Dependency of Contact Angles on Three-Phase Contact Line: A Review

The wetted area of a sessile droplet on a practical substrate is limited by the three-phase contact line and characterized by contact angle, contact radius and drop height. Although, contact angles of droplets have been studied for more than two hundred years, there are still some unanswered questions. In the last two decades, it was experimentally proven that the advancing and receding contact angles, and the contact angle hysteresis of rough and chemically heterogeneous surfaces, are determined by interactions of the liquid and the solid at the three-phase contact line alone, and the interfacial area within the contact perimeter is irrelevant. However, confusion and misunderstanding still exist in this field regarding the relationship between contact angle and surface roughness and chemical heterogeneity. An extensive review was published on the debate for the dependence of apparent contact angles on drop contact area or the three-phase contact line in 2014. Following this old review, several new articles were published on the same subject. This article presents a review of the novel articles (mostly published after 2014 to present) on the dependency of contact angles on the three-phase contact line, after a short summary is given for this long-lasting debate. Recently, some improvements have been made; for example, a relationship of the apparent contact angle with the properties of the three-phase line was obtained by replacing the solid–vapor interfacial tension term, γSV, with a string tension term containing the edge energy, γSLV, and curvature of the triple contact line, km, terms. In addition, a novel Gibbsian thermodynamics composite system was developed for a liquid drop resting on a heterogeneous multiphase and also on a homogeneous rough solid substrate at equilibrium conditions, and this approach led to the same conclusions given above. Moreover, some publications on the line energy concept along the three-phase contact line, and on the “modified” Cassie equations were also examined in this review.

Droplet Impact on a Moving Thin Film with Pseudopotential Lattice Boltzmann Method

The pseudopotential lattice Boltzmann method (LBM) with a tunable surface tension term is applied to study a droplet impact on a moving thin film. The Re effects of dimensionless parameters on the upstream and downstream crown evolution are studied, including Reynolds number (Re), Weber number (We), liquid film thickness, and horizontal velocity of the liquid film. The movement of the liquid film causes the asymmetry development of the upstream and downstream crown. Both the instability of upstream and downstream crowns increases with the increase of Re and We, and the upstream crown becomes more prone to break up. And a critical value of film thickness exists with the height of the upstream and downstream liquid crowns reaches the maximum value. And the velocity of liquid film restrains the development of the height of the upstream and downstream crowns, but it promotes the growth of the crown radius.

SPH Simulation of an Air-Assisted Atomizer Operating at High Pressure: Influence of Non-Newtonian Effects

A twin-fluid atomizer configuration is predicted by means of the 2D weakly-compressible Smooth Particle Hydrodynamics (SPH) method and compared to experiments. The setup consists of an axial liquid jet fragmented by a co-flowing high-speed air stream (Ug ≈ 60 m/s) in a pressurized atmosphere up to 11 bar (abs.). Two types of liquid are investigated: a viscous Newtonian liquid (μl = 200 mPas) obtained with a glycerol/water mixture and a viscous non-Newtonian liquid (μl, apparent. ≈ 150 mPas) obtained with a carboxymethyl cellulose (CMC) solution. 3D effects are taken into account in the 2D code by introducing (i) a surface tension term, (ii) a cylindrical viscosity operator and (iii) a modified velocity accounting for the divergence of the volume in the radial direction. The numerical results at high pressure show a good qualitative agreement with experiment, i.e. a correct transition of the atomization regimes with regard to the pressure, and similar dynamics and length scales of the generated ligaments. The predicted frequency of the Kelvin-Helmholtz instability needs a correction factor of 2 to be globally well recovered with the Newtonian liquid. The simulation of the non-Newtonian liquid at high pressure shows a similar breakup regime with finer droplets compared to Newtonian liquids while the simulation at atmospheric pressure shows an apparent viscosity similar to the experiment.

Analysis of a model for foam improved oil recovery

During improved oil recovery (IOR), gas may be introduced into a porous reservoir filled with surfactant solution in order to form foam. A model for the evolution of the resulting foam front known as ‘pressure-driven growth’ is analysed. An asymptotic solution of this model for long times is derived that shows that foam can propagate indefinitely into the reservoir without gravity override. Moreover, ‘pressure-driven growth’ is shown to correspond to a special case of the more general ‘viscous froth’ model. In particular, it is a singular limit of the viscous froth, corresponding to the elimination of a surface tension term, permitting sharp corners and kinks in the predicted shape of the front. Sharp corners tend to develop from concave regions of the front. The principal solution of interest has a convex front, however, so that although this solution itself has no sharp corners (except for some kinks that develop spuriously owing to errors in a numerical scheme), it is found nevertheless to exhibit milder singularities in front curvature, as the long-time asymptotic analytical solution makes clear. Numerical schemes for the evolving front shape which perform robustly (avoiding the development of spurious kinks) are also developed. Generalisations of this solution to geologically heterogeneous reservoirs should exhibit concavities and/or sharp corner singularities as an inherent part of their evolution: propagation of fronts containing such ‘inherent’ singularities can be readily incorporated into these numerical schemes.

A Droplet Model in Dropwise Condensation With the Presence of Non-Condensable Gases

A droplet model is proposed with respect to molecular clustering to describe the state of steam molecules before condensing on the cooled solid surface in steam condensation process, and also the model is used to account for the mechanism of steam dropwise condensation in the presence of non-condensable gases (NCG). The mathematical model is presented based on the Dillmann and Meier’s homogeneous nucleation theory. Correction of surface tension term is conducted to match the physical and chemical characteristics of condensation process, and the heat transfer model considering the effect of interfacial effects was used to calculate the mean temperature of clusters. The model predicted results of Gibbs free energy at different subcooling degrees and different saturated temperatures were given. And the ratio of heat transfer coefficient with to that without NCG with different fractions of NCG was also obtained. The results show that the non-condensable gases reduce the condensation rates in the same way as shown by experimental results in the literature. That confirms the validity of the model.

INFLUENCE OF SURFACE TENSION AND BOTTOM TOPOGRAPHY ON INTERNAL WAVES

Following the global strategy introduced recently by Bona, Lannes and Saut in Ref. 7, we derive here in a systematic way, and for a large class of scaling regimes, asymptotic models for the propagation of internal waves at the interface between two layers of immiscrible fluids of different densities, under the rigid lid assumption, the presence of surface tension and with uneven bottoms. The full (Euler) model for this situation is reduced to a system of evolution equations posed spatially on ℝd, d = 1, 2, which involve two nonlocal operators. The different asymptotic models are obtained by expanding the nonlocal operators and the surface tension term with respect to suitable small parameters that depend variously on the amplitude, wavelengths and depth ratio of the two layers. Furthermore, the consistency of these asymptotic systems with the full Euler equations is established.

An integrable equation governing short waves in a long-wave model

The dynamics of a nonlinear and dispersive long surface capillary-gravity wave model equation is studied analytically in its short-wave limit. We exhibit its Lax pair and some non-trivial conserved quantities. Through a change of functions, an unexpected connection between this classical surface water-wave model and the sine-Gordon (or sinh-Gordon) equation is established. Numerical and analytical studies show that in spite of integrability their solutions can develop singularities and multivaluedness in finite time. These features can be traced to the fact that the surface tension term in the energy involves second-order derivatives. It would be interesting to see in an experiment whether such singularities actually appear, for which surface tension would be specifically responsible.

The Hybrid Particle-Level-Set Method Applied to Two-Phase Flows

The hybrid particle-level-set method was proposed by Enright et al. (2002) for improving the efficiency and accuracy of the original level-set method. In this paper, we examine the discretization of the surface-tension term and of the reinitialization procedure with respect to their effect on the mass-conservation properties of the overall method. We apply the hybrid particle-level-set method for the computation of two-phase flows and assess its performance with different parameter choices in comparison with the original level-set method.