Contact Angle Boundary Condition Research Articles

Accurate contact angle boundary conditions for the Cahn–Hilliard equations

The contact angle dynamics between a two-phase interface and a solid surface is important in physical interpretations, mathematical modeling, and numerical treatments. We present a novel formulation based on a characteristic interpolation for the contact angle boundary conditions for the Cahn–Hilliard equation. The new scheme inherits characteristic properties, such as the mass conservation, the total energy decrease, and the unconditionally gradient stability. We demonstrate the accuracy and robustness of the proposed contact angle boundary formulation with various numerical experiments. The numerical results indicate a potential usefulness of the proposed method for accurately calculating contact angle problems.

Dynamics of Partial Wetting

The kinetics of partial wetting was investigated for the case of spreading of cylindrical and axisymmetric drops over a horizontal solid surface under the action of capillary forces. The Brochard-Wyart and de Gennes (1992) relation, having a cut off of molecular size below which the continuum theory breaks down, is used to provide a dynamic contact angle boundary condition. The dynamic contact angle relation and the viscous dissipation are shown to be valid even in the case of axisymmetric spreading when using the approximation of a wedge flow pattern near the leading edge. To obtain an analytical solution for the dynamics of spreading, the departure of the outer region solution from the spherical cap profile, near the inflection point, was neglected, and the accuracy of the approximation was examined. The analytical solution obtained is a powerseries expression. Simple short-time and long-time asymptotic solutions are also provided. The model for the dynamics of partial wetting does not require any fitting parameter. The results are found to agree favorably with the available published experimental data even at dynamic contact angles as large as about 90°, which is beyond the range of applicability of the lubrication theory. The model matches very closely the dynamics of spreading obtained with a previous model by Chebbi (2010), in which a numerically similarity solution was used to account for the deviation from the spherical cap profile near the inflection point.

A Model for the Dynamics of Partial Wetting

A model is proposed for the dynamics of spreading of a liquid drop under the action of capillary forces in the case of partial wetting. Deviation from the spherical cap profile is taken into account. The spreading kinetics is obtained using a dynamic contact angle boundary condition in which the velocity of the contact line is given as a function of the dynamic contact angle and a cut off of molecular size below which the continuum theory breaks down. Results are found to be in very good agreement with published experimental data even at high dynamic contact angles, well beyond the range of applicability of the lubrication theory. Differences between the spreading and dewetting cases are discussed. Further investigation is needed to treat the dewetting case.

Sharp interface immersed-boundary/level-set method for wave–body interactions

A sharp interface Cartesian grid method for the large-eddy simulation of two-phase turbulent flows interacting with moving bodies is presented. The overall approach uses a sharp interface immersed boundary formulation and a level-set/ghost–fluid method for solid–fluid and fluid–fluid interface treatments, respectively. A four-step fractional-step method is used for velocity–pressure coupling, and a Lagrangian dynamic Smagorinsky subgrid-scale model is adopted for large-eddy simulations. A simple contact angle boundary condition treatment that conforms to the immersed boundary formulation is developed. A variety of test cases of different scales ranging from bubble dynamics, water entry and exit, landslide-generated waves, to ship hydrodynamics are performed for validation. Extensions for high Reynolds number ship flows using wall-layer models are also considered.

Modeling of an oblique impact of solder droplet onto a groove with the impact point to be offset from the groove surfaces interface

A three-dimensional VOF model has been developed to predict the oblique impact of a molten solder droplet onto a groove with the impact point to be offset from the groove surfaces interface. The contact angle boundary condition and the surface tension are considered in the present model. High speed videography system is used to record the impact dynamics. Asymmetric solder deformation is observed during the impact of the solder droplet onto the groove. The simulated results are in good agreement with the experiment. The presented model can accurately predict the characteristic geometry of the solder droplet, the interaction between the gas and the solder, the wall shear stress acting on the pads, and the energy variations during the impingement.

Dependency of the apparent contact angle on nonisothermal conditions

The dynamic behavior of liquids in partly filled containers is influenced to a large extend by the angle between the gas-liquid phase boundary and the solid container wall at the contact line. This contact angle in turn is influenced by nonisothermal conditions. In the case of a cold liquid meniscus spreading over a hot solid wall, the contact angle apparently becomes significantly larger. In this paper we want to establish a quantitative equation for this enlargement, both from experimental and numerical data. Our findings can be used to build a subgrid model for computations, where the resolution is not sufficient to resolve the boundary layers. This might be the case for large containers which are exposed to low accelerations and where the contact angle boundary condition determines the position of the free surface. These types of computation are performed, for example, to solve propellant management problems in launcher and satellite tanks. In this application, the knowledge of the position of the free surface is very important for the withdrawal of liquid and the calculation of heat and mass transfer.

Height functions for applying contact angles to 2D VOF simulations

AbstractRecent work on a consistent representation of surface tension and on the accurate computation of interface curvature has extended the applicability of the volume‐of‐fluid, or VOF, method to surface tension‐driven phenomena. We have extended these methodologies to interfaces near solid surfaces; specifically, we show how height functions can be used to enforce a contact angle boundary condition at a contact line, for the full range of contact angles. As such, this work may be viewed as following up on the methodology of Renardy et al. (J. Comput. Phys. 2001; 171:243–263). This approach yields accurate values of curvature and the surface tension force at contact lines, values that converge with mesh refinement. The efficacy of the methodology is demonstrated using 2D examples of static fluid configurations in equilibrium and dynamic contact line phenomena that approach known equilibrium configurations. Copyright © 2007 John Wiley & Sons, Ltd.

Behavior of some CMC capillary surfaces at convex corners

We construct examples of nonparametric surfaces z = h(x,y) of zero mean curvature which satisfy contact angle boundary conditions in a cylinder in R3 over a convex domain O with corners. When the contact angles for two adjacent walls of the cylinder differ by more than p-2a, where 2a is the opening angle between the walls, the (bounded) solution h is shown to be discontinuous at the corresponding corner. This is exactly the behavior predicted by the Concus?Finn conjecture. These examples currently constitute the largest collection of capillary surfaces for which the validity of the Concus?Finn conjecture is known and, in particular, provide examples for all contact angle data satisfying the condition above for opening angles 2a in (p / 2,p).

Numerical study of three-dimensional free surface dynamics

The dynamic problem of three-dimensional free surface is numerically studied in this paper. The ALE (Arbitrary Lagrange-Euler) kinematic description is introduced into the control equation system. The ALE description method is used to track free surface. Accurate formulations for calculating the normal vector on the free surface are presented. The discrete numerical equations by finite element method are developed by Galerkin weighted residual method. The boundary condition about free-surface tension is represented in the form of weak integration that can be computed by a differential geometry method derived in the present paper. The effect of contact angle is incorporated in the numerical algorithm. Furthermore, the numerical computations are performed and the comparison between computational and analytical results validated the effectiveness of the method. The results of this paper provide a fundamental understandings of the dynamics of liquid free surfaces, in which the surface tension and contact angle boundary conditions are taken into account. Finally, numerical simulation of large-scale amplitude sloshing of liquid in a cylindrical container is performed and a numerical analysis of the effect of an annular ring-shaped rigid damping baffle on liquid sloshing oscillations in a cylindrical tank is also carried out.

A robust algorithm for the simultaneous parameter estimation of interfacial tension and contact angle from sessile drop profiles

The pendant and sessile drop profile analysis using the finite element method (PSDA-FEM) is an algorithm which allows simultaneous determination of the interfacial tension ( γ) and contact angle ( θ c ) from sessile drop profiles. The PSDA-FEM algorithm solves the nonlinear second-order spherical coordinate form of the Young–Laplace equation. Thus, the boundary conditions at the drop apex and contact position of the drop with the substrate are required to solve for the drop profile coordinates. The boundary condition at the position where the drop contacts the substrate may be specified as a fixed contact line or fixed contact angle. This paper will focus on the fixed contact angle boundary condition for sessile drops on a substrate and how this boundary condition is used in the PSDA-FEM curve-fitting algorithm. The PSDA-FEM algorithm has been tested using simulated drop shapes with and without the addition of random error to the drop profile coordinates. The random error is varied to simulate the effect of camera resolution on the estimates of γ and θ c values obtained from the curve-fitting algorithm. The error in the experimental values for γ from sessile drops of water on acrylic and Mazola corn oil on acrylic falls within the predicted range of errors obtained for γ values from simulated sessile drop profiles with randomized errors that are comparable in magnitude to the resolution of the experimental setup.

An Effective Implementation of Surface Tension Using the Marker and Cell Method for Axisymmetric and Planar Flows

This work presents a method for simulating axisymmetric and planar free-surface flows dominated by surface tension forces. The surface tension effects are incorporated into the free surface boundary conditions through the computation of the capillary pressure. The required curvature is evaluated by fitting a least squares curve to the free surface using the tracking markers in the cell and in its close neighbors. To avoid short wavelength perturbations on the free surface, a mass-conserving local four-point stencil filter is employed. This filter is an extension of the trapezoidal subgrid undulations removal (TSUR) method. The TSUR technique consists of modifying the positions of two ``internal' markers of the four-point stencil in such a way that the surface length and the curvature are minimized, while still preserving the volume. The computation of the curvature is modified at cells adjacent to solid boundaries in order to apply contact angle boundary conditions. To identify neighboring cells efficiently, an implementation is effected through a dual representation of the cell data: in addition to a matrix representation, a list structure is also employed which permits the representation of specific groups of cells and associated data. This technique was implemented in the GENSMAC code, and it has been proved to be robust. The code is shown to produce accurate results when compared with exact solutions of selected fluid dynamical problems involving surface tension. Additionally, it is demonstrated that the method is applicable to complex free-surface flows, such as containers filling, splashing drops, and bursting bubbles.

The self-similar solution for draining in the thin film equation

We investigate self-similar solutions of the thin film equation in the case of zero contact angle boundary conditions on a finite domain. We prove existence and uniqueness of such a solution and determine the asymptotic behaviour as the exponent in the equation approaches the critical value at which zero contact angle boundary conditions become untenable. Numerical and power-series solutions are also presented.

Anomalous Exponents and Dipole Solutions for the Thin Film Equation

We investigate similarity solutions of the thin film equation. In particular we look at solutions on the half-line $x\ge0$ with compact support and zero contact angle boundary conditions in x=0. Such dipole solutions feature an anomalous exponent and are therefore called similarity solutions of the second kind. Using a combination of phase space analysis and numerical simulations, we numerically construct trajectories representing these solutions, at the same time obtaining broader insight into the nature of the four-dimensional phase space. Additional asymptotic analysis provides further information concerning the evolution to self-similarity.

A note on the contact angle boundary condition for Monge-Ampère equations

We give a simple proof of a result of Xinan Ma concerning a necessary condition for the solvability of the two-dimensional Monge-Ampère equation subject to the contact angle or capillarity boundary condition. Our technique works for more general Monge-Ampère equations in any dimension, and also extends to some other boundary conditions.

On a three-dimensional volume tracking model of droplet impact

A three-dimensional model has been developed of droplet impact onto asymmetric surface geometries. The model is based on RIPPLE, and combines a fixed-grid control volume discretization of the flow equations with a volume tracking algorithm to track the droplet free surface. Surface tension is modeled as a volume force acting on fluid near the free surface. Contact angles are applied as a boundary condition at the contact line. The results of two scenarios are presented, of the oblique impact of a 2 mm water droplet at 1 m/sec onto a 45° incline, and of a similar impact of a droplet onto a sharp edge. Photographs are presented of such impacts, against which the numerical results are compared. The contact angle boundary condition is applied in one of two ways. For the impact onto an incline, the temporal variation of contact angles at the leading and trailing edges of the droplet was measured from photographs. This data is applied as a boundary condition to the simulation, and an interpolation scheme proposed to evaluate contact angles between the leading and trailing edges. A simpler model is then proposed, for contact angle as a function of contact line velocity, and applied to both geometries. The model requires values of only two contact angles, at a rapidly advancing and a rapidly receding contact line. Simulation results compare well with photographic data.

Damped Oscillations of a Liquid/Gas Surface upon Step Reduction in Gravity

The reorientation and settling of a liquid/gas interfacein a right circular cylinder upon step reduction in gravity is investigated numerically. A modie ed dynamic contact angle boundary condition is implemented where the hysteresis parameter ° is employed as an adjustable parameter representing the degree of relative slip at the contact line. The effect of the boundary condition on the axial interface oscillation is studied and the dependence thereon of the resonant frequency and settling time is given. Results are shown regarding the behavior of the dynamic contact angle and the penetration of the surface oscillations into the bulk liquid. The numerical results aretuned via comparisons with existing experimentaldata, and a method forestimating theappropriateboundary condition providing good agreement is suggested. An accompanying scale analysis depicts ° as a function of the Ohnesorge number, the cylinder radius, and the static contact angle. These independent parameters are measures of the system damping, characteristic surface dee ection, and effective stiffness of the interface. Applications of the results may be made to spacecraft e uids management and the use of spacecraft and/or drop towers for e uids experimentation.

Dynamics of Liquid Spreading on Solid Surfaces

Using simple scaling arguments and a precursor film model, we show that the appropriate macroscopic contact angle θ during the slow spreading of a completely or partially wetting liquid under conditions of viscous flow and small slopes should be described by tan θ = [tan3 θe − 9 log ηCa]1/3 where θe is the static contact angle, Ca is the capillary number, and η is a scaled Hamaker constant. Using this simple relation as a boundary condition, we are able to quantitatively model, without any empirical parameter, the spreading dynamics of several classical spreading phenomena (capillary rise, sessile, and pendant drop spreading) by simply equating the slope of the leading order static bulk region to our dynamic contact angle boundary condition without performing a matched asymptotic analysis for each case independently as is usually done in the literature.

The interaction of a moving fluid/fluid interface with a flat plate

A well-known technique for metering a multiphase flow is to use small probes that utilize some measurement principle to detect the presence of different phases surrounding their tips. In almost all cases of relevance to the oil industry, the flow around such local probes is inviscid and driven by surface tension, with negligible gravitational effects. In order to study the features of the flow around a local probe when it meets a droplet, we analyse a model problem: the interaction of an infinite, initially straight, interface between two inviscid fluids, advected in an initially uniform flow towards a semi-infinite thin flat plate oriented at 90° to the interface. This has enabled us to gain some insight into the factors that control the motion of a contact line over a solid surface, for a range of physical parameter values.The potential flows in the two fluids are coupled nonlinearly at the interface, where surface tension is balanced by a pressure difference. In addition, a dynamic contact angle boundary condition is imposed at the three-phase contact line, which moves along the plate. In order to determine how the interface deforms in such a flow, we consider the small- and large-time asymptotic limits of the solution. The small-time and linearized large-time problems are solved analytically, using Mellin transforms, whilst the general large-time problem is solved numerically, using a boundary integral method.The form of the dynamic contact angle as a function of contact line velocity is the most important factor in determining how an interface deforms as it meets and moves over the plate. Depending on this, the three-phase contact line may, at one extreme, hang up on the leading edge of the plate or, at the other extreme, move rapidly along the surface of the plate. At large times, the solution asymptotes to an interface configuration where the contact line moves at the far-field velocity.

Finite element simulation of combined buoyancy and thermocapillary driven convection in open cavities

AbstractThermocapillary‐induced and buoyancy‐driven convective flows that commonly occur in crystal growth are numerically simulated using Galerkin finite element method. The physical domain comprises of a open cavity with aspect ratio one and differentially heated vertical walls. The top gas–melt interface is free to deform subject to 90° contact angle boundary conditions at the two vertical walls. The unsteady two‐dimensional Navier–Stokes equations are discretized in time using Chorin‐type splitting scheme and pressure is determined from the Poisson's equation. The free surface is taken to be resting on vertical spines and its evolution in time is determined from the kinematic free surface equation. The governing equations for heat and momentum are solved in the Arbitrary Lagrangian Eulerian frame of reference to handle the moving boundary. The influence of Grashof number, Marangoni number, Bond number, Ohnesorge number and Prandtl number on the flow field and heat transfer is investigated.

Effective slip in numerical calculations of moving-contact-line problems

For many coating flows, the profile thickness h, near the front of the coating film, is governed by a third-order ordinary differential equation of the form h m = f(h), for some given f(h). We consider here the case of dry wall coating which allows for slip in the vicinity of the moving contact-line. For this case, one such model equation, due to Greenspan, f h = -1 + (1 + α)/(h 2 + α), where α is the slip coefficient. The equation is solved using a finite difference scheme, with a contact angle boundary condition prescribed at the moving contact-line. Using the maximum thickness of the profile as the control parameter, we show that there is a direct relationship between the effective Greenspan slip coefficient and the grid-spacing of the numerical scheme used to solve the model equation. In doing so, we show that slip is implicitly built into the numerical scheme through the finite grid-spacing. We also show why converged results with finite film thickness cannot be obtained if slip is ignored.