Session A18: Drops: General

Against the wind: pendant drop periodicity in ascending flow

When exposed to an ascending flow, pendant drops oscillate at magnitudes determined by windspeed, drop diameter, and needle diameter. In this study, we investigate the retention stability and oscillations of pendant drops in a vertical wind tunnel. Oscillation is captured by a high-speed camera for a drop Reynolds number Re = 100-3000. Drops at Re ⪅ 1000 oscillate up to 12 times that of the drops with high Re. Increasing windspeed enables larger volume drops to remain attached to the needles above Re = 500. We categorize drop dynamics into 7 behavioral modes according to the plane of rotation and deformation of shape. Video frame aggregation permits the determination of a static, characteristic shape of our highly dynamic drops. Such a shape provides a hydraulic diameter and the evaluation of the volume swept by the oscillating drops with time. The maximum swept volume per unit drop volume occurs at Re = 600, corresponding to a peak in angular velocity.    

The shape and dynamics of an axisymmetric pendant drop within uniform flow

Droplet shape and dynamics pose not only a canonical problem of multiphase phenomena, but also hold significant importance in various industrial applications. The theory of pendant drops is essential in engineering systems such as inkjet printing, surface cleaning, and optical tensiometry, as well as experimental techniques involving droplet-particle interactions. Despite extensive research on the hydrostatics of pendant drops under gravity, the influence of outer flow disturbances remains poorly understood. This study aims to incorporate aerodynamic effects in the description of pendant drop behavior. As a simplistic approach, irrotational flow parallel to the axis of a symmetric drop is obtained by means of a distribution of singularity elements within the drop. The equilibrium shape of the drop is subsequently determined from a numerical model, wherein the flow field is coupled with the Young-Laplace equation. The model is compared with droplet images acquired from high-speed shadowgraph in a vertical wind tunnel. Results are analyzed based on relevant dimensionless numbers, and dynamic behavior including droplet deformation, instability, and oscillation are also discussed.   

Modeling the capture of fine particles by swelling emulsions

Fast agglomeration by emulsion binders to capture fine, hydrophobic particles was recently developed as an alternative to froth flotation. It involves mixing a suspension of particles in water with a saltwater-in-oil emulsion. The droplets in the emulsion phase swell due to an osmotic flux caused by the presence of salt. In this work, we developed a framework to model this particle-capture method, consisting of (i) using mobility dynamics to investigate the binary interactions between swelling droplets and particles, (ii) an extension of classical collision theory to moving interfaces, which we use to calculate the capture rate of fine particles, (iii) a continuous diffusion-limited swelling model for a single droplet, and (iv) a permeation network model for the swelling of a droplet agglomerate, for which the mass transfer is limited by the permeation of water within the internal oil layers. Our results indicate that drop swelling substantially influences particle capture, even for low swelling rates and high Péclet numbers, where the swelling of a single droplet is governed by a concentration boundary layer and displays a universal behavior at short times. For droplet agglomerates, a similar trend for swelling is observed, and structure-dependent effective diffusivities and Péclet numbers can be determined by matching the results with the effective-diffusion model.   

A Theory of the Size Distribution of Raindrops

Rain, as a natural precipitation phenomenon, has captivated human curiosity for millennia. To gain a deeper understanding of rain characteristics, extensive research has been conducted on raindrop diameters through systematic experimental and theoretical studies. In this work (an advancement over an earlier model we shared in 2021) we combine a deterministic approach based on thermodynamics and fluid dynamics with a stochastic methodology. A key aspect of our approach involves utilizing energy balance, dimensional analysis, and scale analysis principles to ascertain the possible, impossible, and expected raindrop diameters as a function of rain intensity. By considering the first law of thermodynamics, accounting for the work of fluid dynamic forces, and augmenting it with a stochastic description of random disturbances originating from the raindrop ensemble and atmospheric flow, we can predict the functional form of ground-level size distributions of raindrops. Our results exhibit good agreement with numerous published measurements for various rain intensities across diverse global locations. The versatility of our approach holds promise for broad applications in climate models, sprays, aerosols, bubbles, particles, and even rainfall on other planets.   

On the dynamics of freezing sessile droplets: Frost halo formation

The freezing of a supercooled sessile droplet unveils fascinating physics, characterized by the progression of the liquid-ice interface, the formation of a cusp-like morphology at the tip of the droplet and often by the emergence of a frost halo on the underlying substrate. Here, we focus on the latter which has been verified experimentally but has not been theoretically explored. We consider the freezing of a thin volatile sessile droplet on a sub-cooled substrate and develop a comprehensive model based on lubrication theory. It is demonstrated that the vapor that is present in the atmosphere comes into contact with the sub-cooled substrate resulting to considerable condensation near the contact line at early stages of the freezing process, forming a frost halo in the droplet periphery. Liquid volatility also influences the shape of the liquid-ice interface and aspect ratio of the droplet, as it affects the total remaining mass of ice and the thermal profile inside the droplet. The latter is determined by the conductivity of the droplet, the thermal resistance of the substrate, convection to the atmosphere and, crucially, evaporative cooling. We conduct a detailed parametric study and discuss about the effects of ambient conditions, the latent heat of vaporization and melting, and the thermal properties of liquid, ice and the substrate on the frost halo formation and the dynamics of the freezing process in general.   

Rectilinear magnetophoresic oscillation of oil droplet dispersed in immiscible paramagnetic carrier phase

Unbounded single oil droplet in paramagnetic rare-earth salt solution is studied superposing a magnetic field. The resulting Bond and Morton number is located in a parametric region where the droplet moves rectilinearly when the motion is driven by gravity along. When released from an initially quiescent carrier phase, the dominating buoyancy, counter acting the hydrodynamic forces, drives the droplet moving upward until it reaches the near field of a permanent magnetic assembly. At a critical position, the Kelvin force supersedes buoyancy and droplet starts to oscillate periodically. A hydrodynamically quiescent end state is expected after the momentum is fully damped and the droplet settles at an equilibrium position where the Kelvin force and buoyancy is equivalent. The damping kinematics allows us to quantify the history term. We find, surprisingly, the hydrodynamic force where the memory kernel decays faster, instead of t ^-1/2,  following an e^-t with a pre-factor scaled by external viscous dissipation time scale and hydrodynamic length scale. Furthermore, a refined Kelvin force formulation is proposed to improve the current volumeless Lagrangian force model. The latter is routinely used for magnetophoresis modelling where an accurate prediction of the motion of the dispersed phase in magnetic field is of primary importance.   

Elasto-capillary wetting on deformable solids

Elasto-capillary wetting is a fluid-structure interaction (FSI) phenomenon, wherein the solid deformation is driven by the capillary forces at fluid-fluid interfaces. Elasto-capillary wetting is relevant in various micro- and nano-scale applications, such as inkjet printing, microfabrication, microfluidics, and biophysics. Here, we develop a high-fidelity elasto-capillary FSI model that can simulate systems with three immiscible fluids and a nonlinear elastic solid. We use this model to investigate static and dynamic elasto-capillary wetting problems, including the spontaneous droplet transport of both single and multi-component fluids on solids with space-varying stiffness (droplet durotaxis). Our findings indicate that the dynamics of droplets can be effectively controlled by varying the wettability and solid stiffness. Our results suggest more efficient ways to transport droplets in various applications, such as self-cleaning of surfaces and energy harvesting.   

Drop Self-Propulsion on Low Friction Lubricant Surfaces

In recent years, lubricated surfaces have raised significant attention because of their interesting wetting properties. Amongst these surface types, Slippery Liquid-Infused Porous Surfaces (SLIPS) have many applications in heat exchange phenomena, droplet motion, microfluidics and other engineering or medical applications. Recent work has focused on the modelling of the behaviour of water droplets on SLIPS, considering the effect of the lubricant choice, the droplet equilibrium condition through the liquid Young’s law contact angle, surface hysteresis and evaporation phenomena. Very few works have focused on the possibility of stabilizing two different lubricants of the same surface for the creation of a composite SLIP surface. We present a simple method for creating composite SLIP surfaces, with two different lubricants stabilized on the surface. Composite SLIP surfaces opens up interesting possibilities related to having a surface with different wetting properties and virtually no hysteresis. We exemplify this with a study on the motion of a water droplet on a composite SLIP surface having a wedge-like pattern with one lubricant inside and one outside the wedge. We observe droplet motion driven by the difference in wettability of the two lubricants to water, with droplets travelling distances of the order of a centimetre. A theoretical model for the droplet position along the wedge depending on the wedge angle and the lubricant selection is proposed. This provides excellent fits to the experimental data for droplet position with time.    

Experiments on deformable oil droplets in turbulence

In this work, we experimentally investigate the momentum flux and breakup frequency in low-concentration oil in water mixtures. We use six high-speed cameras to capture the droplets dynamics in homogeneous and isotropic turbulence (HIT). The experiments are done in the TU Wien Jet-Stirred Turbulence Tunnel: two opposite jet arrays generate local HIT with zero-mean flow within an octagonal test section. The Taylor-Reynolds number ranges from 100 to 300, featuring energy dissipation rates of Ο(0.1 m²s^-3). We control the turbulence intensity and the droplets diameter to set the Weber number (We) to Ο(1), while changing the droplets viscosity tunes the Ohnesorge number (Oh) to Ο(0.1–10). Different methyl phenyl silicone oils are used, with the kinematic viscosity ranging from 20 to 3000 cSt; the oils are density-matched with the carrier phase (distilled water). To assess the complex interplay between the two phases, we simultaneously measure the interface shape and the surrounding flow structures. The interface reconstruction is based on the virtual-camera method (Masuk et al. 2019), and flow statistics are obtained via Shake-the-Box. As results, we discuss the 3D reconstruction of the filaments formed before the breakup, how the droplets inner viscosity affects the deformation rate, and how the breakup frequency scales with We/(1+Oh).