplet volume fraction, the liquid turbulence intensity in the axial direction increases while that in the wall-normal direction decreases in the central region of the pipe. In addition, the turbulence modification in the present flow is strongly dependent on the droplet Reynolds number; however, the interaction of droplet-induced turbulences is significant due to vortex shedding, particularly at high droplet Reynolds numbers and higher droplet volume fraction.     

同轴气流式液体射流分裂液滴粒径研究

针对同轴气流式液体射流分裂液滴粒径预测模型缺乏的现状,结合射流线性稳定性理论,建立了基于临界模数的同轴气流式黏性液体射流分裂液滴粒径表达式,在此基础上,分别研究了气流旋拧(气流同时存在轴向和周向运动)及流体物性(气体可压缩性、液体黏性、气液密度比和表面张力)对液滴粒径的影响规律.研究发现:周围气流轴向引射作用和同轴旋转...     

Effect of the contact-line dynamics on the natural oscillations of a cylindrical droplet

The natural oscillations of a cylindrical droplet of an inviscid liquid surrounded by a different liquid and bounded in the axial direction by solid planes are studied. The motion of the contact line is described using an effective boundary condition. The dependence of the frequency and damping ratio on the capillary parameter is found. It is shown that the fundamental frequency of the translation mode vanishes beginning from a certain value of the capillary parameter. Depending on the ratio of the radial and axial dimensions of the droplet, the fundamental frequency of the axisymmetric mode and modes higher than the translation mode can vanish in a certain range of the capillary parameter. This dependence of the natural oscillation frequencies on the problem parameters allows one to determine the capillary parameter. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 78–86, September–October, 2007.     

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.     

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.     

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.     

Drift of a reacting droplet due to the chemoconcentration capillary effect

The motion of a droplet with a first-order chemical reaction taking place at its surface with the participation of a surfactant dissolved in the external medium is considered. Approximate expressions are obtained for the velocity and other characteristics of the autonomous motion of the droplet caused by the surface capillary forces due to the nonuniform distribution of the surfactant over the surface of the moving droplet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 51–61, May–June, 1990.     

An experimental method for the investigation of droplet oscillations in a gaseous medium

An experimental method for the investigation of droplet oscillations in a gaseous medium is presented. The droplets are produced using vibrating orifice droplet generators. Experiments are carried out with droplets in the diameter range from 91 m to 288 m using propanol-2, water, and n-hexadecane; the gaseous host medium is air. Oscillatory motions of the fundamental mode n = 2 and of the first higher order mode n = 3 occur during the disintegration of the liquid jet produced by the droplet generator. The periodical production of the droplets allows the observation and evaluation of each phase of the motion under quasi-steady conditions. Surface energies are determined from the droplet shapes on photos. The periods of the oscillations are found to be very close to the prediction of the linear theory.     

Application of maximum entropy method for droplet size distribution prediction using instability analysis of liquid sheet

This paper describes the implementation of the instability analysis of wave growth on liquid jet surface, and maximum entropy principle (MEP) for prediction of droplet diameter distribution in primary breakup region. The early stage of the primary breakup, which contains the growth of wave on liquid–gas interface, is deterministic; whereas the droplet formation stage at the end of primary breakup is random and stochastic. The stage of droplet formation after the liquid bulk breakup can be modeled by statistical means based on the maximum entropy principle. The MEP provides a formulation that predicts the atomization process while satisfying constraint equations based on conservations of mass, momentum and energy. The deterministic aspect considers the instability of wave motion on jet surface before the liquid bulk breakup using the linear instability analysis, which provides information of the maximum growth rate and corresponding wavelength of instabilities in breakup zone. The two sub-models are coupled together using momentum source term and mean diameter of droplets. This model is also capable of considering drag force on droplets through gas–liquid interaction. The predicted results compared favorably with the experimentally measured droplet size distributions for hollow-cone sprays.     

Droplet formation in the rupture of a liquid film under the action of a thermal load

The deformation and rupture of a liquid film, suspended between two solid walls, under the action of a localized thermal load is considered. For studying this process, a two-dimensional model is used, which describes the motion of a thin layer of a viscous non-isothermal liquid under microgravity conditions. For modeling the dynamics of the liquid, the Navier-Stokes equations in the “vorticity—stream function” variables are used. A numerical analysis of the influence of thermal loads on the deformation and the mechanism of the rupture of freely suspended films is performed. Is is shown that for a certain width of the thermal beam acting on the film free surface the rupture of the film may occur with the formation of a droplet. The results of the solution of model problems are presented.     

Particle image velocimetry and infrared thermography in a levitated droplet with nanosilica suspensions

Preferential accumulation and agglomeration kinetics of nanoparticles suspended in an acoustically levitated water droplet under radiative heating has been studied. Particle image velocimetry performed to map the internal flow field shows a single cell recirculation with increasing strength for decreasing viscosities. Infrared thermography and high speed imaging show details of the heating process for various concentrations of nanosilica droplets. Initial stage of heating is marked by fast vaporization of liquid and sharp temperature rise. Following this stage, aggregation of nanoparticles is seen resulting in various structure formations. At low concentrations, a bowl structure of the droplet is dominant, maintained at a constant temperature. At high concentrations, viscosity of the solution increases, leading to rotation about the levitator axis due to the dominance of centrifugal motion. Such complex fluid motion inside the droplet due to acoustic streaming eventually results in the formation of a ring structure. This horizontal ring eventually reorients itself due to an imbalance of acoustic forces on the ring, exposing larger area for laser absorption and subsequent sharp temperature rise.     

Abnormal rotation of a deformed liquid crystal droplet immersed in an isotropic fluid after transient flow

The morphology evolution of liquid crystal droplets immersed in an isotropic fluid in flow field is found to be different from flexible polymer droplets. In this paper, we investigated the retraction of a liquid crystal droplet after transient flow. It is found that the liquid crystal droplet will rotate during the shape recovery, which has never been observed for an isotropic droplet. The factors that influence the rotational angle of a single liquid crystal droplet during retraction progress were studied, including the temperature, the dimension of the droplets, the time of shear flow, the shear rate, the flow type, and the properties of liquid crystal molecules. The rotation of liquid crystal droplet during shape recovery is ascribed to both the bulk elasticity of liquid crystal droplets and the anisotropic properties of the interface between liquid crystal and isotropic fluid.     

Unsteady Liquid Jets

The behaviour of unsteady liquid jets in a gas atmosphere is mainly governed by the conservation of momentum and the interaction with the environment. In this article it will be shown that many of the particular effects in the propagation and desintegration of unsteady jets are simply explained by the conservation of initial momentum. Many of the distortions and peculiar shapes of the liquid elements of the jet can be explained by a time and space development of weak initial distortions of momentum in travelling waves during propagation. The first part of the paper is devoted to the droplet formation in liquid jets of moderate Weber numbers in the order of ten. In this regime and higher the surface tension is not the governing influence of instability of jet decomposition and hence of the droplet formation. If there is a high frequency distortion of the velocity which is artificially implemented in our experiments by a special device, but under usual conditions also existing by turbulence or source distortions of the nozzle flow, the concentration of mass in packets in a kind of shock formation is the main reason for droplet formation. The frequency of droplets in space and time is given by the frequency of the distortion. The amplitude and phase is also given by the initial values of the driver for these quantities. For large amplitudes and low frequencies the jets are decomposed in very peculiar shapes, which can be very simply explained by the collision of packets of concentrated mass on the axis of the jet motion. The whole range of phenomena can be understood in a theoretically elegant manner by a Lagrangian transform of the nonlinear equations of motion together with some plausible considerations about the lateral motion. Solutions for the droplet formation and the formation of liquid films in different shapes can be achieved this way. As an illustration of the whole range of beautiful and regularly reproducible details of jet decomposition some photographs will be shown which summarize the phenomena and give an idea of the importance of systematic investigation of the sometimes very confusing appearance of fluid jets.     

Modelling and simulation of sprays in laminar flames

In the present paper we model and numerically simulate steady, laminar, premixed spray flames. The gasphase is described in Eulerian form by the equations governing the conservation of overall mass, momentum, energy and species mass. The liquid phase is described in Lagrangian form by the overall continuity equation, which reduces to an equation for the droplet size, the equations of motion, the energy equation and a droplet density function transport equation. The latter is the so-called spray equation, which, at any position in the chemically reacting flowfield, describes the joint distribution of droplet size, droplet velocity and droplet temperature. Herein the spray equation is solved using a Monte Carlo method. Detailed models of the exchange of mass, momentum and energy between the gaseous and the liquid phase are taken into account. The results presented in this paper are for an octane-air flame, where small amounts of liquid octane in form of a liquid spray are added to a fresh, unburnt gaseous octane-air mixture.Presented at Euromech Colloquium 324: The Combustion of Drops, Sprays and Aerosols, 25th–27th July 1994, Marseilles, France.     

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.     

On the dynamic behavior of a liquid droplet impacting upon an inclined heated surface

The dynamic behavior of a water droplet impinging upon a heated surface was shown to be significantly different, depending on the normal momentum of the impinging droplet before impact. This experimental study focused mainly on the effects of the impinging angle of the droplet on impact dynamics and its dependence on surface temperature. At the surface temperature of the nucleate boiling regime, disintegration of the droplet did not occur, whereas the deforming droplet adhered to the surface. The liquid film was spread and contracted several times on the horizontal surface, but the expanded droplet merely slipped without noticeable contraction on the inclined surfaces. In the film boiling regime, the impinging droplet spread over the surface as a liquid film separated from the surface by the vapor produced. Depending on the magnitude of the normal momentum of droplet, disintegration into several irregular shapes of liquid elements occurred in the case of the horizontal and 30°-inclined surfaces. The impinging droplet in the case of the 60°-inclined surface did not break up and tended to recover its original spherical shape. Received: 16 February 1999/Accepted: 9 November 1999     

Problems of Hydrodynamics for a Triaxial Ellipsoid

Problems of motion of a triaxial ellipsoid in an ideal liquid and in a viscous liquid in the Stokes approximation and also equilibrium shapes of the rotating gravitating liquid mass are considered. Solutions of these problems expressed via four quadratures depending on four parameters are significantly simplified because they are expressed via the only function of two arguments. The efficiency of the proposed approach is demonstrated by means of analyzing the velocity and pressure fields in an ideal liquid, calculating the added mass of the ellipsoid, determining the viscous friction, and studying the equilibrium shapes and stability of the rotating gravitating capillary liquid. The pressure on the triaxial ellipsoid surface is expressed via the projection of the normal to the impinging flow velocity. The shape of an ellipsoid that ensures the minimum viscous drag at a constant volume is determined analytically. A simple equation in elementary functions is derived for determining the boundary of the domains of the secular stability of the Maclaurin ellipsoids. An approximate solution of the problem of equilibrium and stability of a rotating droplet is presented in elementary functions. A bifurcation point with non-axisymmetric equilibrium shapes branching from this point is found.