Impact force and spreading characteristics of droplet impact on cylindrical surfaces
State Key Laboratory of Engines, Tianjin University（天津大学内燃机国家重点实验室）；National Industry-Education Platform of Energy Storage, Tianjin University（天津大学国家储能产教融合平台）
Droplet impact phenomena are ubiquitous in both nature and industry. Existing studies of droplet impact have focused on the kinematics of droplet impact on flat surfaces, whereas research on cylindrical surfaces remains relatively limited, particularly from a force-based perspective. Here, droplet impact on cylindrical surfaces is studied by numerical simulation, with particular attention to the spreading behavior and impact force acting on the wall. In the deposition mode, a single peak appears in the impact force curve, which corresponds to the rapid transfer of the droplet's initial momentum. In the rebound mode, two distinct peaks are observed, and the second peak arises from the reaction force during the retraction process. Increasing the surface wettability causes the asymmetry coefficient, the ratio of the maximum spreading lengths in the azimuthal and axial directions of the cylinder, to first decrease and then gradually approach a constant, while the effect of the surface wettability on the initial impact force is negligible. As the Weber number We increases, the first dimensionless peak of the impact force $F_{p1}^{*}$ approaches a constant, and the relationship can be expressed as $F_{p1}^{*}=β_1+{β_2}{{We}^{-1}}$ (where ${β}_1$ and ${β}_2$ are constants). The dimensionless maximum spreading area, dimensionless maximum spreading length, dimensionless maximum spreading angle, and asymmetry coefficient all exhibit power-law relationships with the Weber and Ohnesorge numbers. Furthermore, an increase in the diameter ratio of the cylinder and the droplet leads to a reduction in the asymmetry coefficient and an increase in the first dimensionless peak of the impact force.
Existing energy balance models, which estimate maximum droplet spreading, insufficiently capture the droplet spreading from low to high Weber and Reynolds numbers and contact angles. This is mainly due to the simplified definition of the viscous dissipation term and incomplete modeling of the maximum spreading time. In this particular research, droplet impact on a smooth sapphire surface is studied for seven glycerol concentrations between 0% - 100%, and 294 data points are acquired using high-speed photography. Fluid properties such as density, surface tension, and viscosity are also measured…
We examine theoretically the effects of random topographical substrates on the motion of two-dimensional droplets via appropriate statistical approaches. Different random substrate families are represented as stationary random functions. The variance of the droplet shift at both early times and in the long-time limit is deduced and the droplet footprint is found to be a normal random variable at all times. It is shown that substrate roughness decreases droplet wetting, illustrating also the tendency of the droplet to slide without spreading as equilibrium is approached. Our theoretical predict…
The numerical prediction of droplet base radius and contact angle depends on the choice of characteristic radius and height of droplet. In this study, a developed model on the basis of lubrication approximation is used to investigate the effect of characteristic parameters on the spreading of droplets over substrates. Based on two main stages of spreading (initial and equilibrium), characteristic radius and height were first evaluated. The model predictions were then compared with experimental results in literature for both impermeable and permeable substrates. The study provides evidence that…
The dynamics of thin volatile droplets comprising of binary mixtures deposited on a heated substrate are investigated. Using lubrication theory, we develop a novel one-sided model to predict the spreading and retraction of an evaporating sessile axisymmetric droplet formed of a volatile binary mixture on a substrate with high wettability. A thin droplet with a moving contact line is considered, taking into account the variation of liquid properties with concentration as well as the effects of inertia. The parameter space is explored and the resultant effects on wetting and evaporation are eval…
Compared to smooth surfaces, droplet spreading on porous surfaces is more complex and has relevance in many engineering applications. In this work, we investigate the infiltration dynamics of molten sand droplets on structured porous surfaces using the multiphase many-body dissipative particle dynamics (mDPD) method. We carry out three-dimensional simulations with different equilibrium contact angles and surface porosities. The temporal evolution of the radius of the wetted area follows a power law, as in the case of a smooth surface. The infiltration rate on the other hand is dictated by the …
If a droplet is placed on a substrate with a conical shape it spontaneously starts to spread in the direction of a growing fibre radius. We describe this capillary spreading dynamics by developing a lubrication approximation on a cone and by the perturbation method of matched asymptotic expansions. Our results show that the droplet appears to adopt a quasi-static shape and the predictions of the droplet shape and spreading velocity from the two mathematical models are in excellent agreement for a wide range of slip lengths, cone angles and equilibrium contact angles. At the contact line region…
This paper exploits the theory of geometric gradient flows to introduce an alternative regularization of the thin-film equation. The solution properties of this regularization are investigated via a sequence of numerical simulations whose results lead to new perspectives on thin-film behavior. The new perspectives in large-scale droplet-spreading dynamics are elucidated by comparing numerical-simulation results for the solution properties of the current model with corresponding known properties of three different alternative models. The three specific comparisons in solution behavior are made …
In nature, high-speed rain drops often impact and spread on curved surfaces e.g. tree leaves. Although a drop impact on a surface is a traditional topic for industrial applications, drop-impact dynamics on curved surfaces in natural situations are less known about. In the present study, we examine the time-dependent spreading dynamics of a drop onto a curved hydrophobic surface. We also observed that a drop on a curved surface is spreads farther than one on a flat surface. To understand the spreading dynamics, a new analytical model is developed based on volume conservation and temporal energy…
The Cox-Voinov law on dynamic spreading relates the difference between the cubic values of the apparent contact angle (theta) and the equilibrium contact angle to the instantaneous contact line speed (U). Comparing spreading results with this law requires accurate data of theta and U during the entire process. We consider the case when gravitational forces are negligible and transform the general Cox law in a relationship for the temporal evolution of the spreading radius. For cap-shaped droplets, this enables a comparison of experimental or computational results with Cox theory without the ne…
We present a novel experimental system that can be used to study the dynamics of picoliter droplet spreading over substrates with topographic variations. We concentrate on spreading of a droplet within a recessed stadium-shaped pixel, with applications to the manufacture of POLED displays, and find that the sloping side wall of the pixel can either locally enhance or hinder spreading depending on whether the topography gradient ahead of the contact line is positive or negative. Locally enhanced spreading occurs via the formation of thin pointed rivulets along the side walls of the pixel throug…
This paper represents a theoretical and an experimental study of the spreading dynamics of a liquid droplet, generated by a needle free deposition system called the liquid needle droplet deposition technique. This technique utilizes a continuous liquid jet generated from a pressurized dosing system which generates a liquid drop on a substrate to be characterized by optical contact angle measurements. Although many studies have explored the theoretical modelling of the droplet spreading scenario, a theoretical model representing the spreading dynamics of a droplet, generated by the jet impact a…
Plants and insects use slender conical structures to transport and collect small droplets, which are propelled along the conical structures due to capillary action. These droplets can deposit a fluid film during their motion, but despite its importance to many biological systems and industrial applications the properties of the deposited film are unknown. We characterise the film deposition by developing an asymptotic analysis together with experimental measurements and numerical simulations based on the lubrication equation. We show that the deposited film thickness depends significantly on b…
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）
The spreading under surface tension and gravity of a droplet of yield-stress fluid over a thin film of the same material is studied. The droplet converges to a final equilibrium shape once the driving stresses inside the droplet fall below the yield stress. Scaling laws are presented for the final radius and complemented with an asymptotic analysis for shallow droplets. Moreover, numerical simulations using the volume-of-fluid method and a regularized constitutive law, and experiments with an aqueous solution of Carbopol are presented.
When a droplet impacts a fabric mesh at a sufficiently high impact velocity, it not only spreads over the fabric but also penetrate its pores. To determine the influence of this liquid penetration of the fabric on droplet spreading on thin fabric meshes, we measured the droplet spreading ratio on fabric with and without an underlying substrate using a high-speed camera. For fabrics without a substrate, the droplet spreading ratio is reduced as the fabric penetration by the liquid reduces the droplet volume spreading on top of the fabric. Using entropic lattice Boltzmann simulations, we find th…
Droplets coalescing on flat surfaces tend to end up with the smaller droplet migrating into the larger one. We report a counter-intuitive droplet coalescence pattern on a superhydrophilic cylindrical wire, where the larger droplet is pulled toward the smaller one. Consequently, the center of the combined mass significantly moves toward, often beyond, the original location of the smaller droplet. This phenomenon occurs primarily because the viscous friction that a droplet experiences on a cylindrical wire is not positively correlated with the size of the droplet, unlike the droplets coalescing …
When a stationary bubble and a stationary droplet immersed in a liquid pool are brought in contact with each other, they form a bubble-droplet aggregate. Its equilibrium morphology and stability largely depend on the combination of different components' surface tensions, known as spreading factor. In this study, we look at the interaction between a rising bubble and a stationary droplet to better understand the dynamics of coalescence and rising as well as morphological changes for the bubble-droplet aggregate. A systematic study is conducted on the interaction processes with various bubble si…
Water microdroplet impact at velocities up to 100 m/s for droplet diameters from 12 to 100 um is studied. This parameter range covers the transition from capillary-limited to viscosity-limited spreading of the impacting droplet. Splashing is absent for all measurements; the droplets always gently spread over the surface. The maximum spreading radius is compared to several existing models. The model by Pasandideh-Fard et al. agrees well with the measured data, indicating the importance of a thin boundary layer just above the surface, in which most of the viscous dissipation in the spreading dro…
We describe a setup to perform systematic studies on the spreading of droplets of complex fluids under microgravity conditions. Tweaking the gravitational acceleration under which droplets are deposited provides access to different regimes of the spreading dynamics, quantified through the Bond number. In particular, microgravity allows to form large droplets while remaining in the regime where surface tension effects and internal driving stresses are predominant over hydrostatic forces. The VIP-DROP2 experimental module provides a versatile platform to study a wide range of complex fluids thro…
This fluid dynamics video shows how an oil-in-water emulsion stabilized by an ionic surfactant spreads on the free surface of a layer of pure water. The spreading shows two intriguing features: a transparent area surrounding the source of oil droplets, and a fast retraction of the layer of oil droplets on itself once the source has emptied. We show that the dynamics of spreading are strongly connected to the interfacial/bulk properties of the surfactant.
Experimental and model-assisted analysis of lamella thinning and breakup in diesel-surrogate fuel-droplet wall impingement under Leidenfrost conditions
We report the generation of directed self-propelled motion of a droplet of aniline oil with a velocity on the order of centimeters per second on an aqueous phase. It is found that, depending on the initial conditions, the droplet shows either circular or beeline motion in a circular Petri dish. The motion of a droplet depends on volume of the droplet and concentration of solution. The velocity decreases when volume of the droplet and concentration of solution increase. Such unique motion is discussed in terms of Marangoni-driven spreading under chemical nonequilibrium. The simulation reproduce…
The energy budget and dissipation mechanisms during droplet impact on solid surfaces are studied numerically and theoretically. We find that for high impact velocities and negligible surface friction at the solid surface (i.e. free-slip), about one half of the initial kinetic energy is transformed into surface energy, independent of the impact parameters and the detailed energy loss mechanism(s). We argue that this seemingly universal rule is related to the deformation mode of the droplet and is reminiscent of pipe flow undergoing a sudden expansion, for which the head loss can be calculated b…
Combined dynamic-kinematic validation of droplet-wall impact modeling
Moscow Aviation Institute (National Research University)（莫斯科航空学院（国立研究型大学））；School of Energy and Power Engineering, Tomsk Polytechnic University（托木斯克理工大学能源与动力工程学院）；Center for Materials Technologies, Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院材料技术研究中心）
We report an experimental investigation of spreading then shrinking of a surfactant droplet at the air-water interface with fingering instability appearing at the edge of the droplet stain. We find out that a droplet of a surfactant on the water shows three regimes of spreading, shrinking and resting at the appropriate parameters values. These regimes can be distinguished by measuring the mean square displacement of the droplet parts, fractal dimension of stain, or radius of fingerless part of the stain (to measure it, we define the angular box-counting dimension), versus time. The shrinking r…
Would a raindrop impacting on a coarse beach behave differently from that impacting on a desert of fine sand? We study this question by a series of model experiments, where the packing density of the granular target, the wettability of individual grains, the grain size, the impacting liquid, and the impact speed are varied. We find that by increasing the grain size and/or the wettability of individual grains the maximum droplet spreading undergoes a transition from a capillary regime towards a viscous regime, and splashing is suppressed. The liquid-grain mixing is discovered to be the underlyi…
When a droplet is brought in contact with an undercooled surface, it wets the substrate and solidifies at the same time. The interplay between the phase transition effects and the contact-line motion, leading to its arrest, remains poorly understood. Here we reveal the early solidification patterns and dynamics of spreading hexadecane droplets. Total internal reflection (TIR) imaging is employed to temporally and spatially resolve the early solidification behaviour. With this, we determine the conditions leading to the contact-line arrest. We quantify the overall nucleation behaviour, \textit{…
The surfactant-driven spreading of droplets is an essential process in many applications ranging from coating flow technology to enhanced oil recovery. Despite the significant advancement in describing spreading processes in surfactant-laden droplets, including the exciting phenomena of superspreading, many features of the underlying mechanisms require further understanding. Here, we have carried out molecular dynamics simulations of a coarse-grained model with a force-field obtained from the statistical associating fluid theory to study droplets laden with common and superspreading surfactant…
We present results on the leveling of polymer microdroplets on thin films prepared from the same material. In particular, we explore the crossover from a droplet spreading on an infinitesimally thin film (Tanner's law regime) to that of a droplet leveling on a film thicker than the droplet itself. In both regimes, the droplet's excess surface area decreases towards the equilibrium configuration of a flat liquid film, but with a different power law in time. Additionally, the characteristic leveling time depends on molecular properties, the size of the droplet, and the thickness of the underlyin…
The interaction between planar shock waves and droplets, involved the evolution of high transient unsteady wave structures and the induced cavitation process, occurs widely in nature and industry. In this paper, a combination of theoretical analysis and high-resolution numerical simulation is employed to study the inherent characteristics of the interaction. A multi-component two-phase compressible flow model, coupled with the phase transition procedure, is used to capture the spatiotemporal evolution of wave structures and cavitation behaviours, including the inception, growth and collapse of…
Even a small fraction of nanoparticles in fluids affects the splashing behavior of a droplet upon impact on a smooth surface. Nanofluid drop impact onto a smooth sapphire substrate is experimentally investigated over wide ranges of Reynolds ($10^2<\mathrm{Re}<10^4$) and Weber ($50<\mathrm{We}<500$) numbers for three nanofluid mass concentrations (0.01%, 0.1%, 1%) using high-speed photography. Nanofluids are prepared by diluting a commercial Al$_2$O$_3$-water nanofluid in aqueous glycerol solutions without dispersants. In total, 30 samples are prepared and 1799 data points are acquired. Every s…
We study spreading dynamics of nematic liquid crystal droplets within the framework of the long-wave approximation. A fourth order nonlinear parabolic partial differential equation governing the free surface evolution is derived. The influence of elastic distortion energy and of imposed anchoring variations at the substrate are explored through linear stability analysis and scaling arguments, which yield useful insight and predictions for the behaviour of spreading droplets. This behaviour is captured by fully nonlinear time-dependent simulations of three dimensional droplets spreading in the …
From biological tissues to layers of paint, macroscopic non-porous materials with the capacity to swell when brought in contact with an appropriate solvent are ubiquitous. Here, we study experimentally and theoretically one of the conceptually simplest of such systems, the swelling of a thin hydrogel layer by a single water drop. Using a bespoke experimental setup, we observe fast absorption leading to a radially spreading axisymmetric blister. Employing a linear poroelastic framework and thin-layer scalings, we develop a non-linear one-dimensional diffusion equation for the evolution of the b…
