Spherical Cap Shape Research Articles

Modeling the effect of shape deformation induced by gravity on the evaporation of pendant and sessile drops

Pendant and sessile drops form a spherical cap only in the absence of gravity. The effect of gravity on drop shape is often neglected on the basis of the assumption that the drop size is smaller than the capillary length [Lc=(σ/gρ)1/2], although the deformation may not be fully negligible even in those cases. This paper focuses on evaluation of the effect that deformation due to gravity has on the evaporation characteristics of pendant and sessile drops. The drop shape is described by the Bashforth–Adams equation, a non-linear second order ordinary differential equation, which is solved numerically using a Runge–Kutta method with variable time steps. Under quasi-steady approximation, the species and energy conservation equations in the gas phase have analytical solutions, even for temperature-dependent gas thermophysical properties, once the solution of a basic Laplace problem is known. The Laplace equation is solved in axial symmetric geometry by using COMSOL Multiphysics®, for a wide range of drop sizes and contact angles, yielding vapor distribution, vapor fluxes, and evaporation rates. Comparison with the results from drops of same size in microgravity (i.e., having a spherical cap shape) shows that the effect is also perceptible for drops with a size smaller than the capillary length and that it can become quite important for those with larger sizes. Complementary results are found for sessile and pendant drops with respect to wall wettability, suggesting that the phenomenon can be analyzed using a unitary approach.

Improved-FSDT-based solid-shell element for buckling analysis of plate, spherical cap, and cylindrical shell of FG porous materials

This study's objective is to propose a full three-dimensional finite element modeling for examining the buckling behaviors of functionally graded (FG) porous materials with plates, spherical cap, and cylindrical shapes. An improved first-order shear deformation theory (IFSDT) is utilized in order to develop the three-dimensional solid-shell element. The governing equations are developed in a way that offers a parabolic distribution of the transverse shear strains through the FG porous shell thickness and a zero condition of the transverse shear stresses on the top and bottom surfaces. The given IFSDT solid-shell element has the ability to model any kind of shell structures and incorporates a three-dimensional material law. In this study, two types of porosity distribution functions are considered. The performance of the current IFSDT solid-shell element has been demonstrated by examining the buckling behavior of Functionally Graded Porous plates, spherical caps, and cylindrical shells. The effect of the porosity distribution, porosity coefficient, power-law index, geometrical design parameters and boundary conditions on the buckling behaviors of FG porous shells structures is evaluated.

Bubbly flows in a column with submerged hollow fiber membranes

Experiments of bubbly flows in a column with submerged hollow fiber membranes are carried out to discuss the relation between bubbly flow characteristics and fouling control. Two kinds of liquids, water and xanthan gum-water solution (XG-W), are used as the liquid phases. The relaxation rates of membranes are set at 0.0 or 0.5%. The number of small bubbles with ellipsoidal shapes in water are larger than that in XG-W, while large bubbles with spherical cap shapes are remarkably observed in XG-W in spite of the presence of membranes. The rate of large bubbles is enhanced in XG-W due to bubble coalescence. The gas holdup in water is higher than that in XG-W. The gas holdup becomes higher due to the presence of membranes and membrane relaxation in both water and XG-W. For discussion on fouling control due to bubbly flows, experiments are carried out to remove pseudo sludge attached to hollow fiber membranes. The relaxation of membranes enhances the removal rate of pseudo sludge. The mean removal rate, which is the mean value of the removal rate of each membrane, increases with increasing the gas holdup. This relation is well expressed by a linear regression in the present experimental range regardless of the liquid property and membrane relaxation, that is, the gas holdup has a prospect of an indicator for estimating fouling probability.

Single-Bubble Rising in Shear-Thinning and Elastoviscoplastic Fluids Using a Geometric Volume of Fluid Algorithm.

The motion of air bubbles within a liquid plays a crucial role in various aspects including heat transfer and material quality. In the context of non-Newtonian fluids, such as elastoviscoplastic fluids, the presence of air bubbles significantly influences the viscosity of the liquid. This study presents the development of an interface-capturing method for multiphase viscoelastic fluid flow simulations. The proposed algorithm utilizes a geometric volume of fluid (isoAdvector) approach and incorporates a reconstructed distance function (RDF) to determine interface curvature instead of relying on volume fraction gradients. Additionally, a piecewise linear interface construction (PLIC) scheme is employed in conjunction with the RDF-based interface reconstruction for improved accuracy and robustness. The validation of the multiphase viscoelastic PLIC-RDF isoAdvector (MVP-RIA) algorithm involved simulations of the buoyancy-driven rise of a bubble in fluids with varying degrees of rheological complexity. First, the newly developed algorithm was applied to investigate the buoyancy-driven rise of a bubble in a Newtonian fluid on an unbounded domain. The results show excellent agreement with experimental and theoretical findings, capturing the bubble shape and velocity accurately. Next, the algorithm was extended to simulate the buoyancy-driven rise of a bubble in a viscoelastic shear-thinning fluid described by the Giesekus constitutive model. As the influence of normal stress surpasses surface tension, the bubble shape undergoes a transition to a prolate or teardrop shape, often exhibiting a cusp at the bubble tail. This is in contrast to the spherical, ellipsoidal, or spherical-cap shapes observed in the first case study with a bubble in a Newtonian fluid. Lastly, the algorithm was employed to study the buoyancy-driven rise of a bubble in an unbounded elastoviscoplastic medium, modeled using the Saramito-Herschel-Bulkley constitutive equation. It was observed that in very small air bubbles within the elastoviscoplastic fluid, the dominance of elasticity and capillary forces restricts the degree of bubble deformation. As the bubble volume increases, lateral stretching becomes prominent, resulting in the emergence of two tails. Ultimately, a highly elongated bubble shape with sharper tails is observed. The results show that by applying the newly developed MVP-RIA algorithm, with a tangible coarser grid compared to the algebraic VOF method, an accurate solution is achieved. This will open doors to plenty of applications such as bubble columns in reactors, oil and gas mixtures, 3D printing, polymer processing, etc.

The movement and shape change characteristics of a bubble passing through a liquid-liquid interface

In order to study the movement and shape change characteristic of bubble when passing through the interface of two kinds of liquids with different viscosity, the free rising process of a single bubble in static stratified liquids was numerically simulated with the volume-of-fluid method. The results show that, when the initial height of bubble rising is the same, the rising velocity, deformation increase with the increase of bubble radius. When the maximum intensity of the vortex in the bubble is distributed at the top of the bubble, the top of the left and right sides and the bottom of the left and right sides, the bubble shape is spherical, ellipsoid and spherical cap shape respectively. At different initial heights, the bubble trajectory shows three different shapes ? linear, spiral, and C-shaped. The relation-ship between the bubble aspect ratio and rising height is predicted when different radius bubble passing through the interface. The amount of liquid B (lower layer) carried by the bubble increases with the increase of the bubble?s initial radius, and the amount of liquid carried by bubbles in C-shaped trajectory is higher than that in spiral trajectory.

Corrosion of steel in a CO2-containing solution droplet generated in wet gas pipelines studied by scanning Kelvin probe

A scanning Kelvin probe (SKP) was used to study corrosion of X52 pipeline steel under a CO2-containing solution droplet simulating the water condensate generated in interior of wet gas pipelines. The Volta potential measured by SKP was indicative of the feature of corrosion occurring under the droplet. Three potential zones were observed from the droplet center towards the transition and the side of the droplet, which were -0.1 V, -0.4 V and 0.1 V (Kelvin probe, kp), respectively, after 4 h of testing. The droplet side had the highest Volta potential, which was attributed to corrosion scale generation and precipitation due to a limited solution volume and achieved solubility limit of FeCO3 scale. The transition zone possessed the most negative Volta potential due to a higher corrosion activity than the droplet center with more supply and dissolution of CO2, while the droplet center had the thickest solution. The SKP measurements could estimate the volume evolution of the solution droplet with time at a relatively accurate scale. The solution droplet could maintain its topographic feature, i.e., a spherical cap shape, within 4 h of testing in this work. The increase in droplet concentration due to solution evaporation during testing affected the Volta potential, but slightly only.

Finite element analysis of the influence of cyclic strain on cells anchored to substrates with varying properties.

The response of cytoskeleton to mechanical cues plays a pivotal role in understanding several aspects of cellular growth, migration, and cell-cell and cell-matrix interactions under normal and diseased conditions. Finite element analysis (FEA) has become a powerful computational technique to study the response of cytoskeleton in the maintenance of overall cellular mechanics. With the revelation of role of external mechanical microenvironment on cell mechanics, FEA models have also been developed to simulate the effect of substrate stiffness on the mechanical properties of cancer cells. However, the models developed so far model cellular response under static mode, whereas in physiological condition, cells always experience dynamic loading conditions. To develop a more accurate model of cell-extracellular matrix (ECM) interactions, this paper models the cytoskeleton and other parts of the cell by beam and solid elements respectively, assuming spherical morphology of the cell. The stiffness and roughness of extracellular matrix were varied. Furthermore, static and dynamic sinusoidal loads were applied through a flat plate indenter on the cell along with providing sinusoidal strain at the substrate. It is observed that due to axial loading, cell reaches a plastic region, and when the sinusoidal loading is added to the axial load, the cell experiences permanent deformation. Degradation of the cytoskeleton elements and a physiologically more relevant spherical cap shape of the cell were also considered during the analysis. This study suggests that asperity topology of the substrate and indirect cyclic load can play a significant role in the shape alterations and motion of a cell.

The concept of elasto-visco-plasticity and its application to a bubble rising in yield stress fluids

• New simulations are presented that include the elasticity of yield stress materials, as oldroyd first suggested. • The inverted teardrop bubble shape and the negative wake are predicted. • Numerical results indicate the importance of elasticity in yield stress materials. • A satisfactory agreement is found between experiments and simulations for the bubble shape and rising velocity. • The critical yield stress value for bubble entrapment increases by elastic forces. Based on the original ideas by Oldroyd in 1947, Saramito in 2007 introduced a generally acceptable model for yield stress fluids that also accounts for their elasticity. We use this model to examine the buoyancy-driven rise of a bubble through Carbopol and predict the inverted teardrop shape that bubbles often attain when they translate through an elastoviscoplastic material. Such shapes are common in viscoelastic materials, but it is impossible to predict them using viscoplastic models without elasticity. Our results also show that generally, the fore-aft symmetry is lost and the negative wake is formed in the trailing edge of the bubble, even when creeping flow conditions are approached. We show that elastic and plastic forces are dominant in the smaller bubbles, whereas significant shear and extensional thinning occurs in the larger ones, making inertia forces dominant. The agreement of our predicted bubble shapes and rise velocities with their experimental counterparts is very satisfactory either for the smaller or the larger bubbles from those we examined. The former have an inverted teardrop shape and the latter have either an oblate or spherical cap shape. A deviation is found in the bubble radius for which the transition from the inverted teardrop shape to the oblate shape occurs. This deviation is attributed primarily to insufficient rheological characterization of the materials, particularly in terms of their nonlinear elasticity and their extensional properties.

Evaporation of Binary-Mixture Liquid Droplets: The Formation of Picoliter Pancakelike Shapes.

Small multicomponent droplets are of increasing importance in a plethora of technological applications ranging from the fabrication of self-assembled hierarchical patterns to the design of autonomous fluidic systems. While often far away from equilibrium, involving complex and even chaotic flow fields, it is commonly assumed that in these systems with small drops surface tension keeps the shapes spherical. Here, studying picoliter volatile binary-mixture droplets of isopropanol and 2-butanol, we show that the dominance of surface tension forces at small scales can play a dual role: Minute variations in surface tension along the interface can create Marangoni flows that are strong enough to significantly deform the drop, forming micron-thick pancakelike shapes that are otherwise typical of large puddles. We identify the conditions under which these flattened shapes form and explain why, universally, they relax back to a spherical-cap shape toward the end of drop lifetime. We further show that the formation of pancakelike droplets suppresses the "coffee-ring" effect and leads to uniform deposition of suspended particles. The quantitative agreement between theory and experiment provides a predictive capability to modulate the shape of tiny droplets with implications in a range of technologies from fabrication of miniature optical lenses to coating, printing, and pattern deposition.

From single bubble to global mixing: Added mass, drift, wake

The goal was to present an approach where the micro-scale events related to single-particle motion directly translate onto higher levels and produce large-scale observable phenomena related to transport and mixing. Three multiphase hydrodynamic effects were considered, analysed and discussed: added mass, drift, wake. Although the results are of general validity, the spherical-cap shape (cap bubble) was used as the definite object for demonstration. It was shown how the unsteady inertial effect of added mass (coefficient C), irrelevant for mixing, is related to the steady kinematic concept of Darwin drift (coefficient D) that is directly associated with mixing. It was shown how the inviscid drift and the viscous wake operate and contribute to the resulting circulation and mixing. List of currently available formulas for C was completed, which are usable for estimation of D.

Modeling of elastoplastic behavior of freestanding square thin films under bulge testing

Bulge testing is an experimental technique applied to the characterization of thin films, with which the mechanical properties are obtained employing numerical and analytical approaches. For rectangular and circular thin films, classical methods have been well adopted to calculate the stresses and strains at the central point without the dependency of its properties. It is due to that the equilibrium conditions are dependent on the curvatures generated at the maximum deflection point. Therefore, the kinematics of the bulged surface can be represented by a spherical cap and cylindrical shapes in these films. For square thin films, the deflected surface presents a complex displacement field that implies the development of analytical models for the estimation of these. In this sense, equations to represent the equibiaxial stress state (central point) for freestanding square thin films are developed and applied in this study. The curvatures are determined using a reviewed nonlinear deflection field. Virtual experiments were driven by applying finite element analysis to validate the stress and strain models through parametric analysis. In order to impose the residual stress in thin films, two approaches of modeling for the simulations are described. In order to validate the applicability of the proposed equations, different cases were presented for the linear elastic analysis, and two examples were developed to characterize elastoplastic behavior (bilinear and nonlinear) from load–deflection curves. The obtained results show that the developed models for square thin films are in good agreement with the finite element simulations since these predicted the stress and strain states with good accuracy.

Forced oscillation dynamics of surface nanobubbles.

Surface nanobubbles have potential applications in the manipulation of nanoscale and biological materials, waste-water treatment, and surface cleaning. These spherically capped bubbles of gas can exist in stable diffusive equilibrium on chemically patterned or rough hydrophobic surfaces, under supersaturated conditions. Previous studies have investigated their long-term response to pressure variations, which is governed by the surrounding liquid's local supersaturation; however, not much is known about their short-term response to rapid pressure changes, i.e., their cavitation dynamics. Here, we present molecular dynamics simulations of a surface nanobubble subjected to an external oscillating pressure field. The surface nanobubble is found to oscillate with a pinned contact line, while still retaining a mostly spherical cap shape. The amplitude-frequency response is typical of an underdamped system, with a peak amplitude near the estimated natural frequency, despite the strong viscous effects at the nanoscale. This peak is enhanced by the surface nanobubble's high internal gas pressure, a result of the Laplace pressure. We find that accurately capturing the gas pressure, bubble volume, and pinned growth mode is important for estimating the natural frequency, and we propose a simple model for the surface nanobubble frequency response, with comparisons made to other common models for a spherical bubble, a constant contact angle surface bubble, and a bubble entrapped within a cylindrical micropore. This work reveals the initial stages of growth of cavitation nanobubbles on surfaces, common in heterogeneous nucleation, where classical models based on spherical bubble growth break down.

Compaction Dynamics during Progenitor Cell Self-Assembly Reveal Granular Mechanics

Summary We study the self-assembly dynamics of human progenitor cells in agarose microwells that are used for production of chondrogenic organoids. Using image analysis on time-lapse microscopy, we estimate the aggregate area in function of time for a large number of aggregates. In control conditions, the aggregate radius follows an exponential relaxation that is consistent with the dewetting dynamics of a liquid film. Upon introducing Y-27632 Rho kinase inhibitor, the compatibility with the liquid model is lost, and slowed-down relaxation dynamics are observed. We demonstrate that these aggregates behave as granular piles undergoing compaction, with density relaxation that follows a stretched exponential. Using simulations with an individual cell-based model, we construct a phase diagram of cell aggregates that suggests that the aggregate in presence of Rho kinase inhibitor approaches glass transition.

Magnetic resonance imaging of interaction and coalescence of two bubbles injected consecutively into an incipiently fluidized bed

Rapid magnetic resonance imaging (MRI) was used to visualize and quantify the interaction of two bubbles injected into an incipiently fluidized bed. The particle size, bubble sizes and the vertical and horizontal separations between bubbles were varied to understand their effects on bubble behavior. Image analysis quantified the size, shape and position of the bubbles over time. Bubbles were found to either (a) coalesce, (b) influence one another without coalescing or (c) have a collapse of a lower bubble due to influence from the upper bubble. In all cases, the lower bubbles were much more influenced in their rise by the upper bubbles than vice-versa, as lower bubbles accelerated toward the wakes of upper bubbles. Upper bubbles developed spherical cap shapes, while lower bubbles elongated vertically elongated as they rose. The experimental data provided here will serve as excellent benchmarks to challenge the assumptions made in computational and theoretical models.

Magnetic resonance imaging of single bubbles injected into incipiently fluidized beds

Rapid magnetic resonance imaging is used to investigate the volume, shape and rise velocity of single isolated bubbles injected into incipiently fluidized beds as well as the particle velocity field surrounding these bubbles. The volume of gas injected and particle size are varied to investigate their effects on bubble behavior. Image processing is used to quantify data into plots which can be compared directly with predictions from analytical and numerical models. Results show that more injected gas leaks into the interstitial flow in beds of large particles than beds of small particles because of the higher permeability to gas flow. Bubbles develop from a taller shape with a large wake angle into a wider shape with a smaller wake angle as they develop from a spherical shape to a spherical cap shape as they rise. Bubble rise velocities for fully formed bubbles are consistent with correlations in the literature.

Spontaneous Displacement of High Viscosity Micrometer Size Oil Droplets from a Curved Solid in Aqueous Solutions.

Spontaneous displacement of high viscosity (∼103 Pa·s) micrometer size oil droplets from a curved solid in aqueous solutions was investigated. For high viscosity oils, the dynamic droplet shape was found to deviate significantly from a spherical cap shape due to the considerable viscous force in the oil phase. The displacement dynamics of high viscosity droplets were analyzed using molecular kinetic and hydrodynamic models. The molecular kinetic model was found to describe the dynamic displacement well for the droplets of small departure from the spherical cap shape, while the hydrodynamic model is more applicable to the droplets of higher three-phase contact line displacement velocities and hence larger deviation of the droplets from the spherical cap shape.

Thermodynamic modelling of InAs/InP(0 0 1) growth towards quantum dots formation by metalorganic vapor phase epitaxy

Quantum Dots (QDs) are considered as an efficient building block of many optoelectronic applications, such as semiconductor laser, photodetector, whereby their physical dimension is the key parameter to be controlled. In this work, we have studied experimentally the growth of InAs QDs on InP(0 0 1) substrate by MOVPE and established a theoretical model explaining the observed epitaxial behaviour. In variance with the classical Stranski-Krastanov growth, we show that during the growth there is an intermediate stage whereby although first large 3D islands are formed, an increased density of small QDs is formed concurrent with the shrinkage of the large 3D islands. As a result, the growth can be divided into three regimes: 2D layer growth, large 3D islands growth and QDs growth. To explain this evolution, a thermodynamic model has been developed accounting for the process driven by surface energy, elastic relaxation energy and inter-island interaction energy. It will be shown that the balance between the surface energy and the elastic relaxation energy provides the transition from 2D layer to large 3D islands (around at 3 ML of InAs growth). This model also supports the energetically favourable truncated pyramidal shape for large 3D islands and the spherical cap shape for QDs. We show in this paper that a balance between surface energy, elastic relaxation energy and inter-island elastic interaction explains the volume shrinkage of the early large 3D islands towards the formation of small QDs in high density.

Lattice Boltzmann simulation of droplet dynamics on granular surfaces with variable wettability.

Soil-composing particles undergo wettability changes, impacting hydraulic and mechanical processes such as erosion and landslides. Such processes evolve at very small scales, typically at the particle level. Here we capture the evolution of liquid interfaces in a single particle and several particles with the lattice Boltzmann (LB) method. The paper presents a three-dimensional LB study on the droplet dynamics on a layer of uniformly packed spherical particles with varying size and intrinsic contact angle (CA) aimed at mimicking conditions comparable to those of real soils. The numerical droplet is initialized close to the granular surface and deposited by gravity. Three spreading and infiltration behaviors were identified: a droplet with a stable apparent CA, a droplet with a metastable apparent CA before infiltration, and immediate infiltration. The results showed that the formation of a droplet with a stable or metastable spherical-cap shape depends on the particle size and the intrinsic CA. Furthermore, the initial wetted zone expansion was found to be governed by inertial effects with its behavior characterized by a power law. Finally, the apparent CA, which is closely related to the intrinsic CA, was found to be influenced by the particle size due to a significant portion of the droplet being embedded into the granular surface for the larger particles and reducing the apparent CA. This paper provides a basis for future research targeting the behavior of droplet interaction with granular surfaces with variable intrinsic CAs (from wettable to superhydrophobic) such as soils and other granular materials for industrial applications. The numerical approach implemented can also be extended to model other dynamic processes for a droplet, such as evaporation, high-velocity impacting, and lateral sliding.

Analytical Model for Diffusive Evaporation of Sessile Droplets Coupled with Interfacial Cooling Effect.

Current analytical models for sessile droplet evaporation do not consider the nonuniform temperature field within the droplet and can overpredict the evaporation by 20%. This deviation can be attributed to a significant temperature drop due to the release of the latent heat of evaporation along the air-liquid interface. We report, for the first time, an analytical solution of the sessile droplet evaporation coupled with this interfacial cooling effect. The two-way coupling model of the quasi-steady thermal diffusion within the droplet and the quasi-steady diffusion-controlled droplet evaporation is conveniently solved in the toroidal coordinate system by applying the method of separation of variables. Our new analytical model for the coupled vapor concentration and temperature fields is in the closed form and is applicable for a full range of spherical-cap shape droplets of different contact angles and types of fluids. Our analytical results are uniquely quantified by a dimensionless evaporative cooling number Eo whose magnitude is determined only by the thermophysical properties of the liquid and the atmosphere. Accordingly, the larger the magnitude of Eo, the more significant the effect of the evaporative cooling, which results in stronger suppression on the evaporation rate. The classical isothermal model is recovered if the temperature gradient along the air-liquid interface is negligible ( Eo = 0). For substrates with very high thermal conductivities (isothermal substrates), our analytical model predicts a reversal of temperature gradient along the droplet-free surface at a contact angle of 119°. Our findings pose interesting challenges but also guidance for experimental investigations.

Prediction of dynamic contact angles and bubble departure diameters in pool boiling using equilibrium thermodynamics

A model is proposed for predicting dynamic contact angles and bubble departure diameters in pool boiling based on principles of equilibrium thermodynamics. The method is shown to give good agreement with experimental measurements for pool boiling over a wide range of pressures, liquid superheats and working fluids.The model suggests that for fast-growing bubbles associated with boiling at high Jakob Numbers (>100) the bubble shape is dominated by fluid drag forces rather than surface tension forces, resulting in the bubbles having a predominantly hemispherical shape during growth. For slower-growing bubbles associated with lower Jakob numbers (<50) surface tension forces appear to dominate over fluid drag forces, resulting in the bubbles adopting a spherical cap shape with a contact angle that remains close to the thermodynamic contact angle. This observation is in accordance with experiment.The model implies that in high pressure boiling the thermodynamic contact angle for the system has a controlling influence on bubble departure sizes. Analysis of departure diameters in high pressure boiling of water supports the view that the thermodynamic contact angle is strongly dependent on temperature, and that it is this temperature dependence that is responsible for the small bubble departure sizes observed in high pressure boiling. Previous models of bubble departure have been unable to explain the pressure dependence of bubble departure size. A correlation of the thermodynamic contact angle against temperature for water is proposed.The current model of bubble departure requires a means of calculating the bubble growth rate, as the latter determines the fluid drag forces acting on an attached bubble. A new predictive method for bubble growth is developed for this purpose, based on Scriven’s solution for conduction controlled bubble growth, with corrections to take account of non-uniform liquid superheat, microlayer evaporation, the effect of growth rate on the bubble shape, and the cooling effect of the microlayer on the heated surface. The predictive method gives satisfactory agreement with bubble growth data for a wide range of pool boiling experiments and also with results of microscale CFD simulations of the growth of individual bubbles.