Young-Laplace Law Research Articles

A simple mechanical apparatus for measuring the surface tension of soap bubbles

We report on the construction of a simple apparatus for measuring the small overpressure inside soap bubbles with sizes in the cm range. The apparatus uses Archimedes' principle and it can be easily calibrated. Thus, surface tensions of soap films can be measured quantitatively. The apparatus is also well suited to demonstrate the Young–Laplace law in front of an audience such as a classroom. The design of the apparatus is subject to a criterion for stability, which we discuss in detail and that can be discussed with the audience. The setup of the apparatus requires components that are commonly available in households or school labs and it can be set up by students on their own.

Modeling and simulation of the metal transfer on GMAW-S process

The GMAW-S process occurs from sequences of arcing and short-circuit welding periods. For modeling purposes, many researchers believe that the metal transfer occurs only in the period of short circuit (consideration to be taken into account in the present work), in which the link between the electrode and the welding pool is generated through a liquid metal bridge. From this exact moment, it can be observed changes in the metal bridge geometry due to electromagnetic (pinch effect), gravitational, and surface tension forces. This paper is aimed to model the dynamic geometry of metal bridge during the short-circuit period, considering the fusion and feed rate electrode wire balance. This model was represented by state equations building from applying Kirchoff’s, Lorentz, Young–Laplace law, Bernoulli’s principle, the equation of fluid continuity, and concepts of analytic geometry. The model implementation was developed in Simulink and could be used to study the metal transfer control, as well as, the appropriate welding parameters selection, aiming to achieve a higher level of welding quality.

Influence of different confining matrices on negative pressure in liquid n-heptane investigated using positronium bubbles as a probe

HypothesisThe negative pressure in liquids under the concave meniscus of nanometer size can be observed experimentally. This allows verification of the predictions of the macroscopic Young-Laplace law, which has so far been performed only on the basis of theoretical calculations. The deviation of the negative pressure from the Young-Laplace law allows to get information about the structure of the porous matrix. ExperimentsThe properties of n-heptane confined in nanometer-sized spaces are monitored in situ due to the particular ability of positronium to form subnanometer bubbles in liquids. Positron annihilation lifetime spectroscopy allows to detect their sizes, which in turn are used to determine surface tension and negative pressure of the investigated liquid. These results are obtained by means of an improved quantum-mechanical model approximating the o-Ps bubble with a 1.3 eV finite potential well. FindingsThe dependence of the negative pressure on the curvature of the concave meniscus is found, which follows the Young-Laplace law with a good accuracy for model cylindrical pores. The differences between the pressure in the liquid confined in pores of various sizes and shapes in different materials allow inference about several phenomena (e.g. pore blocking in materials with a complex pore structure or swelling of the polymer).

Volume scavenging of networked droplets

A system of N spherical-cap fluid droplets protruding from circular openings on a plane is connected through channels. This system is governed by surface tension acting on the droplets and viscous stresses inside the fluid channels. The fluid rheology is given by the Ostwald–de Waele power law, thus permitting shear thinning. The pressure acting on each droplet is caused by capillarity and given in terms of the droplet volume via the Young–Laplace law. Liquid is exchanged along the network of fluid conduits due to an imbalance of the Laplace pressures between the droplets. In this way some droplets gain volume at the expense of others. This mechanism, christened “volume scavenging,” leads to interesting dynamics.Numerical experiments show that an initial droplet configuration is driven to a stable equilibrium exhibiting 1 super-hemispherical droplet and N−1 sub-hemispherical ones when the initial droplet volumes are large. The selection of this “winning” droplet depends not only on the channel network and the fluid volume, but also notably on the fluid rheology. The rheology is also observed to drastically change the transition to equilibrium. For smaller droplet volumes the long-term behavior is seen to be more complicated since the types of equilibria differ from those arising for larger volumes. These observations motivate our analytical study of equilibria and their stability for the corresponding nonlinear dynamical system. The identification of equilibria is accomplished by locating the zeros of a mass polynomial, defined through the constant volume/mass constraint. The key tool in our stability analysis is a pressure–volume work functional, related to the total surface area, which serves as a Lyapunov function for the dynamical system. This functional is useful since equilibria are typically not hyperbolic and linearization techniques not available. Equilibria will be shown to be hierarchically organized in terms of size of the pressure–volume work functional. For larger droplet volumes this ordering exhibits one hierarchy of equilibria. Two hierarchies exist when the volumes are smaller. The minimizing equilibria in either case are asymptotically stable.

Stresses in curved nematic membranes.

Ordering configurations of a director field on a curved membrane induces stress. In this work, we present a theoretical framework to calculate the stress tensor and the torque as a consequence of the nematic ordering; we use the variational principle and invariance of the energy under Euclidean motions. Euler-Lagrange equations of the membrane as well as the corresponding boundary conditions also appear as natural results. The stress tensor found includes attraction-repulsion forces between defects; likewise, defects are attracted to patches with the same sign in Gaussian curvature. These forces are mediated by the Green function of the Laplace-Beltrami operator of the surface. In addition, we find nonisotropic forces that involve derivatives of the Green function and the Gaussian curvature, even in the normal direction to the membrane. We examine the case of axial membranes to analyze the spherical one. For spherical vesicles we find the modified Young-Laplace law as a consequence of the nematic texture. In the case of spherical cap with defect at the north pole, we find that the force is repulsive with respect to the north pole, indicating that it is an unstable equilibrium point.

A new splitting scheme to the discrete Boltzmann equation for non-ideal gases on non-uniform meshes

We present a novel numerical procedure for solving the discrete Boltzmann equations (DBE) on non-uniform meshes. Our scheme is based on the Strang splitting method where we seek to investigate two-phase flow applications. In this note, we investigate the onset of parasitic currents which arise in many computational two-phase algorithms. To the best of our knowledge, the results presented in this work show, for the first time, a spectral element discontinuous Galerkin (SEDG) discretization of a discrete Boltzmann equation which successfully eliminates parasitic currents on non-uniform meshes. With the hope that this technique can be used for applications in complex geometries, calculations are performed on non-uniform mesh distributions by using high-order (spectral), body-fitting quadrilateral elements. Validation and verification of our work is carried out by comparing results against the classical 2D Young–Laplace law problem for a static drop.

X-ray CT analysis of pore structure in sand

Abstract. The development of microfocused X-ray computed tomography (CT) devices enables digital imaging analysis at the pore scale. The applications of these devices are diverse in soil mechanics, geotechnical and geoenvironmental engineering, petroleum engineering, and agricultural engineering. In particular, the imaging of the pore space in porous media has contributed to numerical simulations for single-phase and multiphase flows or contaminant transport through the pore structure as three-dimensional image data. These obtained results are affected by the pore diameter; therefore, it is necessary to verify the image preprocessing for the image analysis and to validate the pore diameters obtained from the CT image data. Moreover, it is meaningful to produce the physical parameters in a representative element volume (REV) and significant to define the dimension of the REV. This paper describes the underlying method of image processing and analysis and discusses the physical properties of Toyoura sand for the verification of the image analysis based on the definition of the REV. On the basis of the obtained verification results, a pore-diameter analysis can be conducted and validated by a comparison with the experimental work and image analysis. The pore diameter is deduced from Young–Laplace's law and a water retention test for the drainage process. The results from previous study and perforated-pore diameter originally proposed in this study, called the voxel-percolation method (VPM), are compared in this paper. In addition, the limitations of the REV, the definition of the pore diameter, and the effectiveness of the VPM for an assessment of the pore diameter are discussed.

Axisymmetric multiphase lattice Boltzmann method for generic equations of state

We present an axisymmetric lattice Boltzmann model based on the Kupershtokh et al. multiphase model that is capable of solving liquid–gas density ratios up to 103. Appropriate source terms are added to the lattice Boltzmann evolution equation to fully recover the axisymmetric multiphase conservation equations. We validate the model by showing that a stationary droplet obeys the Young–Laplace law, comparing the second oscillation mode of a droplet with respect to an analytical solution and showing correct mass conservation of a propagating density wave.

Effects of surface tension on the adhesive contact between a hard sphere and a soft substrate

In the present paper, the adhesive contact between a rigid sphere and a compliant elastic substrate with surface tension is analyzed. Using the minimum potential energy principle and considering both the effects of surface tension and adhesion, explicit solutions are obtained for the contact radius and the indent depth for the case without external loading. It is found that surface tension evidently alters the pressure distribution in the contact region and tends to decrease both the contact radius and the indent depth. The present model establishes a bridge between the JKR model for adhesive contact problems and the Young-Laplace's law of surface tension.

A path-conservative finite volume scheme for compressible multi-phase flows with surface tension

The accurate simulation of compressible multi-phase flows with surface tension effects is currently still one of the most challenging problems in computational fluid dynamics (CFD). The basic difficulties are the capturing of the correct interface dynamics between the two fluids as well as the computation of the interface curvature. In this paper, we present a novel path-conservative finite volume discretization of the continuum surface force method (CSF) of Brackbill et al. to account for the surface tension effect due to curvature of the phase interface. This is achieved in the context of a diffuse interface approach, based on the seven equation Baer–Nunziato model of compressible multi-phase flows. Such diffuse interface methods for compressible multi-phase flows including capillary effects have first been proposed by Perigaud and Saurel. In the CSF method, the surface tension effect is replaced by a volume force, which is usually integrated as a classical volume source term. However, since this source term contains the gradient of a color function that is convected with the flow velocity, we propose to integrate the CSF source term as a non-conservative product and not simply as a source term, following the ideas on path-conservative finite volume schemes put forward by Castro and Parés.For that purpose, we use the new generalized Osher-type Riemann solver (DOT), recently proposed by Dumbser and Toro and compare it with a path-conservative Roe and Rusanov scheme. Via numerical evidence we can show that if the curvature computation is exact, our new scheme is well-balanced for a steady circular bubble in equilibrium according to the Young–Laplace law. This means that the pressure jump term across the interface in the momentum equation is exactly balanced with the surface tension force.We apply our scheme to several one- and two-dimensional test problems, including 1D Riemann problems, an oscillating droplet, as well as a deforming droplet initially attached to a wall and which is subject to gravity and surface tension forces. Finally, we also check the rising of a gas bubble due to buoyancy forces. We compare our numerical results with those published previously in the literature.

Entropic lattice Boltzmann method for multiphase flows: Fluid-solid interfaces.

The recently introduced entropic lattice Boltzmann model (ELBM) for multiphase flows [A. Mazloomi M., S. S. Chikatamarla, and I. V. Karlin, Phys. Rev. Lett. 114, 174502 (2015)] is extended to the simulation of dynamic fluid-solid interface problems. The thermodynamically consistent, nonlinearly stable ELBM together with a polynomial representation of the equation of state enables us to investigate the dynamics of the contact line in a wide range of applications, from capillary filling to liquid drop impact onto a flat surfaces with different wettability. The static interface behavior is tested by means of the liquid column in a channel to verify the Young-Laplace law. The numerical results of a capillary filling problem in a channel with wettability gradient show an excellent match with the existing analytical solution. Simulations of drop impact onto both wettable and nonwettable surfaces show that the ELBM reproduces the experimentally observed drop behavior in a quantitative manner. Results reported herein demonstrate that the present model is a promising alternative for studying the vapor-liquid-solid interface dynamics.

Laplace pressure evolution and four instabilities in evaporating two-grain liquid bridges

Dynamic variables characterizing evolution during evaporation of capillary bridge between two spheres are analyzed. The variables include: average Laplace pressure, pressure resulting force, surface tension force and total capillary force calculated based on the previously reported geometrical variables using Young-Laplace law [1,2]. This is the first time to our knowledge that Laplace pressure is calculated from the measured bridge curvatures along the process of evaporation and compared to experimental measurement data. A comparison with the experimental data from analogous capillary bridge extension tests is also shown and discussed.The behavior of evaporating liquid bridges is seen as strongly dependent on the grain separation. Initial negative Laplace pressure at small separations is seen to significantly augment during an advanced stage of evaporation, but to turn into positive pressure, after an instability toward the end of the process, and prior to rupture. At larger separations the pressure is positive all the time, changing a little, but rupturing early. Rupture in all cases occurs at positive pressure. However, because of the evolution of the surface area of contact, the resultant total capillary forces are always tensile, and decreasing toward zero in all cases. Comparison between measured total resultant capillary forces and those calculated from the Young-Laplace law is very good, except for some discrepancies at very small separations (below 50μm). Up to four consecutive instabilities of capillary bridge are seen developing at some sphere separations. They are: re-pinning-induced suction (pressure) instability; Rayleigh nodoid/catenoid/unduloid unstable transition, associated with zero-pressure; Rayleigh unduloid/cylinder unstable transition, associated with the formation of a liquid-wire; and lastly, a pinching instability of the liquid-wire, associated with the bridge rupture. Rupture of the bridges is seen at large separations to occur quite early, at only 1/4–1/3 of the initial water volume evaporated. At smallest separations, rupture occurs in a seemingly unstable way when water evaporates from the bridge thinnest section of the neck.

On sharp interface limits for diffuse interface models for two-phase flows

We discuss the sharp interface limit of a diffuse interface model for a two-phase flow of two partly miscible viscous Newtonian fluids of different densities, when a certain parameter \epsilon > 0 related to the interface thickness tends to zero. In the case that the mobility stays positive or tends to zero slower than linearly in \epsilon we will prove that weak solutions tend to varifold solutions of a corresponding sharp interface model. But, if the mobility tends to zero faster than \epsilon^3 we will show that certain radially symmetric solutions tend to functions, which will not satisfy the Young-Laplace law at the interface in the limit.

Local instability of a plate with a circular nanohole under uniaxial tension

The problem of the influence of surface tension on the local instability of a plate under tension is investigated by the example of an infinite plate with a circular nanohole. The method for solving the problem under consideration is based on a previously developed approach for solving problems on local buckling of a thin uniaxially stretched elastic sheet with variously shaped macroholes. The critical (Euler) load, at which the instability occurs, is sought by the Ritz method in framework of the linearized Karman set of equations based on the principle of virtual motions as the smallest positive load giving the minimum of the potential energy of the plate deformation. The generalized plane stress state of the plate with a nanohole is found for its application taking into account surface stresses. The linearized Gurtin-Murdoch relationships for the surface elasticity, the generalized Young-Laplace law, and the Goursat-Kolosov complex potentials are used when deriving analytical dependences for the stress in this plane problem. The deflection of the plate is represented by a double row satisfying the damping condition at infinity and ensuring freedom of motion of hole edges. Numerical calculations are performed for the aluminum plate. Two variants of the elastic properties of a circular hole are considered. Zones of compressing circumferential tensions near the edges of the circular hole with radii 1, 2, and 4 nm under uniaxial tension are constructed. It is shown that the account of the surface stresses decreases the critical load, which has a tendency to decrease in both cases with a decrease in the cut radius in a nanometer variation range.

Driving a planar model system into the 3(rd) dimension: generation and control of curved pore-spanning membrane arrays.

The generation of a regular array of micrometre-sized pore-spanning membranes that protrude from the underlying surface as a function of osmotic pressure is reported. Giant unilamellar vesicles are spread onto non-functionalized Si/SiO(2) substrates containing a highly ordered array of cavities with pore diameters of 850 nm, an interpore distance of 4 μm and a pore depth of 10 μm. The shape of the resulting pore-spanning membranes is controlled by applying an osmotic pressure difference between the bulk solution and the femtoliter-sized cavity underneath each membrane. By applying Young-Laplace's law assuming moderate lateral membrane tensions, the response of the membranes to the osmotic pressure difference can be theoretically well described. Protruded pore-spanning membranes containing the receptor lipid PIP(2) specifically bind the ENTH domain of epsin resulting in an enlargement of the protrusions and disappearance as a result of ENTH-domain induced defects in the membranes. These results are discussed in the context of an ENTH-domain induced reduction of lateral membrane tension and formation of defects as a result of helix insertion of the protein in the bilayer.

Experimental Evaluation of the Pore-Access Size Distribution of Sands

A simple experimental method is presented for evaluation of the pore-access size distribution of sands and, more generally, of cohesionless granular materials. The water-retention curve of a sand specimen is evaluated using a technique similar to the hanging column method (ASTM 2008) [ASTM D6836: Standard Test Methods for Determination of the Soil Water Chararcteristic Curve for Desorption Using a Hanging Column, Pressure Extractor, Chilled Mirror Hygrometer, and/or Centrifuge, Annual Book of ASTM Standards, ASTM International, West Conshohocken, PA, 2008] by application of a step-by-step variation of suction and measure of the corresponding changes in water content. The pore-access size distribution is then evaluated from the water-retention curve using the Young-Laplace law, which links the suction to the pore-access radius. The efficiency of this experimental method is demonstrated by studying the influence of various parameters such as the shape and the size of the grains, the spreading of the grain-size distribution, and also the relative density of the specimen on the pore-size distribution.

A Young-Laplace law for black hole horizons

Black hole horizon sections, modeled as marginally outer trapped surfaces (MOTS), possess a notion of stability admitting a spectral characterization. Specifically, the ``principal eigenvalue'' ${\ensuremath{\lambda}}_{o}$ of the MOTS-stability operator (an elliptic operator on horizon sections) must be nonnegative. We discuss the expression of ${\ensuremath{\lambda}}_{o}$ for axisymmetric stationary black hole horizons and show that, remarkably, it presents the form of the Young-Laplace law for soap bubbles in equilibrium, if ${\ensuremath{\lambda}}_{o}$ is identified with a formal pressure difference between the inner and outer sides of the ``bubble.'' In this view, which endorses the existing fluid analogies for black hole horizons, MOTS-stability is interpreted as a consequence of a pressure increase in the black hole trapped region.

Simulations of Janus droplets at equilibrium and in shear

Janus droplets are compound droplets that consist of two adhering drops of different fluids that are suspended in a third fluid. We use the Shan-Chen lattice Boltzmann method for multicomponent mixtures to simulate Janus droplets at rest and in shear. In this simulation model, interfacial tensions are not known a priori from the model parameters and must be determined using numerical experiments. We show that interfacial tensions obtained with the Young-Laplace law are consistent with those measured from the equilibrium geometry. The regimes of adhering, separated, and engulfing droplets were explored. Two different adhesion geometries were considered for two-dimensional simulations of Janus droplets in shear. The first geometry resembles two adhering circles with small overlap. In the second geometry, the two halves are semicircular. For both geometries, the rotation rate of the droplet depends on its orientation. The width of the periodic simulation domain also affects the rotation rate of both droplet types up to an aspect ratio of 6:1 (width:height). While the droplets with the first geometry oscillated about the middle of the domain, the droplets of the second geometry did not translate while rotating. A four-pole vortex structure inside droplets of the second geometry was found. These simulations of single Janus droplets reveal complex behaviour that implies a rich range of possibilities for the rheology of Janus emulsions.

Pore Scale Models for Imbibition of CO2 Analogue Fluids in Etched Micro-model||junctions Using Micro-fluidic Experiments and Direct Flow Calculations

We investigate both drainage and spontaneous imbibition processes at the pore scale using a combination of micro- fluidic experiments and lattice-Boltzmann (LB) flow calculations. First, we have fabricated a range of specifically designed etched micro-models to investigate the role of pore shape and throat width on fluid displacement. These designs include junctions with both equal and unequal channel widths in order to achieve a range of capillary entry pressures. All models were etched into Poly(methyl methacrylate) (PMMA) and chemically treated to create a hydrophobic surface. The displacement process is captured via a high-speed video microscope under ambient conditions. The experimental results were then directly compared with LB simulations. For the drainage experiments, we observe that the fluid displacement in the junction follows the Young-Laplace Law. For the case of spontaneous imbibition, however, the models with unequal channel widths display different displacement behaviour. Our experimental observations are confirmed in detail by Lattice Boltzmann Method (LBM) simulations, lending credibility to our observations. Instead of following Young-Laplace filling rules, we observe that the throat in closest proximity fills up first. This has potentially important consequences for calculation of residual saturation of CO2 at the core scale, which is determined by spontaneous imbibition of brine following CO2 injection.

Comparison of the Young-Laplace Law and Finite Element Based Calculation of Ventricular Wall Stress: Implications for Postinfarct and Surgical Ventricular Remodeling

Both the Young-Laplace law and finite element (FE) based methods have been used to calculate left ventricular wall stress. We tested the hypothesis that the Young-Laplace law is able to reproduce results obtained with the FE method. Magnetic resonance imaging scans with noninvasive tags were used to calculate three-dimensional myocardial strain in 5 sheep 16 weeks after anteroapical myocardial infarction, and in 1 of those sheep 6 weeks after a Dor procedure. Animal-specific FE models were created from the remaining 5 animals using magnetic resonance images obtained at early diastolic filling. The FE-based stress in the fiber, cross-fiber, and circumferential directions was calculated and compared to stress calculated with the assumption that wall thickness is very much less than the radius of curvature (Young-Laplace law), and without that assumption (modified Laplace). First, circumferential stress calculated with the modified Laplace law is closer to results obtained with the FE method than stress calculated with the Young-Laplace law. However, there are pronounced regional differences, with the largest difference between modified Laplace and FE occurring in the inner and outer layers of the infarct borderzone. Also, stress calculated with the modified Laplace is very different than stress in the fiber and cross-fiber direction calculated with FE. As a consequence, the modified Laplace law is inaccurate when used to calculate the effect of the Dor procedure on regional ventricular stress. The FE method is necessary to determine stress in the left ventricle with postinfarct and surgical ventricular remodeling.