The formation of a liquid bridge during the coalescence of drops

This paper examines a mathematical model for the coalescence of two viscous liquid volumes in an inviscid gas or in a vacuum which removes the pressure singularity at the instant of impact inherent in the classical formulation of the continuum model. The very early stages of coalescence are examined in order to study the formation of the liquid bridge in two cases: (i) for two infinitely long, coalescing liquid cylinders; and (ii) for two coalescing spheres. Numerical solutions are computed for the velocity and pressure fields in the flow in both cases, and they confirm the removal of the pressure singularity. Also, the free-surface position at small times is determined.     

On elastocapillarity: A review

Elastocapillary phenomena involving elastic deformation of solid structures coupled with capillary effects of liquid droplets/films can be observed in a diversity of fields,e.g.,biology and microelectromechanical systems(MEMS).Understanding the physical mechanisms underlying these phenomena is of great interest for the design of new materials and devices by utilizing the effects of surface tension at micro and nano scales.In this paper,some recent developments in the investigations on elastocapillary phenomena are briefly reviewed.Especially,we consider the deformation,adhesion,self-assembly,buckling and wrinkling of materials and devices induced by surface tensions or capillary forces.The main attention is paid to the experimental results of these phenomena and the theoretical analysis methods based on continuum mechanics.Additionally,the applications of these studies in the fields of MEMS,micro/nanometrology,and biomimetic design of advanced materials and devices are discussed.     

Interfacial tension reduction in PBT/PE/clay nanocomposite

We investigated the effect of organically modified nanoclay (organoclay) on immiscible polymer blends [polybutylene terephthalate (PBT)/polyethylene (PE)] with a special focus on the role of clay as a compatibilizer. When organoclay (Nanofil 919; Sud-Chemie, Inc.) is added to the blend, the clay first locates at the interface and then selectively locates in the PBT phase due to its affinity with PBT. This results in effective size reduction and narrowed size distribution of the dispersed phase. However, with a small amount of organoclay, it is observed that the clay locates at the interface regardless of its affinity for a specific component to minimize the chemical potential. The interfacial tension change of the blend with the addition of organoclay was quantitatively predicted from extensional force measurement. When the blend is subjected to an extension, the interfacial tension functions as a resistance against drop deformation. When we added organoclay to the blend, the extensional force was significantly reduced, which means that the contribution of the interfacial tension to the total force is reduced. For a 10/90 PBT/PE blend, the interfacial tension was reduced from 5.76 to 0.14 cN m⁻¹ when 1 wt% of organoclay was added. This interfacial tension reduction arises from the localization of the organoclay at the interface and its nonhomogeneous distribution along the interface, suppressing the coalescence between the droplets, which is a role of a compatibilizer. Conclusively, the immiscible polymer blends can be compatibilized with organoclay. The organoclay changes the blend morphology by interfacial tension reduction due to the localization of the organoclay at the interface and by the viscosity ratio change due to the selective localization by its affinity to a specific component in the blend.     

An asymptotic iteration method for the numerical analysis of near-critical free-surface flows

Satisfying the boundary conditions at the free surface may impose severe difficulties to the computation of turbulent open-channel flows with finite-volume or finite-element methods, in particular, when the flow conditions are nearly critical. It is proposed to apply an iteration procedure that is based on an asymptotic expansion for large Reynolds numbers and Froude numbers close to the critical value 1.The iteration procedure starts by prescribing a first approximation for the free surface as it is obtained from solving an ODE that has been derived previously by means of an asymptotic expansion (Grillhofer and Schneider, 2003). The numerical solution of the full equations of motion then gives a surface pressure distribution that differs from the constant value required by the dynamic boundary condition. To determine a correction to the elevation of the free surface we next solve an ODE that is obtained from the asymptotic analysis of the flow with a prescribed pressure disturbance at the free surface. The full equations of motion are then solved for the corrected surface, and the procedure is repeated until criteria of accuracy for surface elevation and surface pressure, respectively, are satisfied.The method is applied to an undular hydraulic jump as a test case.     

A boundary integrodifferential equation arising from the linearized theory of water waves

The prototype problem of the linearized irrotational gravity-capillarity water waves due to a Havelock doublet in a uniform planar stream is solved via a layer ansatz and a boundary integral formulation. As an effect of surface tension, two distinct flow regimes arise, separated by a critical speed where a resonance occurs. The transcirtical flow is described by including the viscous vorticity diffusion at the free boundary.

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Sommario Si risolve, nell' ambito di una formulazione integrale al contorno, il problema linearizzato delle onde di gravità irrotazionali generate da un dipolo di Havelock in presenza di tensione superficiale. Si hanno due distinti regimi di flusso separati da una velocità critica. Il regime transcritico, in cui si ha un fenomeno di risonanza, viene descritto tenendo conto della presenza di uno strato limite viscoso sulla frontiera libera.

     

Boundary-conforming mapping applied to computations of highly deformed solidification interfaces

A new boundary-conforming mapping is developed for the calculation of highly deformed cellular solidification interfaces in a model of directional solidification of a binary alloy. The mapping is derived through a variational fomulation that is designed so that the grid penetrates the grooves between cells along the interface without causing a loss of ellipticity of the mapping equations. A finite element/Newton method is presented for simultaneous solution of the free boundary problem described by the solutal model of directional solidification and the mapping equations. Results are compared to previous calculations and demonstrate the importance of accurate representation of the interface shape for understanding the solution structure.     

Micro capillary pumped loop system for a cooling high power device

This work discusses the operation of a capillary-driven two-phase loop, configured on a micro capillary pumped loop (MCPL) system without an external power supply but capable of automatic heat transmission. The MCPL device, fabricated using MEMS (microelectricomechanical system) technology, was tested and yielded the following results: first, the proposed design of a new MCPL system with a water reservoir operating at low pressures is feasible and requires no additional power supply and instead relies on automatic heat transmission. Second, the issue of depriming in a MCPL was effectively controlled, the endurance of MCPL for the depriming problem can be executed by yielding input heat fluxes of 185.2 W/cm² at an evaporator temperature of 165 °C, thus revealing that this model provides excellent cooling performance. Third, the effective operation range was determined and its successful operation was confirmed for MCPL. The ease of starting up increased with the temperature of the reservoir. Finally, two-phase tension that originated in the groove structures in the evaporator and condenser was confirmed to control the movement of the fluids throughout the system and verified to be effective in improving cooling efficiency.     

Multiple interacting circular nano-inhomogeneities with surface/interface effects

A two-dimensional problem of multiple interacting circular nano-inhomogeneities or/and nano-pores is considered. The analysis is based on the Gurtin and Murdoch model [Gurtin, M.E., Murdoch, A.I., 1975. A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323.] in which the interfaces between the nano-inhomogeneities and the matrix are regarded as material surfaces that possess their own mechanical properties and surface tension. The precise component forms of Gurtin and Murdoch's three-dimensional equations are derived for interfaces of arbitrary shape to provide a basis for critical review of various modifications used in the literature. The two-dimensional specification of these equations is considered and their representation in terms of complex variables is provided. A semi-analytical method is proposed to solve the problem. Solutions to several example problems are presented to: (i) examine the difference between the results obtained with the original and modified Gurtin and Murdoch's equations, (ii) compare the results obtained using Gurtin and Murdoch's model and those for a problem of nano-inhomogeneities with thin membrane-type interphase layers, and (iii) demonstrate the effectiveness of the approach in solving problems with multiple nano-inhomogeneities.     

Nonlinear wave propagation and reflection — Comparing the numerics with the analytics

The accuracy of numerical methods needs always a special attention. In this paper, analytical and numerical methods have been compared to describe the initial stage of nonlinear propagation and reflection of longitudinal ultrasonic waves. The perturbation method has been used to derive the analytical solution and the finite difference scheme to find the numerical solution for multiple free-boundary reflections of a harmonic burst at ultrasonic frequencies. The comparison of results at relatively small nonlinearities reveals a good qualitative and quantitative agreement between the analytical and numerical solutions. The method for determining analytically the exact region of interaction for counter-propagating waves is outlined in detail. At higher frequencies and larger nonlinear effects some quantitative differences between analytical and numerical results appear. The results are applicable in modelling nonlinear wave motion, including NDT and nonlinear one-dimensional vibrations.     

Influence of film thickness and cross-sectional geometry on hydrophilic microchannel condensation

Condensation in hydrophilic microchannel is strongly influenced by the channel cross-sectional geometry and the condensing surfaces hydrophobicity, which govern the evolution of the liquid film. This work makes progress on studying the relationship between channel geometry and condensation through flow regime visualizations, film-thickness measurements with optical interferometery, and temperature profile measurements with heat flux distribution construction. The hydrophilic microchannels have aspect ratios ranging from 1 to 5 and hydraulic diameters from 100 μm through 300 μm. The experimental measurement qualitatively matches the prediction of previous theoretical models accounting for the surface tension effect, which highlights the importance of surface tension force and channel geometry in the microchannel condensation. Pressure drop and mean heat flux measurements show that a larger channel is favorable for minimizing the pressure drop, while a smaller channel size and higher aspect ratio are desirable for maximizing the mean heat flux. The optimization of the channel geometry for a given application lies in the trade-off between these two factors.     

A new strategy for the numerical modeling of a weld pool

Welding processes involve high temperatures and metallurgical and mechanical consequences that must be controlled. For this purpose, numerical simulations have been developed to study the effects of the process on the final structure. During the welding process, the material undergoes thermal cycles that can generate different physical phenomena, like phase changes, microstructure changes and residual stresses and distortions. But the accurate simulation of transient temperature distributions in the part needs to carefully take account of the fluid flow in the weld pool. The aim of this paper is thus to propose a new approach for such a simulation taking account of surface tension effects (including both the “curvature effect” and the “Marangoni effect”), buoyancy forces and free surface motion.The proposed approach is validated by two numerical tests from the literature: a sloshing test and a plate subjected to a static heat source. Then, the effects of the fluid flow on temperature distributions are discussed in a hybrid laser/arc welding example.     

An investigation of falling liquid films on a vertical heated/cooled plate

The temperature field and flow patterns of a liquid film flowing over a vertical uniformly heated surface have been experimentally investigated. Our experiments show that this film flow is sensitive to the heating conditions. When the film is cooled by the substrate, its surface area increases, and when it is heated its surface area decreases. The analysis attributed the changing properties of the flow to lateral Marangoni effect, i.e. to surface tension gradient transverse to the flow. The influence of the viscosity variations on the non-isothermal liquid film flow was also considered and compared with that of the surface tension variations. It was shown that the contraction or extension of the films was mainly caused by the lateral surface tension gradient that might be determined by the viscosity variations.     

Experimental and numerical study of Marangoni convection in shallow liquid layers

An experimental and numerical study of thermal Marangoni convection in shallow liquid layers was carried out for a range of temperature differences and layer depths. This was done to permit earth based experiments to be undertaken in situations where Marangoni convection dominated the flow. Particle image velocimetry (PIV) was used to obtain the flow patterns and velocity vectors. The experimental results were compared to numerical models created using FLUENT V6. Both results are in good agreement. The liquid free surface profile due to the presence of the menisci is shown to be critical for good quantitative validation. The layer depths are also proven to be shallow enough for Marangoni convection to dominate over buoyancy under normal gravity conditions.     

On boundary potential energies in deformational and configurational mechanics

This contribution deals with the implications of boundary potential energies, i.e. in short surface, curve and point potentials, on deformational and configurational mechanics. Within the realm of deformational mechanics the surface/curve potentials are allowed in the most general case to depend on the deformation, the surface/curve deformation gradient and the spatial surface normal/curve tangent and are parametrised in the material placement and the material surface normal/curve tangent. The point potentials depend on the deformation and are parametrised in the material placement. From the configurational mechanics perspective the roles of fields and parametrisations are reversed. By considering variational arguments based on the kinematics of deforming surfaces/curves, in particular the relevant surface/curve stresses and distributed forces contributing to (localized) deformational and configurational force balances at surfaces/curves/points, which extend the common traction boundary conditions, are derived. Thereby, dissipative distributed configurational forces that are energetically conjugate to configurational changes are introduced as definitions. The (localized) force balances at surfaces/curves/points together with the contributing stresses and distributed forces within deformational and configurational mechanics display an intriguing duality. The resulting dissipative configurational tractions at the boundary are exemplified for some illustrative cases of boundary potentials.     

New solutions for surface tension driven spreading of a thin film

The standard fourth-order non-linear PDE modelling the flow of thin fluid film subject to surface tension is studied. The Lie group method is used to reduce the model equation from a fourth-order PDE to a fourth-order ODE. Analytical solutions are obtained for certain cases. Where analytical progress cannot be made, we determine numerical solutions.     

Numerical study on magneto-convection of cold water in an open cavity with variable fluid properties

The aim of the present study is to understand the problem of buoyancy and thermocapillary induced convection of cold water near its density maximum in an open cavity with temperature dependent properties in the presence of uniform external magnetic field. The governing equations are solved by the finite volume method. The results are discussed for various values of reference temperature parameter, density inversion parameter, Rayleigh, Hartmann and Marangoni numbers. It is observed that the temperature of maximum density leaves strong effects on fluid flow and heat transfer due to the formation of bi-cellular structure. Convection heat transfer is enhanced by thermocapillary force when buoyancy force is weakened.     

Accounting for local capillary effects in two-phase flows with relaxed surface tension formulation in enriched finite elements

This paper introduces a numerical method able to deal with a general bi-fluid model integrating capillary actions. The method relies first on the precise computation of the surface tension force. Considering a mathematical transformation of the surface tension virtual work, the regularity required for the solution on the evolving curved interface is weakened, and the mechanical equilibrium of the triple line can be enforced as a natural condition. Consequently, contact angles of the liquid over the solid phase result naturally from this equilibrium. Second, for an exhaustive representation of capillary actions, pressure jumps across the interface must be accounted for. A pressure enrichment strategy is used to properly compute the discontinuities in both pressure and gradient fields. The resulting method is shown to predict nicely static contact angles for some test cases, and is evaluated on complex 3D cases.