A Simple Weighted Integration Method for Calculating Surface Tension Force to Suppress Parasitic Flow in the Level Set Approach

Parasitic flows may occur in the numerical simulation of incompressible multiphase flow due to errors in the calculation of surface tension terms, specifically for the curvature and unit normal vector. An improved method for calculating the surface tension based on the level set approach is proposed, in which the contribution of not only the center node but also the rest area of a control volume to the calculation of surface tension is considered in a balanced manner. The weighted integration method (WIM) is more consistent with the concept of a banded interface in the level set method. It is applied to the temporal evolution of a two-dimensional neutrally buoyant liquid drop and a buoyancy driven deformable bubble in an immiscible fluid for the validation of WIM. The results show that the parasitic flows are evidently suppressed by the weighted integration method. The weight factors for WIM in 3-D cases are also suggested.

A Simple Weighted Integration Method for Calculating Surface Tension Force to Suppress Parasitic Flow in the Level Set Approach

Parasitic flows may occur in the numerical simulation of incompressible multiphase flow due to errors in the calculation of surface tension terms, specifically for the curvature and unit normal vector. An improved method for calculating the surface tension based on the level set approach is proposed, in which the contribution of not only the center node but also the rest area of a control volume to the calculation of surface tension is considered in a balanced manner. The weighted integration method (WIM) is more consistent with the concept of a banded in- terface in the level set method. It is applied to the temporal evolution of a two-dimensional neutrally buoyant liquid drop and a buoyancy driven deformable bubble in an immiscible fluid for the validation of WIM. The results show that the parasitic flows are evidently suppressed by the weighted integration method. The weight factors for WIM in 3-D cases are also suggested.

Surface tension of molten tin investigated with sessile drop method

The surface tension of molten tin was determined by a set of self-developed digital equipment with sessile drop method at oxygen partial pressure of 1.0×10 -6MPa under different temperatures, and the dependence of surface tension of molten tin on temperature was also discussed. The emphasis was placed on the comparison of surface tension of the same molten tin sample measured by using different equipments with sessile drop method. Results of the comparison indicate that the measurement results with sessile drop method under the approximate experimental conditions are coincident, and the self-developed digital equipment for surface tension measurement has higher stability and accuracy. The relationships of surface tension of molten tin and its temperature coefficient with temperature and oxygen partial pressure were also elucidated from the thermodynamic equilibrium analysis.

A new method of fast measuring surface tension of melt cast iron and its application in graphite shape identification

Surface tension is one of important physical features of melt alloy. Many properties of melt alloy, such as graphite shape of cast iron and modified microstructure of aluminum alloy, can be evaluated by means of surface tension. In order to evaluate and control the melt quality in-situ melting operation, the authors advanced a new method and developed an automatic device for fast measuring surface tension of melt alloy and applied it to the practice of rapid identifying graphite shape of cast iron. In this paper, the principle of fast measuring surface tension, the construction of the automatic measurement device and the examples of evaluating graphite shape of cast iron based on the new method and device are discussed.

CALCULATION OF SURFACE TENSION OF LIQUIDS BY MONTE CARLO METHOD

In this paper a method has been proposed for the calculation of cohesive work and thus sur-face tension of liquids by the Monte Carlo computer simulation method with the use of Kihara po-tential. The surface tension values calculated for liquid argon, oxygen and nitrogen are in fairagreement with the experimental data.

Modeling bubble column reactor with the volume of fluid approach:Comparison of surface tension models

This work aims at comparing surface tension models in VOF(Volume of Fluid) modeling and investigating the effects of gas distributor and gas velocity. Hydrodynamics of a continuous chain of bubbles inside a bubble column reactor was simulated. The grid independence study was first conducted and a grid size of 1.0 mm was adopted in order to minimize the computing time without compromising the accuracy of the results. The predictions were validated by comparing the experimental studies reported in the literature. It was found that all surface tension models can describe the bubble rise and bubble plume in a column with slight deviations.

Stream Surface Strip Element Method and Simulation of Three-Dimensional Deformation of Continuous Hot Rolled Strip

A new method,the stream surface strip element method,for simulating the three-dimensional deformation of plate and strip rolling process was proposed.The rolling deformation zone was divided into a number of stream surface(curved surface)strip elements along metal flow traces,and the stream surface strip elements were mapped into the corresponding plane strip elements for analysis and computation.The longitudinal distributions of the lateral displacement and the altitudinal displacement of metal were respectively constructed to be a quartic curve and a quadratic curve,of which the lateral distributions were expressed as the third-power spline function,and the altitudinal distributions were fitted in the quadratic curve.From the flow theory of plastic mechanics,the mathematical models of the three-dimensional deformations and stresses of the deformation zone were constructed.Compared with the streamline strip element method proposed by the first author of this paper,the stream surface strip element method takes into account the uneven distributions of stresses and deformations along altitudinal direction,and realizes the precise three-dimensional analysis and computation.The simulation example of continuous hot rolled strip indicates that the method and the model accord with facts and provide a new reliable engineering-computation method for the three-dimensional mechanics simulation of plate and strip rolling process.

Surface tension of liquid Au-Bi-Sn alloys

The surface tension of a promising lead-free solder Au-Bi-Sn alloys was investigated both by the sessile-drop method and calculation. Experimental measurements were carried out for two cross-sections with the constant gold to bismuth ration of 1:1 and 1:2. For all the investigated compositions, decrease of the surface tension is observed with increasing temperature. Meanwhile, the surface tension values were also calculated based on Butler’s equation, with using the newest research on thermodynamics data of Au-Bi-Sn ternary system. Compared with the experimental results, a good agreement was obtained.

Surface Tension of Molten Ni and Ni-Co Alloys

Surface tension of molten Ni and Ni-Co (5 and 10 mass fraction) alloys was measured at the temperature range of 1773-1873 K using an improved sessile drop method with an alumina substrate in an Ar+3%H2 atmosphere. The error of the data obtained was analyzed. The surface tension of molten Ni and Ni-Co (5 and 10 mass fraction) alloys decreases with increasing temperature. The influence of Co on the surface tension of Ni-Co alloys is little in the studied Co concentration range.

Lattice Boltzmann simulation of liquid vapor system by incorporating a surface tension term

In this study, we investigate the pseudopotential multiphase model of lattice Boltzmann method（LBM） and incorporate a surface tension term to implement the particle interaction force. By using the Carnahan–Starling（CS） equation of state（EOS） with a proper critical pressure–density ratio, a density ratio over 160000 is obtained with satisfactory numerical stability. The added surface tension term offers a flexible choice to adjust the surface tension strength. Numerical tests of the Laplace rule are conducted, proving that smaller spurious velocity and better numerical stability can be acquired as the surface tension becomes stronger. Moreover, by wall adhesion and heterogeneous cavitation tests, the surface tension term shows its practical application in dealing with problems in which the surface tension plays an important role.

Diffusion-controlled Adsorption Kinetics at Air/Solution Surface Studied by Maximum Bubble Pressure Method

In studying the diffusion-controlled adsorption kinetics of aqueous surfactant solutions at the air/solution surface by means of the maximal bubble pressure method, Fick's diffusion equation for a sphere should be used. In this paper the equation was solved by means of Laplace transformation under different initial and boundary conditions. The dynamic surface adsorption F(t) for a surfactant solution, which was used to describe the diffusion-controlled adsorption kinetics at the solution surface, was derived. Different from the planar surface adsorption, the dynamic surface adsorption F(t) for the short time consists of two terms: one is the same as Ward-Tordai equation and the other reflects the geometric effect caused by the spherical bubble surface. This effect should not be neglected for the very small radius of the capillary. The equilibrium surface tension γeq and the dynamic surface tension γ(t) of aqueous C10E6 [CH3(CH2)9(OCH2CH2)6OH] solution at temperature 25℃ were measuredby means of Wilhelmy plate method and maximal bubble pressure method respectively. As t→ 0, the theoreticalanalysis is in good agreement with experimental results and the dependence of γ(t) on is linear.

Surface tension of molten Ni-(Cr,Co,W) alloys and segregation of elements

Surface tension of molten Ni-(Cr,Co,W) alloys was measured at the temperature of 1 773-1 873 K in an Ar+3%H2 atmosphere using an improved sessile drop method.The segregation of Cr,Co and W in alloy was calculated and analyzed using Butler's equation.The results show a good agreement between measured and calculated data.The surface tension of molten Ni-(Cr,Co,W) alloys decreases with increasing temperature.In Ni-(Cr,Co,W) alloys,the element with lower surface tension tends to segregate on the surface of molten alloy while that with higher surface tension tends to segregate inside of the molten alloy.The larger the differences in surface tension,atom radius and electron configuration between solvent and solute are,the more significant the segregation is.As a result,Ni segregates onto the surface and Co and W segregate inside the alloys.

Third-Order Stokes Wave Solutions of the Free Surface Capillary- Gravity Wave and the Interfacial Internal Wave

Based on the Stokes wave theory, the capillary-gravity wave and the interfacial internal wave in two-layer constant depth＇s fluid system are investigated. The fluids are assumed to be incompressible, inviscid and irrotational. The third-order Stokes wave solutions are given by using a perturbation method. The results indicate that the third-order solutions depend on the surface tension, the density and the depth of each layer. As expected, the first-order solutions are the linear theoretical results （the small amplitude wave theoretical results）. The second-order and the third-order solutions describe the nonlinear modification and the nonlinear interactions. The nonlinear impact appears not only in the n （n〉~2） times＇ high frequency components, but also in the low frequency components. It is also noted that the wave velocity depends on the wave number, depth, wave amplitude and surface tension.

Surface tension effects on the behaviour of a rising bubble driven by buoyancy force

In the inviscid and incompressible fluid flow regime,surface tension effects on the behaviour of an initially spherical buoyancy-driven bubble rising in an infinite and initially stationary liquid are investigated numerically by a volume of fluid （VOF） method. The ratio of the gas density to the liquid density is 0.001, which is close to the case of an air bubble rising in water. It is found by numerical experiment that there exist four critical Weber numbers We1,~We2,~We3 and We4, which distinguish five different kinds of bubble behaviours. It is also found that when 1≤We2, the bubble will finally reach a steady shape, and in this case after it rises acceleratedly for a moment, it will rise with an almost constant speed, and the lower the Weber number is, the higher the speed is. When We 〉We2, the bubble will not reach a steady shape, and in this case it will not rise with a constant speed. The mechanism of the above phenomena has been analysed theoretically and numerically.

On study of horizontal thin film flow of Sisko fluid due to surface tension gradient

The present paper is concerned with the steady thin film flow of the Sisko fluid on a horizontal moving plate, where the surface tension gradient is a driving mechanism. The analytic solution for the resulting nonlinear ordinary differential equation is obtained by the Adomian decomposition method （ADM）. The physical quantities are derived including the pressure profile, the velocity profile, the maximum residue time, the stationary points, the volume flow rate, the average film velocity, the uniform film thickness, the shear stress, the surface tension profile~ and the vorticity vector. It is found that the velocity of the Sisko fluid film decreases when the fluid behavior index and the Sisko fluid parameter increase, whereas it increases with an increase in the inverse capillary number. An increase in the inverse capillary number results in an increase in the surface tension which in turn results in an increase in the surface tension gradient on the Sisko fluid film. The locations of the stationary points are shifted towards the moving plate with the increase in the inverse capillary number, and vice versa locations for the stationary points are found with the increasing Sisko fluid parameter. Furthermore, shear thinning and shear thickening characteristics of the Sisko fluid are discussed. A comparison is made between the Sisko fluid film and the Newtonian fluid film.

The Determination of Surface Tension of Traditional Chinese Medicine Tablet

In this paper we aimed to determine the surface tension of traditional Chinese medicine (TCM) tablet and tried to provide a key reference parameter to control tablet film coating quality. Using contact angle tester, the Chinese medicine tablet, a famous Fufang Danshen tablet CP as a model example, was determined by Zisman critical surface tension method. Our study showed that the critical surface tension of Fufang Danshen tablet CP was determined to be 18.00 dyne/cm. The Zisman method, hence, can determine successfully the critical surface tension of Chinese medicine tablet and will become an effective manner to control tablet film coating quality.

Morphology observation of dipalmitoyl phosphatidyl choline (DPPC) monolayer on water surface by dropping method

Preparation of DPPC lipid monolayer in water trough has been done by dropping method and compared with compression method. Monolayer was studied by surface pressure isotherm, fluorescence microscopy, Brewster angle microscopy, and infrared external reflection spectroscopy. Results of these measurements showed that dropping method gave better results compared to compression method. In dropping method, transition from liquid expanded state to liquid condensed is gradual compared to sharp one in compressed method. During monolayer formation, adjustment and interaction between hydrophilic part of lipid and water and among hydrophobic part of lipid molecule are slow, stable, and more natural as worked out from surface area versus pressure isotherm. At a given molecular area, surface pressure is less compared to compression method thus monolayer is in more fluidic state in dropping method than compression method. The observation was supported by all techniques described above.

Effect of Initial Geometry on Surface Flatness after Arc Welding Analyzed with MPS Method

The flow behavior in a typical arc welding process is very complex and features rapid melting and solidification. In this work, the flow in the melted region was determined by the effect of surface tension, viscosity, and gravity. Two-dimensional change of the geometry of the melted region was analyzed using the MPS （moving particle semi-implicit） method to melt a small upper plate in a hole at the center of a base plate. The relation between the initial geometry of the upper plate and the surface flatness after welding was calculated. The calculation results showed that the upper plate with a triangular notch at its center minimized the surface unevenness after welding. The depth of notch and thickness of the upper plate were optimized.

The Theoretical Study of O_2 Adsorption on NiTi (100) and (110) Surfaces

The discrete variational Xα method (DV-Xα) within the framework of density-functional theory was applied to study O 2 molecule adsorption on NiTi (100) and (110) surfaces.The bond order and charge distribution between Ti and O atoms for two possible O 2 molecule adsorption ways on NiTi(100) and (110) surfaces were calculated.It is found that the adsorption way for O-O bond perpendicular to NiTi surface is preferred to that for O-O bond parallel to NiTi surface,and O 2 molecule only interacted with one nearest surface titanium atom during the adsorption process.Mulliken population and the partial density of state analysis show that the interaction between Ti and O atoms is mainly donated by O 2p and Ti 4s electrons on NiTi(110) surface,O 2p and Ti 4s,4p electrons on NiTi(100) surface,respectively.The total density of state analysis shows that NiTi(100) surface is more favorable for O 2 molecule adsorption.

Three-dimensional multi-relaxation-time lattice Boltzmann front-tracking method for two-phase flow

We developed a three-dimensional multi-relaxation-time lattice Boltzmann method for incompressible and immiscible two-phase flow by coupling with a front-tracking technique. The flow field was simulated by using an Eulerian grid, an adaptive unstructured triangular Lagrangian grid was applied to track explicitly the motion of the two-fluid interface, and an indicator function was introduced to update accurately the fluid properties. The surface tension was computed directly on a triangular Lagrangian grid, and then the surface tension was distributed to the background Eulerian grid. Three benchmarks of two-phase flow, including the Laplace law for a stationary drop, the oscillation of a three-dimensional ellipsoidal drop, and the drop deformation in a shear flow, were simulated to validate the present model.