Saffman and Turner (1957) argued that the collision rate for droplets in turbulence increases as the turbulent strain rate increases. But the numerical simulations of Dhanasekaran et al. (2021) in a steady straining flow show that the Saffman-Turner model is oversimplified because it neglects droplet-droplet interactions. These result in a complex dependence of the collision rate on the strain rate and on the differential settling speed. Here we show that this dependence is explained by a sequence of bifurcations in the collision dynamics. We compute the bifurcation diagram when strain is aligned with gravity, and show that it yields important insights into the collision dynamics. First, the steady-state collision rate remains non-zero in the limit Kn $\to0$, contrary to the common assumption that the collision rate tends to zero in this limit (Kn is a non-dimensional measure of the mean free path of air). Second, the non-monotonic dependence of the collision rate on the differential settling speed is explained by a grazing bifurcation. Third, the bifurcation analysis explains why so-called "closed trajectories" appear and disappear. Fourth, our analysis predicts strong spatial clustering near certain saddle points, where the effects of strain and differential settling cancel
We investigate the evaporation of a two-dimensional droplet on a solid surface. The solid is flat but with smooth chemical variations that lead to a space-dependent local contact angle. We perform a detailed bifurcation analysis of the equilibrium properties of the droplet as its size is changed, observing the emergence of a hierarchy of bifurcations that strongly depends on the particular underlying chemical pattern. Symmetric and periodic patterns lead to a sequence of pitchfork and saddle-node bifurcations that make stable solutions to become saddle nodes. Under dynamic conditions, this cha…
To understand the behavior of composite fluid particles such as nucleated cells and double-emulsions in flow, we study a finite-size particle encapsulated in a deforming droplet under shear flow as a model system. In addition to its concentric particle-droplet configuration, we numerically explore other eccentric and time-periodic equilibrium solutions, which emerge spontaneously via supercritical pitchfork and Hopf bifurcations. We present the loci of these solutions around the codimenstion-two point. We adopt a dynamical system approach to model and characterize the coupled behavior of the t…
Recent studies have demonstrated the significant roles of droplet self-spin motion in affecting the head-on collision of binary droplets. In this paper, we present a computational study by using the Volume-of-Fluid (VOF) method to investigate the spin-affected droplet separation of off-center collisions, which are more probable in reality and phenomenologically richer than head-on collisions. Different separation modes are identified through a parametric study with varying spinning speed and impact parameter. A prominent finding is that increasing the droplet spinning speed tends to suppress t…
Effects of correlated collisions and intermittency on the growth of lucky droplets
Max Planck Institute for Dynamics and Self-Organization（马克斯·普朗克动力学与自组织研究所）；Faculty of Physics, University of Göttingen（哥廷根大学物理学院）；Theoretical Physics I, University of Bayreuth（拜罗伊特大学理论物理学I）
This paper makes a correction to the collision rates of small droplets in turbulent fluid derived by Saffman and Turner(1956). Not only the distortion but also the rotation of the fluid is taken into account between two close droplets. A rotation reference is fixed on one drop, and the fluxes of the other drops moving towards the fixed one are carried out based on this new reference. The behaviors of turbulent flow are analyzed within the smallest eddies under the rotation reference, and a correction is made to the collision rates by multiplying a factor sqrt(2).
Effects of spinning motion on the bouncing and coalescence between a spinning droplet and a non-spinning droplet undergoing the head-on collision were numerically studied by using a Volume-of-Fluid method. A prominent discovery is that the spinning droplet can induce significant non-axisymmetric flow features for the head-on collision of equal-size droplets composed of the same liquid. Specifically, a non-axisymmetric bouncing was observed, and it is caused by the conversion of the spinning angular momentum into the orbital angular momentum. This process is accompanied by the rotational kineti…
We investigate the effect of electrical charge on collisions of hydrodynamically interacting, micron-sized water droplets settling through quiescent air. The relative dynamics of charged droplets is determined by hydrodynamic interactions, particle and fluid inertia, and electrostatic forces. We analyse the resulting relative dynamics of oppositely charged droplets by determining its fixed points and their stable and unstable manifolds. The stable manifold of a saddle point forms a separatrix that separates colliding trajectories from those that do not collide. The qualitative conclusions from…
On Free Moving Micron-Sized Droplet-Particle Collisions
School of Chemical and Process Engineering, University of Leeds（利兹大学化学与工艺工程学院）；School of Natural Sciences, Department of Mathematics, University of Manchester（曼彻斯特大学自然科学学院数学系）；School of Mathematics, Statistics, Chemistry and Physics, Murdoch University（默多克大学数学、统计、化学和物理学院）
Water droplets coalesce into larger ones in atmospheric clouds to form rain. But droplets on collision courses do not always coalesce due to the cushioning effects of the air between them. The extent to which these so-called hydrodynamic interactions reduce coalescence rates is embodied in the collision efficiency, which is often small and is not generally known. In order to characterize the mechanisms that determine the collision efficiency, we exploited new time-resolved three-dimensional droplet tracking techniques to measure the positions of cloud droplet pairs settling through quiescent a…
We study theoretically and numerically spatial distribution and collision rate of droplets that sediment in homogeneous isotropic Navier-Stokes turbulence. It is assumed that typical turbulent accelerations of fluid particles are much smaller than gravity. This was shown to imply that the particles interact weakly with individual vortices and, as a result, form a smooth flow in most of the space. In weakly intermittent turbulence with moderate Reynolds number, rare regions where the flow breaks down can be neglected in the calculation of space averaged rate of droplet collisions. However, incr…
The off-center collision of binary bouncing droplets of equal size was studied numerically by a volume-of-fluid (VOF) method with two marker functions, which has been validated by comparing with available experimental results. A non-monotonic kinetic energy recovery with varying impact parameters was found based on the energy budget analysis. This can be explained by the prolonged entanglement time and the enhanced internal-flow-induced viscous dissipation for bouncing droplets at intermediate impact parameters, compared with those at smaller or larger impact parameters. The universality of th…
A model that predicts the outcome of collisions between droplets and particles in terms of the distribution of the droplet volume post-collision is lacking, in contrast to the case for droplet-droplet interactions. Taking existing models that successfully predict the outcomes (coalescence, stretching or reflexive separation) and post-separation characteristics (sizes, numbers and velocities of the resulting droplets) of droplet-droplet collisions and adapting them to take into account an inextensible, non-deformable particle with varying wettability characteristics, a new model is presented fo…
It was previously shown that the superdroplet algorithm for modeling the collision-coalescence process can faithfully represent mean droplet growth in turbulent clouds. But an open question is how accurately the superdroplet algorithm accounts for fluctuations in the collisional aggregation process. Such fluctuations are particularly important in dilute suspensions. Even in the absence of turbulence, Poisson fluctuations of collision times in dilute suspensions may result in substantial variations in the growth process, resulting in a broad distribution of growth times to reach a certain dropl…
We develop a data-driven characterization of the pilot-wave hydrodynamic system in which a bouncing droplet self-propels along the surface of a vibrating bath. We consider drop motion in a confined one-dimensional geometry, and apply the {\em Dynamic mode decomposition} (DMD) in order to characterize the evolution of the wave field as the bath's vibrational acceleration is increased progressively. DMD provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of spatio-temporal data. The DMD characterization of the wave field yields a fresh perspect…
Active droplets emit a chemical solute at their surface that modifies their local interfacial tension. They exploit the nonlinear coupling of the convective transport of solute to the resulting Marangoni flows to self-propel. Such swimming droplets are by nature anti-chemotactic and are repelled by their own chemical wake or their neighbours. The rebound dynamics resulting from pairwise droplet interactions was recently analysed in detail for purely head-on collisions using a specific bispherical approach. Here, we extend this analysis and propose a reduced model of a generic collision to char…
Droplet encapsulating bubble: Investigation of droplet spreading dynamics and bubble encapsulation time
School of Mechanical Engineering, Iran University of Science and Technology（伊朗理工大学机械工程学院）；Department of Mechanical Engineering, University of California, Riverside（加州大学河滨分校机械工程系）
We use direct numerical simulation to study the temporal evolution of a perturbation localized on the turbulent layer that typically separates a cloud from the surrounding clear air. Across this shearless layer, a turbulent kinetic energy gradient naturally forms. Here, a finite perturbation in the form of a local initial temperature fluctuation is applied to simulate a hydrodynamic instability inside the background turbulent air flow. A numerical initial value problem for two diametrically opposite types of drop population distributions is then solved. Specifically, we consider a mono-dispers…
The head-on collision of two identical droplets is investigated based on the BGK-Boltzmann equation. Gauss-Hermite quadratures with different degree of precision are used to solve the kinetic equation, so that the continuum (solution truncated at the Navier-Stokes order) and non-continuum (rarefied gas dynamics) solutions can be compared. When the kinetic equation is solved with adequate accuracy, prominent variations of the vertical velocity (the collision is in the horizontal direction), the viscous stress components, and droplet morphology are observed during the formation of liquid bridge,…
Chemically-active droplets exhibit complex avoiding trajectories. While heterogeneity is inevitable in active matter experiments, it is mostly overlooked in their modelling. Exploiting its geometric simplicity, we fully-resolve the head-on collision of two swimming droplets of different radii and demonstrate that even a small contrast in size critically conditions their collision and subsequent dynamics. We identify three fundamentally-different regimes. The resulting high sensitivity of pairwise collisions is expected to profoundly affect their collective dynamics.
There are many applications of multiphase flow in important fields such as biological, chemical and power processes. Bubble coalescence is of a significant importance in simulating multiphase fluid flows. Weber number ($W_e$), Reynolds number (Re) and collision parameter play important role in the coalescence of bubbles. In the present work, front-tracking method is applied to simulate bubble coalescence. Moreover, the results are presented for different collision parameters and changes in the coalescence of bubbles are discussed
We introduce a new Lagrangian particle tracking algorithm that tracks particles in three dimensions to separations between trajectories approaching contact. The algorithm also detects low Weber number binary collisions that result in coalescence as well as droplet break-up. Particles are identified in two-dimensional high-resolution digital images by finding sets of circles to describe the edge of each body. This allows identification of particles that overlap in projection by over 80% even for noisy images and without invoking additional temporal data. The algorithm builds trajectories from t…
New Approaches for the Interface Reconstruction and Surface Force Computation for Volume of Fluid Simulations of Droplet Interaction of Immiscible Liquids
The collision of water and elastic liquid drops with a thin cylinder (thread) is studied. The droplet flight trajectory and the cylinder axis are mutually perpendicular. Attention is focused on the difference between collisions of water drops and drops of elastic fluids. In the experiments, the drop diameter was 3 mm, the diameter of horizontal stainless steel cylinders was 0.4 and 0.8 mm. The drops were formed by slowly pumping liquid through a vertical stainless steel capillary with an outer diameter of 0.8 mm, from which droplets were periodically detached under the action of gravity. The d…
We investigate the role of intense vortical structures, similar to those in a turbulent flow, in enhancing collisions (and coalescences) which lead to the formation of large aggregates in particle-laden flows. By using a Burgers vortex model, we show, in particular, that vortex stretching significantly enhances sharp inhomogeneities in spatial particle densities, related to the rapid ejection of particles from intense vortices. Furthermore our work shows how such spatial clustering leads to an enhancement of collision rates and extreme statistics of collisional velocities. We also study the ro…
In-air microfluidics became a new method for technical production processes with ultra-high throughput formerly performed in micro channels. Direct Numerical Simulations (DNS) provide a valuable contribution for the fundamental understanding of multiphase flow and later application design. This chapter presents a feasibility study with first DNS results of droplet-jet collisions of immiscible liquids using the in-house software Free Surface 3D (FS3D). Two cases were investigated with a setup comparable to experiments by Baumgartner et al. [1], where a droplet chain of a glycerol solution hits …
Predicting the outcome of liquid droplet collisions is an extensively studied phenomenon but the current physics based models for predicting the outcomes are poor (accuracy $\approx 43\%$). The key weakness of these models is their limited complexity. They only account for 3 features while there are many more relevant features that go unaccounted for. This limitation of traditional models can be easily overcome through machine learning modeling of the problem. In an ML setting this problem directly translates to a classification problem with 4 classes. Here we compile a large labelled dataset …
Active droplets swim as a result of the nonlinear advective coupling of the distribution of chemical species they consume or release with the Marangoni flows created by their non-uniform surface distribution. Most existing models focus on the self-propulsion of a single droplet in an unbounded fluid, which arises when diffusion is slow enough (i.e. beyond a critical Péclet number, $\mbox{Pe}_c$). Despite its experimental relevance, the coupled dynamics of multiple droplets and/or collision with a wall remains mostly unexplored. Using a novel approach based on a moving fitted bispherical grid, …
We consider the motion of a droplet bouncing on a vibrating bath of the same fluid in the presence of a central potential. We formulate a rotation symmetry-reduced description of this system, which allows for the straightforward application of dynamical systems theory tools. As an illustration of the utility of the symmetry reduction, we apply it to a model of the pilot-wave system with a central harmonic force. We begin our analysis by identifying local bifurcations and the onset of chaos. We then describe the emergence of chaotic regions and their merging bifurcations, which lead to the form…
Post impingement morphology and dynamics of water droplets on convex cylindrical surfaces has been explored experimentally. Droplet impact and post-impact feature studies have been conducted on hydrophillic and superhydrophobic cylindrical surfaces. Effects of the impact Weber number and target-to-drop diameter ratio have been studied. The post-impact hydrodynamics have been quantified using the wetting fraction, the spreading factor and nondimensional film thickness at the north pole of the target. The observations reveal that the wetting fraction and spread factor increases with an increase …
Employing time-dependent density-functional theory, we have studied dynamical equilibration and binary head-on collisions of quantum droplets made of a $^{39}$K-$^{39}$K Bose mixture. The phase space of collision outcomes is extensively explored by performing fully three-dimensional calculations with effective single-component QMC based and two-components LHY-corrected mean-field functionals. We exhaustively explored the important effect -- not considered in previous studies -- of the initial population ratio deviating from the optimal mean-field value $N_2/N_1 = \sqrt{a_{11} / a_{22}}$. Both …
