Equal-sized Droplets Research Articles

Collision dynamics of binary equal-size droplets at high pressures: a focus on stretching separation and reflexive separation

ABSTRACT Stretching separation and reflexive separation are two collision regimes that give rise to satellite droplets, significantly impacting the droplet size distribution. However, the influence of high pressure on these regimes remains unclear. This paper investigates the collision dynamics of binary equal-size droplets under varying ambient pressures using the volume of fluid method and adaptive mesh refinement. The results show that vortices, generated during droplet motion changes or ligament ruptures, facilitate mass transport within the droplet–droplet deformation. Additionally, this paper summarizes four phases of droplet deformation in the separation regimes, which are delayed in stretching separation and advanced in reflexive separation with increasing ambient pressure. Elevated ambient pressure restricts droplet movement, resulting in tougher ligament fractures, smaller satellite droplets, and an expanded coalescence regime boundary. Unlike previous works, this paper incorporates gas kinetic and dissipation energies during droplet collisions at high pressures, revealing that at 7 MPa, the gas dissipation energy is comparable to the liquid dissipation energy and cannot be neglected. To account for these effects, a pressure-dependent formula is integrated into the composite collision model under the Lagrangian framework, offering enhanced predictions of droplet size within the stretching separation regime.

Non-simultaneous impact of droplet pairs on solid surfaces

This study delves into the dynamics of non-simultaneous droplet impacts on solid substrates, focusing on interactions between identical impacting droplets. Comparisons between non-simultaneous and simultaneous impacts are presented to understand the phenomena comprehensively. An in-house-built microcontroller-based droplet generator releases two equal-sized droplets on demand, allowing for simultaneous or non-simultaneous impacts. The interaction between impacting droplets generates an uprising sheet, whose characteristics vary with time lag between impacts, impact Weber number, and inter-droplet spacing. The evolution of central sheet characteristics, lamellae spreading dynamics, splashing mechanism, and secondary atomization is evaluated. Findings reveal that central sheet morphology varies with the time lag between impacts, transitioning from a two-dimensional (2D) “semilunar” sheet (vertical or inclined) with a linear base to a three-dimensional (3D) sheet with a curved base, increasing the probability of secondary atomization. The temporal evolution of the central sheet position, height, and inclination angle is governed by the momentum of spreading lamellae. A novel scaling law for maximum sheet extension and a theoretical expression for surface liquid spread are proposed, consistent with the measurements. The characteristics of secondary droplets generated during non-simultaneous impacts are similar to those from simultaneous impacts, with the size of the secondary droplets being one order of magnitude larger than those expected from isolated single-droplet impacts.

Three-dimensional lattice Boltzmann model with self-tuning equationof state for multiphase flows.

In this work, the recent lattice Boltzmann (LB) model with self-tuning equationof state (EOS) [Huang et al., Phys. Rev. E 99, 023303 (2019)2470-004510.1103/PhysRevE.99.023303] is extended to three dimensions for the simulation of multiphase flows, which is based on the standard three-dimensional 27-velocity lattice and multiple-relaxation-time collision operator. To achieve the self-tuning EOS, the equilibrium moment is devised by introducing a built-in variable, and the collision matrix is improved by introducing some velocity-dependent nondiagonal elements. Meanwhile, the additional cubic terms of velocity in recovering the Newtonian viscous stress are eliminated to enhance the numerical accuracy. For modeling multiphase flows, an attractive pairwise interaction force is introduced to mimic the long-range molecular interaction, and a consistent scheme is proposed to compensate for the ɛ^{3}-order discrete lattice effect. Thermodynamic consistency in a strict sense is established for the multiphase LB model with self-tuning EOS, and the wetting condition is also treated in a thermodynamically consistent manner. As a result, the contact angle, surface tension, and interface thickness can be independently adjusted in the present theoretical framework. Numerical tests are first performed to validate the multiphase LB model with self-tuning EOS and the theoretical analyses of bulk and surface thermodynamics. The collision of equal-sized droplets is then simulated to demonstrate the applicability and effectiveness of the present LB model for multiphase flows.

Generalization of numerical simulation results on the electrical coalescence threshold for two conducting droplets based on non-dimensional parameters

Electrocoalescence, the physical process underlying the demulsification of a dielectric dispersion medium containing small conductive droplets (e.g., water), involves droplet merging at low electric fields and splashing at higher voltages. Understanding the physics of electrocoalescence is crucial for optimizing industrial electrocoalescers. However, mathematical modeling of these complex, multiphysics phenomena is challenging, and many published results are questionable. In this study, we utilized a previously developed reliable model for computing the threshold between electrical coalescence and non-coalescence. We investigate the applicability of dimensionless parameters, such as the Ohnesorge number and Weber electric number, to describe the coalescence threshold for uncharged droplets of equal size. Using COMSOL Multiphysics software, we analyze the dependency of the threshold electric field strength on water droplet radius and establish an equivalent dimensionless relationship. Our findings reveal that a universal Weber number quite accurately describes the threshold over a wide range of droplet radii, regardless of changes in liquid viscosity and inter-electrode gap. Direct mathematical simulations using up-to-date numerical models enable us to determine non-dimensional parameter values corresponding to the threshold electric field strength, providing generalizable results.

A second-order phase field-lattice Boltzmann model with equation of state inputting for two-phase flow containing soluble surfactants

In this paper, we propose a phase field-lattice Boltzmann (LB) model with an equation of state (EOS) inputting for two-phase flow containing soluble surfactants. In this model, both the order parameter for the phase field and the surfactant concentration are described by second-order partial differential equations, along with Navier–Stokes equations for the flow field. Changes in surfactant concentration do not affect the order parameter distribution; hence, an unwanted sharpening effect cannot arise. Most importantly, in the existing models, the EOS of surface tension is determined by posterior simulation tests instead of being directly set as an input parameter before the simulations. Hence, it is difficult to determine the model parameters in practical applications. To address this issue, we systematically develop a fully analytical EOS for surface tension based on the Gibbs–Duhem equation. Subsequently, an approximate explicit form for EOS is provided by utilizing the Jacobi–Gauss quadrature rule. Furthermore, a multiple-relaxation-time LB scheme is utilized to numerically solve the governing equations of three physical fields. Two benchmark examples are simulated to validate the accuracy of the present model. The consistency between the numerical results and the analytical EOS is verified. Moreover, the dynamics of droplets with surfactant in simple shear flow is investigated, unveiling the profound impact of various factors, such as surfactant bulk concentration, capillary number, and viscosity ratio, on single droplet deformation and two equal-sized droplets interaction. A detailed exploration of the fluid mechanism involved in two-phase flow with soluble surfactants is presented.

NUMERICAL INVESTIGATION OF OFF-CENTER COLLISION BETWEEN TWO EQUAL-SIZED WATER DROPLETS

Droplet collision is a basic phenomenon in numerous natural and industrial processes, while the understanding of collision dynamics is still lacking. In this work, a numerical investigation of the offcenter collision of two equal-sized water droplets is performed with the Weber number of 14 to 196 and impact parameter of 0 to 0.8. The incompressible Navier-Stokes equations are solved by the finite volume method. The volume of fluid (VOF) method and adaptive mesh technique are used to capture the gas-liquid interface. First, by comparing with reliable published experimental data, the reliability of the numerical results is verified. Then, the shape evolution for coalescence, reflexive separation, and stretching separation is described in detail. The effect of the Weber number and impact parameter on the collision of two equal-sized water droplets is analyzed. Moreover, the analysis of the surface energy and kinetic energy is conducted for the collision process. Furthermore, the dimensions of ligament and bridge for high-impact parameter stretching separation are presented quantitatively. Finally, the collision outcome for the simulation cases in this work is depicted and discussed. This work is helpful for fundamentally understanding the mechanism of collision dynamics of droplets, as well as applying the droplet collision model to related processes.

The effect of a temperature-dependent viscosity on cooling droplet-droplet collisions

A detailed understanding of the collision dynamics of liquid droplets is relevant to natural phenomena and industrial applications. These droplets could experience temperature changes altering their physical properties, which affect the droplet collisions. As viscosity is one of the relevant physical properties, this study focuses on the effect of temperature on viscosity, with an Arrhenius temperature dependence, of collisions of two equal-sized droplets using the Volume of Fluid Method. The results show that the higher temperature of the droplets leads to an effectively lower viscosity, leading to increased interface oscillations. This leads to the onset of separation at lower Weber numbers as expected. The local cooling droplets will create a local viscosity profiles, which results in the formation of a ridge upon combination of droplets. In addition, the collision outcomes sometimes cannot be explained solely on basis of an effective viscosity, undermining the usefulness of existing collision regime maps.

Quantized vortex nucleation in collisions of superfluid nanoscopic helium droplets at zero temperature.

We address the collision of two superfluid 4He droplets at non-zero initial relative velocities and impact parameters within the framework of liquid 4He time-dependent density functional theory at zero temperature. Despite the small size of these droplets (1000 He atoms in the merged droplet) imposed by computational limitations, we have found that quantized vortices may be readily nucleated for reasonable collision parameters. At variance with head-on collisions, where only vortex rings are produced, collisions with a non-zero impact parameter produce linear vortices that are nucleated at indentations appearing on the surface of the deformed merged droplet. Whereas for equal-size droplets, vortices are produced in pairs, an odd number of vortices can appear when the colliding droplet sizes are different. In all cases, vortices coexist with surface capillary waves. The possibility for collisions to be at the origin of vortex nucleation in experiments involving very large droplets is discussed. An additional surprising result is the observation of the drops coalescence even for grazing and distal collisions at relative velocities as high as 80 and 40m/s, respectively, induced by the long-range van der Waals attraction between the droplets.

Flow and mixing dynamics in face-to-face and rear-end collisions of pairs of equal-sized droplets

In this work, the behaviors of pairs of equal-sized droplets in rear-end and face-to-face collisions were simulated using the improved lattice Boltzmann method for incompressible two-phase flows. First, the time evolution of the droplet shape was investigated by tracing colored particles, and this was compared between the rear-end and face-to-face collisions. For collinear collisions, the droplet shapes in the rear-end collisions were found to be similar to those in the face-to-face collisions. However, the behaviors of the tracer particles were different: the droplets in the rear-end collisions mixed more easily than those in the face-to-face collisions. For offset collisions, it was found that the rolling motion of the coalesced droplet accelerates the mixing inside it in both face-to-face and rear-end collisions. A new index—the total mixing intensity—was introduced, and the droplet mixing can be quantitatively evaluated by calculating its value. The results indicate that the droplet mixing process of a collinear collision can be characterized by the velocity ratio, which is defined as the ratio of the center-of-mass velocity to the relative impact velocity.

Numerical study of head-on collision of two equal-sized compound droplets

Although on-axis collisions between compound droplets are involved in numerous technological applications, no detailed investigation of such collisions is yet available. To address this problem, the present work uses an axisymmetric front-tracking method to numerically explore the dynamics of on-axis collisions of compound droplets that contain one or more inner droplets. Two identical droplets are placed symmetrically on the midplane of a computational domain and made to make contact with an initial colliding velocity. Various parameters such as the Reynolds number Re, the Weber number We, the size of the inner droplets, the interfacial tension ratio, and the eccentricity are considered. Three primary outcomes are observed: complete coalescence (CC), outer coalescence (OC), and rebound (R) for Re = 4–256 and We = 1–128. CC is when both the inner and outer droplets coalesce, whereas OC is when only the outer droplets coalesce. R is when the droplets come into contact and then bounce back. Increasing Re or decreasing We enhances the CC pattern, as does increasing the size of the inner droplets or the interfacial tension ratio. The influence of the initial distance between the droplets is also investigated. Finally, regime diagrams related to these patterns of collision are also presented.

Numerical investigation of impacting heat transfer of binary droplets on superhydrophobic substrates

The droplet impacting on a substrate is widely encountered in daily life and industrial applications. In this paper, the impact dynamics and heat transfer are numerically simulated for both single droplet and binary droplets on a hot superhydrophobic substrate. The dimensionless numerical model is built up through the transient 2D axisymmetric model with a volume of fluid (VOF) model. The effect of the Weber number, Reynolds number, size ratio, contact angle on the spreading dynamics and heat transfer is investigated in details. It is found that the spreading factor and contact time increase with the increasing Weber number. Besides, the maximum spreading factor and contact time of binary droplets are larger than those of a single droplet under the same Reynolds and Weber numbers. The transient heat transfer rate of a single droplet is larger than that of binary droplets impingement due to larger spreading surface area-to-volume. The total input heat of a single droplet is generally larger than that of binary droplets except at large Weber numbers, and the equal-sized binary droplets have the larger total input heat than un-equal-sized binary droplets. For binary droplets impact on a hot superhydrophobic surface, the hot substrate can promote the spreading and retard the receding due to thermo-capillary effect, and thus will enhance heat transfer between the droplet and the hot substrate. These findings may be helpful in gaining insights into the dynamics and heat transfer of binary droplets impact on the hot substrate.

Numerical study on the engulfing behavior between immiscible droplets in a confined shear flow

This study investigates the engulfing behavior between two equal-sized immiscible droplets suspended in a confined shear flow. The finite element analysis is carried out, where a ternary diffuse-interface model and the Stokes equation are coupled. The numerical results of nearly 60 cases are studied to systematically examine the effect of shear rate, surface tension, and confinement on the engulfing behavior. In general, the partial and complete engulfing result in Janus and compound droplets, respectively, when the shear rate is low. It is found that the partial engulfing regime is significantly affected by the shear rate such that a proper condition for the Janus formation becomes narrower as the shear rate increases. We also note that the shell of a compound droplet can be broken by increasing the initial separation or confinement.

Local deformation and coalescence between two equal-sized droplets in a cross-focused microchannel

• Three flow patterns of head-on coalescence of droplets are observed. • Effects of viscosity ratio and surfactant are explored. • Local nipple of droplet is conductive to coalescence. • Influence factors of the nipple are investigated systematically. • A novel model and a predictive equation of the nipple are proposed. The local deformation and the head-on coalescence dynamics of droplets in a cross-focused microchannel are experimentally investigated. The dynamic process of droplets after collision is divided into squeezing stage and decompressing stage, and the flow patterns are divided into squeezing coalescence, decompressing coalescence and non-coalescence based on the moment when the liquid film ruptures. The local nipples caused by the lubrication force in the thin film arise in the decompressing stage and facilitate coalescence. The effects of the dispersed to continuous-phase viscosity ratio and the concentration of surfactant to the dynamic process, operating range of each flow pattern, and the extent of local deformation are investigated, and the predictive equation of the nipple angle is proposed. This research combines the local deformation with coalescence and provides a reference for further understanding the mechanism of droplet coalescence.

Ultrasonic Atomization: New Spray Characterization Approaches

In particle engineering, spray drying is an essential technique that depends on producing sprays, ideally made of equal-sized droplets. Ultrasonic sprays appear to be the best option to achieve it, and Faraday waves are the background mechanism of ultrasonic atomization. The characterization of sprays in this atomization strategy is commonly related to the relation between characteristic drop sizes and the capillary length produced by the forcing frequency of wavy patterns on thin liquid films. However, although this atomization approach is practical when the intended outcome is to produce sprays with droplets of the same size, drop sizes are diverse in real applications. Therefore, adequate characterization of drop size is paramount to establishing the relations between empirical approaches proposed in the literature and the outcome of ultrasonic atomization in actual operating conditions. In this sense, this work explores new approaches to spray characterization applied to ultrasonic sprays produced with different solvents. The first two introduced are the role of redundancy in drop size measurements to avoid resolution limitation in the measurement technique and compare using regular versus variable bin widths when building the histograms of drop size. Another spray characterization tool is the Drop Size Diversity to understand the limitations of characterizing ultrasonic sprays solely based on representative diameters or moments of drop size distributions. The results of ultrasonic spray characterization obtained emphasize: the lack of universality in the relation between a characteristic diameter and the capillary length associated with Faraday waves; the variability on drop size induced by both liquid properties and flow rate on the atomization outcome, namely, lower capillary lengths produce smaller droplets but less efficiently; the higher sensibility of the polydispersion and heterogeneity degrees in Drop Size Diversity when using variable bin widths to build the histograms of drop size; the higher drop size diversity for lower flow rates expressed by the presence of multiple clusters of droplets with similar characteristics leading to multimodal drop size distributions; and the gamma and log-normal mathematical probability functions are the ones that best describe the organization of drop size data in ultrasonic sprays.

Early stage of delayed coalescence of soluble paired droplets: A numerical study

When two sessile droplets of miscible fluids come into contact, the coalescence process can be significantly delayed owing to the competition between the capillary and Marangoni effects. It is important to reveal the mechanism of the deformation and displacement of the sessile droplets at the early stage of the delayed coalescence, which determines the self-stabilized shape and joint motion of the two droplets later on. In this work, we numerically investigate the early-stage dynamics of the delayed coalescence between two sessile droplets of equal size and laden with aqueous mixtures of different solvent mass fractions. A three-dimensional numerical model is adopted based on lubrication theory and is validated by comparison against previous experimental results. Through simulation, we first showed how the concentration transport is coupled with droplet deformation. Then, we explained the underlying mechanism of delayed coalescence by analyzing the liquid bridge numerically and theoretically. A scaling law for the duration of liquid bridge growth is given and agrees well with the numerical results. Finally, the effects of the solubility on the dynamics are investigated. Our study reveals how the capillary and Marangoni effects dominate the flow during the early stage of the delayed coalesce and thus determine its following behavior.

Large eddy simulation and experiment of shear breakup in liquid-liquid jet: Formation of ligaments and droplets

Understanding the shear breakup in jet flows and the formation of droplets from ligaments is important to determine the final droplet size distribution (DSD). The initial droplet size, which affects the final DSD, is considered to be generated by the shear breakup. Large eddy simulation (LES) was performed to investigate the shear breakup in liquid-liquid jet flows. The explicit Volume of Fluid (VOF) model with the geometric reconstruction scheme was used to capture the oil-water interface. The estimated oil distribution including wave peaks, ligaments, droplets and water streaks were compared to the experiments with a good agreement. The estimated DSD matched with the measurements favorably well. In the simulation, the formation of droplets with a smooth and curved surface from ligaments or sheet-like structures was obtained. Different mechanisms were observed along with the shear layer including the formation of droplets from ligament through the capillary forces, breakage of a droplet into smaller ones and attachment of a droplet to a ligament. The destructive shear forces and resisting surface tension forces were quantified on stretching and retracting ligaments. The influence of internal viscous force was found to be negligible due to low oil viscosity. The critical capillary number was found to be larger than 5.0 for ligaments breaking with the shear breakup. The capillary number was below unity for retracting ligaments. The coalescence of two equal-sized droplets was obtained in the shear breakup region. The shear stress magnitude at the contact region increased more than two folds. The total surface area decreased nearly 20% after the coalescence.

Feature Size Modulation in Dewetting of Nanoparticle-Containing Ultrathin Polymer Films

Ultrathin polymer films (thinner than ∼100 nm) undergo spontaneous rupture on a non-wetting surface with the appearance of random holes due to interfacial van der Waal’s interaction or disjoining pressure. These holes grow in size and coalesce into threadlike morphology, which subsequently disintegrate in an isotropic array of nearly equal-sized droplets, the size (dD) and periodicity (λD) of which scales with the initial film thickness (hF). We show that both λD and dD of the dewetted droplets can be modulated by adding trace amounts of nanoparticles (NPs) in the films. While addition of higher proportion of NPs is known to stabilize the films against dewetting, the presence of a lower amount of NPs leads to non-monotonic variation of λD and dD with hF, with the possible creation of miniaturized dewetted features under certain conditions. We also show that λD and dD of the dewetted films depend on whether dewetting is engendered by thermal annealing of the film beyond the glass transition temperature of the constituent polymer (TG) or by exposing the film to solvent vapor (SV). Our findings reveal that both λD and dD are much larger when the film (both with and without particles) is dewetted in the SV atmosphere, arguably due to penetration of the solvent into the film matrix, resulting in an increase in the effective thickness of the film during dewetting. We show that in SV-mediated dewetting, the much faster dynamics is attributed to solvent penetration through the film and its wetting the substrate, rather than a drastic drop in viscosity. We also observe that SV exposure not only leads to much faster dewetting dynamics, but it also successfully engenders dewetting in films with higher hF, which remain stable when annealed thermally.

Binary collision of CMAS droplets—Part I: Equal-sized droplets

Abstract

Ferro-hydrodynamic interactions between ferrofluid droplet pairs in simple shear flows

A numerical investigation on the interactions between a pair of equal-sized ferrofluid droplets in a simple shear flow subjected to uniform magnetic fields is carried out in this paper. Our numerical model utilizes a finite element method based level set algorithm to trace the topological changes along the droplet interface in two phase flows by coupling the flow and magnetic fields. Systematic numerical simulations with a well-resolved droplet interface are carried out in the Stokes flow limit (i.e., Re=0.03) to explore the effects of magnetic field direction, strength, and initial vertical offset on the pairwise interactions between the droplets. The findings indicate that in a solitary shear flow, a critical capillary number Cacr exists, beyond which the droplets exhibit separation through a sliding-over motion, instead of coalescence. Applying a uniform magnetic field along α=0° (i.e., parallel to the flow direction) results in a faster coalescence between the droplets. At α=90°, a critical magnetic bond number Bocr appears where the droplets show a reversing motion behavior, instead of a passing-over motion leading to coalescence. In contrast, at α=45°, the droplets separate away from each other, following a reversal motion at a capillary number where the droplets usually coalesce with each other in a solitary shear flow. Moreover, the effect of initial vertical offset on the interaction behavior of the droplets is examined, and it is found that the collision event is dependent on the initial vertical separation distance between the droplets. Increasing the vertical separation distance along α=0° initiates faster coalescence, while at α=90°, coalescence is delayed. Furthermore, at α=45°, the droplets undergo a reversing motion and experience larger migration at increased vertical separation distances.

二元海水液滴对心碰撞过程数值模拟

Numerical simulations of head-on binary collisions between equal-size seawater droplets were conducted with the volume of fluid (VOF) method and the adaptive grid technique, to investigate the collision physics and mechanics of seawater droplets in shower cooling towers. Trial simulations of head-on collisions of tetradecane droplets in nitrogen medium were performed firstly to give results in good agreement with those of the previous experiments. The binary collisions of equal-size seawater droplets were simulated under room temperature and normal pressure conditions. The stream field and flow mechanism of seawater droplets collision were analyzed, and the effects of droplet diameters and concentrations on the collision process were studied. Two different types of collision outcomes were identified: the coalescence and the reflexive separation, for which the critical Weber numbers were given. The schematic diagrams for various head-on collision regimes of seawater droplets at various Ohnesorge numbers were also obtained.