Gravity Effect on the Axisymmetric Drop Spreading

The flattening (spreading) of the axisymmetrical drop on a plane horizontal surface under action of gravity force at zero tangential force (no shear at the gas–liquid interface) is investigated analytically and numerically. We determine the exact profile of compressed drop assuming the condition of drop volume conservation. 2D time dependant numerical model, based on a finite difference method, has been developed to describe the hydrodynamics inside the drop. The energy and Navier–Stokes equations are solved within the drop’s analytical profile. Effects of surface tension and thermocapillarity are taken into account. The effect of gravity has been studied to define main features of the drop dynamics. In calculations vector of gravitational acceleration is oriented perpendicularly to the surface, the Bond number is changed in the range from Bo = 0 to Bo = 151.6. Our results show that the gravity has a significant effect on the drop spreading.     

微圆管内R1234ze(E)流动沸腾的环状流特性模拟研究

本文建立了制冷剂R1234ze(E)在微圆管内流动沸腾过程中的环状流模型,对传热和气液两相流动压降进行了模拟研究。综合考虑重力、表面张力及气液界面剪切力的影响,模拟分析了周向液膜不均匀分布特性及该特性对流动与换热的影响,经验证,计算结果与已有实验结果吻合较好,此外还研究了不同因素对环状流区域表面传热系数与压降的影响。模拟结果表明：在流动起始区域,截面液膜厚度的分布受重力作用影响,随着流动沸腾过程的进行,该影响作用开始减弱,且有重力作用时的环状流平均表面传热系数高于无重力作用时的环状流平均表面传热系数,随着重力加速度的增加,环状流的平均表面传热系数不断增大;随着质量流速的增大,表面传热系数与压降均随之增大;随着管径增大,表面传热系数与压降均随之减小。     

A finite element‐based level set method for fluid–elastic solid interaction with surface tension

A numerical method for simulating fluid–elastic solid interaction with surface tension is presented. A level set method is used to capture the interface between the solid bodies and the incompressible surrounding fluid, within an Eulerian approach. The mixed velocity–pressure variational formulation is established for the global coupled mechanical problem and discretized using a continuous linear approximation in both velocity and pressure. Three ways are investigated to reduce the spurious oscillations of the pressure that appear at the fluid–solid interface. First, two stabilized finite element methods are used: the MINI‐element and the algebraic subgrid method. Second, the surface integral corresponding to the surface tension term is treated either by the continuum surface force technique or by a surface local reconstruction algorithm. Finally, besides the direct evaluation method proposed by Bruchon et al., an alternative method is proposed to avoid the explicit computation of the surface curvature, which may be a source of difficulty. These different issues are addressed through various numerical examples, such as the two incompressible fluid flow, the elastic inclusion embedded into a Newtonian fluid, or the study of a granular packing. Copyright © 2012 John Wiley & Sons, Ltd.     

A finite-element method for interfacial surfactant transport, with application to the flow-induced deformation of a viscous drop

A Galerkin finite-element method is developed for solving the transport equation governing the evolution of the surface concentration of an insoluble surfactant over a stationary or evolving fluid interface. The numerical procedure is implemented on an unstructured three-dimensional surface grid consisting of six-node curved triangular elements. Numerical investigations show that the finite-element method is superior to a previously developed finite-volume method for both convection- and diffusion-dominated transport, and especially when the interfacial grid is coarse and steep gradients arise due to local accumulation. The numerical methods for surface transport are combined with a boundary-element method for Stokes flow, and dynamical simulations are performed to illustrate the possibly significant effect of the surface equation of state relating the surface tension to the surfactant concentration on the deformation of a viscous drop in simple shear flow.     

Fundamental Numerical Simulation of Microbubble Interaction Using Multi-scale Multiphase Flow Equation

Modeling gas–liquid interfaces is a key issue in numerical research on multiphase flows. In particular, the mechanism of bubble coalescence/repulsion has not yet been clarified, and thus far, no valid numerical models have been developed. The mechanism may be related to thermodynamics, electromagnetics, hydrodynamics, and heat and mass transfer. Experimental studies have revealed that the concentration of electrolyte dissolved in water and the ion pairs affect bubble coalescence. Contamination such as ions adsorbed at the interface cannot be ignored when considering interfacial interaction. Interfacial effects may be more significant than gravitational effects especially when considering microbubble interactions. Therefore, it is necessary to consider such interfacial effects on microbubble coalescence when developing a gas–liquid interfacial model. In our previous study, we assumed that the interface exists as a thin fluid membrane with finite thickness. Then, we developed a new gas–liquid interfacial model on the basis of thermodynamic and mathematical approaches. In that study, a new governing equation was derived by incorporating the free energy, including the electrostatic potential due to contamination at the interface, into the conventional Navier–Stokes equation on the basis of a multi-scale concept. In the present study, a numerical test was performed for evaluating the surface tension effect of our new interfacial model. Then, the new governing equation was qualitatively evaluated by simulating the interaction between two microbubbles. In the simulation, we investigated microbubble behavior by considering electrostatic potentials of different magnitudes.     

Surface tension of liquid Cu and anisotropy of its wetting of sapphire

Precise surface tension data of liquid Cu are fundamental for studying its interaction with differently oriented single crystalline sapphire surfaces. For this reason, the surface tension of liquid Cu was measured covering a wide temperature interval of 1058 °C ≤ T ≤ 1413 °C. To avoid contamination of the sample from contact with container walls, the measurement was performed contactlessly in an electromagnetic levitation furnace using the oscillating drop method. A fast digital CMOS-camera (400 fps) recorded top view images of the oscillating sample. From an analysis of the frequency spectrum the surface tension was determined. The measured surface tension of Cu was used to calculate interfacial energies from contact angles of liquid Cu droplets, deposited on the C(0001), A(11-20), R(1-102) and M(10-10) surfaces of sapphire substrates. These were measured by means of the sessile drop method at 1100 °C using a drop dispenser. Within the first minutes of contact with the sapphire substrates, the contact angles of liquid Cu droplets rise to their equilibrium values. From these, in addition to interfacial energies also works of adhesion were determined.     

Surface tension of γ-TiAl-based alloys

Within the Integrated Project IMPRESS, funded by the EU, a concerted action was taken to determine the thermophysical properties of a γ-TiAl-based alloys, suitable for casting of large turbine blades for aero-engines and stationary gas turbines. The challenge was to develop a castable alloy, free of grain refiners and susceptible to heat treatment. Owing to the high reactivity of this class of alloys, many difficulties were encountered to process the liquid phase in a crucible. This prevented also the measurements of specific heat, viscosity and electrical conductivity in the liquid phase. However, surface tension and density could be measured using container-less techniques. For the surface tension determination, both the oscillating droplet method by the electromagnetic levitation as well as a combined method using two methodologies in one test (i.e. the pendant drop and sessile drop) by an advanced experimental complex that has been designed for investigations of high temperature capillarity phenomena were applied. All the quantities have been obtained as a function of temperature, in some cases also in the undercooled liquid. In this article, we report a comparative discussion on the results obtained for the surface tension of Ti–Al–Nb and Ti–Al–Ta alloys, together with the corresponding theoretical values calculated by thermodynamic models.     

High-speed imaging and CFD simulations of a deforming liquid metal droplet in an electromagnetic levitation experiment

Electromagnetic levitation of a liquid metal droplet is of great interest to study gas–liquid metal reactions. An important prerequisite for the evaluation of the overall mass transfer between the gas and metal is to characterize the geometry of the deforming molten droplet, which determines the interfacial reaction area. In this article, the free surface shape and dynamics of a molten 80%Ni–20%Cr droplet is investigated both experimentally and numerically. The frequencies associated to the oscillatory translational motions of the drop and to the vibrations of its free surface are measured using high-speed video image analysis. A 2D transient model is then presented, in which three interacting phenomena are considered: electromagnetic phenomena, the turbulent flow of liquid metal in the drop and the change in the drop shape. The numerical results presented demonstrate the capabilities of the model.     

Thermodynamic study on the melting of nanometer-sized gold particles on graphite substrate

Surface tension plays an important role in lowering the melting temperature of nanometer-sized particles, but whether the surface tension determined in macro scale is valid for the nanometer-sized particles is unclear. Moreover, the melting of the nanometer-sized particles formed on solid substrates can be affected by interfacial tension, but no research has been reported on the effect of substrates on the melting temperature. Therefore, in order to predict the melting temperature of nanometer-sized metallic particles on solid substrates, thermodynamic parameters such as surface tension and interfacial tension should be properly estimated. In the present work, thermodynamic assesment is given on the melting temperature of gold particles in nanometer-size placed on a graphite substrate. Surface tension of liquid gold and the contact angle between liquid gold and the graphite substrate are measured by the constrained drop method and the sessile drop method in macro scale, respectively. Then, the effect of the graphite substrate on the melting temperature of nanometer-sized gold particles are examined by thermodynamic calculations minimizing the total Gibbs free energy, the sum of bulk, surface and interface energies. It is found that the graphite substrate has negligible effect on the melting temperature of nanometer-sized gold particles. Thermodynamic assessments provide that the surface tension of solid gold is 1.339 N/m at 1373 K and that the decrease in the surface tension of liquid gold with size will be considerable for the particles smaller than ~5 nm.     

High-throughput screening of protein surface activity via flow injection analysis-pH gradient-dynamic surface tension detection

Using flow injection analysis (FIA), a pH gradient is blended in real time with a protein sample as the pH-dependent protein surface activity is measured by a dynamic surface tension detector (FIA-pH-DSTD). This instrumental system was developed as a high-throughput method for the screening of protein surface activity at the air/liquid interface as a function of pH. This method utilizes the continuous flow, drop-based dynamic surface tension detector in combination with flow injection sample introduction and blending of a steady-state concentration of protein sample with a pH gradient ranging from pH 2.0 to pH 11.5. Dynamic surface tension is measured through the differential pressure across the air/liquid interface of repeatedly growing and detaching drops. Continuous surface tension measurement is achieved for each eluting drop of 2-s length (2 muL), providing insight into both the kinetic and thermodynamic behaviors of molecular orientation processes at the liquid/air interface. Three-dimensional data are obtained, with surface tension first converted to surface pressure, which is collected as a function of elution time versus drop time. In FIA-pH-DSTD, a commercial pH probe is used to measure pH during elution time, enabling surface pressure throughout drop time to be subsequently plotted as a function of eluting pH. An automated DSTD calibration procedure and data analysis method is applied, which allows simultaneous use of two different solvents, permitting real-time dynamic surface tension data to be obtained. The method was applied to the analysis of 14 commercial purified proteins, yielding characteristic features of surface activity as a function of pH. The reproducibility of the measurement and selectivity advantage of the DSTD was shown for the analysis of serum albumins from various mammalian sources. Several applications were also suggested and discussed in order to show the potential of the method for protein and food chemistry studies and in the study of protein-polymer interactions.     

Computer simulation of evolving capillary bridges in granular media

A numerical method for the simulation of spatially evolving liquid–vapour interfaces in arbitrary two dimensional granular media is presented. Solid- and liquid-phase objects are described by polynomials whose edges evolve according to surface tension forces until a prescribed equilibrium contact angle at three-phase contact points and a constant mean curvature on two-phase contact lines is achieved. The main advantage of the method is the possibility to account for topological transitions (interface coalescence or rupture) and direct calculation of the force acting on solid interfaces due to liquid bridges. The method has been validated by comparing numerical and analytical results for a single pendular liquid bridge and then demonstrated on the simulation of transition from the pendular to funicular and capillary state in a wet particle assembly.     

Simulating Propellant Reorientation of Vehicle Upper Stage in Microgravity Environment

The computational and experimental studies have been performed to investigate the hydrodynamic process of liquid propellant reorientation for the launch vehicle series fuel tanks in microgravity environment. The VOF method was used to simulate the free surface flow of gas-liquid. The process of the liquid propellant reorientation started from initially curved interface at low Bond number. The propellant reorientation flow procedure at high Bond number was obtained from numerical simulation and scale model experiment in drop tower. The numerical results agreed well with the experiments. The results can be used to adjust the engineering reorientation parameters.     

Density and Surface Tension of Liquid Ternary Ni–Cu–Fe Alloys

The density and surface tension of liquid Ni–Cu–Fe alloys have been measured over a wide temperature range, including the undercooled regime. A non-contact technique was used, consisting of an electromagnetic levitator equipped with facilities for optical densitometry and oscillating drop tensiometry. At temperatures above and below the liquidus point, the density and surface tension are linear functions of temperature. The concentration dependence of the density is significantly influenced by a third-order (ternary) parameter in the excess volume. The surface tensions are rather insensitive to substitution of the two transition metals Ni, Fe against each other and depend only on the copper concentration. By numerically solving the Butler equation, the surface tension of the ternary system can be derived from the thermodynamic potentials ^E G of the binary phases (Ni–Cu, Fe–Cu, Ni–Fe) alone.Paper presented at the Seventeenth European Conference on Thermophysical Properties, September 5–8, 2005, Bratislava, Slovak Republic.     

Surface tension of aqueous lithium bromide with heat/mass transfer enhancement additives: the effect of additive vapor transport

Heat/mass transfer enhancement additives used in aqueous lithium bromide absorption chillers are surfactants that lower the surface tension of the working fluid. It has long been speculated that the surface tension characteristics are a key to the enhancement but the point is controversial because some surfactants do not provide enhancement. In the present study, the surface tension of aqueous lithium bromide was measured, with and without various surfactant additives, using a drop weight method. Measurements were also made on water, with and without an additive. The results provide new information that clarifies several confusing aspects of the literature data. The major result is the realization that the surface tension of aqueous lithium bromide is strongly affected by the presence of surfactant vapor around the liquid interface. This apparently explains the large differences in surface tension data found in the literature since no previous studies mentioned the importance of the vapor conditions.     

Thermophysical Properties of Ag and Ag–Cu Liquid Alloys at 1098K to 1573K

The surface tension and density of liquid Ag and Ag–Cu alloys were measured with the sessile drop method. The sessile drop tests were carried out at temperatures from 1098K to 1573 K, on cooling (temperature decreasing stepwise) under a protective atmosphere of high purity Ar (6N). The density of liquid Ag and Ag–Cu alloys decreases linearly with increasing temperature, and an increase in concentration of copper results in a lower density. The surface tension dependence on temperature can be described by linear equations, and the surface tension increases with increasing Cu content. The results of the measurements show good agreement with existing literature data and with thermodynamic calculations made using the Butler equation.     

Viscosity measurement using drop coalescence in microgravity

We present in here validation studies of a new method for application in microgravity environment which measures the viscosity of highly viscous undercooled liquids using drop coalescence. The method has the advantage of avoiding heterogeneous nucleation at container walls caused by crystallization of undercooled liquids during processing. Homogeneous nucleation can also be avoided due to the rapidity of the measurement using this method. The technique relies on measurements from experiments conducted in near zero gravity environment as well as highly accurate analytical formulation for the coalescence process. The viscosity of the liquid is determined by allowing the computed free surface shape relaxation time to be adjusted in response to the measured free surface velocity for two coalescing drops. Results are presented from two sets of validation experiments for the method which were conducted on board aircraft flying parabolic trajectories. In these tests the viscosity of a highly viscous liquid, namely glycerin, was determined at different temperatures using the drop coalescence method described in here. The experiments measured the free surface velocity of two glycerin drops coalescing under the action of surface tension alone in low gravity environment using high speed photography. The liquid viscosity was determined by adjusting the computed free surface velocity values to the measured experimental data. The results of these experiments were found to agree reasonably well with the known viscosity for the test liquid used.     

An improved method of sessile drop for determining the surface tension of liquids

An improved procedure is developed for the processing of images of meridional section of a liquid drop, obtained as a result of realization of the sessile drop method for determining the surface tension of liquid. The procedure provides for the scanning of digital image of droplet, for numerical solution of the Young-Laplace equation, and for the calculation of surface tension, wetting angle, and volume of the drop.     

Effect of dynamic surface tension on the growth rate of the Tonks-Frenkel instability

By means of a numerical analysis of the dispersion equation for capillary motions of a liquid with a charged flat free surface subjected to relaxation of the surface tension, it is shown that the growth rate of the instability of the free surface of the liquid decreases as the characteristic relaxation time of the surface tension increases and the Tonks-Frenkel parameter decreases. The instability itself occurs in a limited range of wave numbers whose width is also determined by the value of the Tonks-Frenkel parameter. Pis’ma Zh. Tekh. Fiz. 25, 61–65 (October 12, 1999)     

A finite element technique combined with gas-liquid two-phase flow calculation for unsteady free surface flow problems

A numerical method based on the finite element method is presented for the analysis of two-dimensional viscous incompressible flows with free surfaces. The problem is formulated as a gas-liquid two-phase flow problem by adding the gaseous region adjacent to liquid region in a solution domain. A free surface is regarded as the surface of density discontinuity between gas and liquid and is not a part of boundaries of a solution domain. The present method has removed the handling of moving boundaries from numerical computations, and it has enabled us to assemble a simple algorithm for stable computations. The motion of the surface of density discontinuity is calculated by solving an advection equation derived from the equation of continuity. For that calculation, a new upwind scheme has been devised by applying the method of characteristic lines. The ability of the present method has been tested by solving three numerical examples: the broken-dam problem, the dynamic behavior of liquid drop squirted from a nozzle, and the Kelvin-Helmholtz instability. Encouraging results have been obtained.