Droplet impingement on a heated substrate is the fundamental process underlying various technologies, ranging from spray cooling to inkjet printing. Understanding the coupled effects of fluid dynamics and heat transfer patterns during droplet jumping, boiling and evaporation, which determine the outcomes of the impingement process, is essential. Here, we developed two-colour planar laser-induced fluorescence and micro-particle image velocimetry technologies to measure quantitatively the velocity and temperature distributions inside the droplet during an impingement process with high temporal and spatial resolution. With our novel measuring system, the hot spots at the solid–liquid interface are discovered for the first time. The influence of contact boiling on the droplet internal mixing, which impedes droplet recoiling and reduces the rebounding velocity, is discussed. A significant enhancement in heat absorption for partially rebounding droplets is discovered, where the impingement heat transfer rate is doubled compared to other vapour-layer-covered droplets. The scaling correlations of viscous dissipation rate and contact time of rebounding droplets, as well as the time variation of droplet temperature rise, are proposed. More detailed patterns inside droplets can be captured by these experimental methods, which will help to reveal more intrinsic mechanisms lying in thermally induced flow, complex fluids and droplet-impacting-based technologies.

The nanoscale is the new frontier of fluid dynamics and its phenomenology can echo at the macroscale as in the canonical example of drop impact on a planar substrate. Unprecedented advances in measurement technology have recently equipped fluid dynamicists with the ability to probe nanoscale effects. The paper by Li et al. (J. Fluid Mech., vol. 785, 2015, R2) uses ultrafast imaging at the hundreds of nanoseconds scale to resolve the first contact between the drop and the substrate and thereby reveal the effect of prescribed nano-roughness on contact line motion.

We report the dynamics of a droplet levitated in a multi-emitter, single-axis acoustic levitator. The deformation and atomisation behaviour of the droplet in the acoustic field displays myriad complex phenomena, in a series of events. These include the primary breakup of the droplet, wherein it exhibits stable levitation, deformation, sheet formation and equatorial atomisation, followed by its secondary breakup, which could be of various types such as umbrella, bag, bubble or multi-stage breakup. A large number of tiny atomised droplets, formed as a result of the primary and secondary breakup, can remain levitated in the acoustic levitator and exhibit aggregation and coalescence. The visualisation of the interfacial instabilities on the surface of the liquid sheet using both side- and top-view imaging is presented. An approximate size distribution of the atomised droplets is also provided. The stable levitation of the droplet is due to a balance of acoustic and gravitational forces while the resulting ellipsoidal shape of the droplet is a consequence of the balance of the deforming acoustic force and the restoring capillary force. Stronger acoustic forces can no longer be balanced by capillary forces, resulting in a highly flattened droplet, with a thin liquid sheet at the edge (equatorial region). The thinning of the sheet is caused by the differential acceleration induced by the increasing pressure difference between the poles and the equator as the sheet deforms. When the sheet thickness reduces to of the order of a few microns, Faraday waves develop at the thinnest region (preceding the rim), which causes the generation of tiny-sized droplets that are ejected perpendicular to the sheet. The corresponding hole formation results in a perforated sheet that causes the detachment of the annular rim, which breaks due to Rayleigh–Plateau (RP) instability. The radial ligaments generated in the sheet, possibly due to Rayleigh–Taylor (RT) instability, break into droplets of different sizes. The secondary breakup exhibits Weber number dependency and includes umbrella, bag, bubble or multi-stage types, ultimately resulting in the complete atomisation of the droplets. Both the primary and the secondary breakup of the droplet involve interfacial instabilities such as Faraday, Kelvin–Helmholtz, RT and RP and are well supported by visual evidence.

This study comprehensively investigates the response of a combusting droplet during its interaction with a high-speed transient flow imposed by a coaxially propagating blast wave. The blast wave is generated using a specially designed miniature shock generator that produces blast waves using the wire-explosion technique, facilitating a wide range of Mach numbers (1.03 < Ms < 1.8). The experiments are performed in two configurations: open field and focused blast wave. The charging voltage and the configuration determine the Mach number (Ms) and flow characteristics. The flame is found to exhibit two major response patterns: partial extinction followed by reignition and full extinction. Increasing the Mach number (Ms > 1.1) makes the droplet flame more vulnerable to extinction. Additionally, the flame exhibits stretching and shedding, followed by reignition at lower Mach numbers (Ms < 1.06). In all cases, the flame base lifts off in response to the imposed flow, and the advection of the flame base interacting with the flame tip results in flame extinction. The entire interaction occurs in two stages: (i) interaction with the blast wave and the decaying velocity profile associated with it, and (ii) interaction with the induced flow behind the blast wave as a result of the entrainment (delayed response). Alongside the flame's response, the droplet also interacts with the flow imposed by the blast wave, exhibiting different response modes including pure deformation, Rayleigh–Taylor piercing bag breakup and shear-induced stripping.

The internal flow within an evaporating sessile droplet has intriguing fluid mechanics important to various microfluidics applications. In the present study, a phenomenon is observed through numerical methods wherein the buoyancy-driven flow structure inside a droplet on a non-wetting substrate transitions from an axisymmetric toroidal vortex flow to a non-axisymmetric single vortex flow with increase in the substrate temperature. As the axisymmetric nature of the droplet flow field and evaporation characteristics are broken, the internal velocity accelerates significantly. The transition, which is attributed to a flow instability inside the droplet, is more prone to occur as the droplet volume or the contact angle increases. The onset of the flow transition is analysed as the amplification of a small perturbation, thereby establishing a correlation between the flow instability and the Rayleigh number (Ra). Specifically, when Ra exceeds some critical value, the onset of the flow transition is observed, which explains the effects of substrate temperature and droplet volume on the internal flow. Next, the influence of the droplet contact angle on the critical Ra was investigated, and the underlying reasons were analysed. Finally, we discuss the heat transfer efficiency within the droplet and analyse why the internal flow tends to transition to a non-axisymmetric flow pattern from an energy minimization perspective.

A viscous, lubrication-like response can be triggered in a thin film of fluid squeezed between a rigid flat surface and the tip of an incoming projectile. We develop a scaling for this viscous approach stage of fluid-mediated normal impact, applicable to soft impactors. Under the assumption of mediating fluid being incompressible, the impacting solid displays two limit regimes: one dominated by elasticity, and the other by inertia. The transition between the two is predicted by a dimensionless parameter, which can be interpreted as the ratio between two time scales that are the time that it takes for the surface waves to warn the leading edge of the impactor of the forthcoming impact, and the characteristic duration of the final viscous phase of the approach. Additionally, we elucidate why nearly incompressible solids feature (a) substantial ‘gliding’ prior to contact at the transition between regimes, (b) the largest size of entrapped bubble between the deformed tip of the impactor and the flat surface, and (c) a sudden drop in entrapped bubble radius past the transition between regimes. Finally, we argue that the above time scale ratio (a dimensionless number) can govern the different dynamics reported experimentally for a fluid droplet as a function of its viscosity and surface tension.

We analyse the effect of drop-deformation-induced change in streamline topology on the scalar transport rate (the Nusselt number $Nu$) in an ambient planar linear flow. The drop-phase resistance is assumed dominant, and the drop deformation is characterised by the capillary number ($Ca$). For a spherical drop ($Ca = 0$) in an ambient planar extension, closed streamlines lead to $Nu$ increasing with the Péclet number ($Pe$), from $Nu_0$, corresponding to purely diffusive transport, to $4.1Nu_0$, corresponding to a large-$Pe$ diffusion-limited plateau. For non-zero $Ca$, we show that the flow field consists of spiralling streamlines densely wound around nested tori foliating the deformed drop interior. Now $Nu$ increases beyond the aforementioned primary plateau, saturating in a secondary one that approaches $22.3Nu_0$ for $Ca \rightarrow 0$, $Pe\,Ca \rightarrow \infty$. The enhancement appears independent of the drop-to-medium viscosity ratio. We further show that this singular dependence, of the transport rate on drop deformation, is generic across planar linear flows; chaotically wandering streamlines in some of these cases may even lead to a tertiary enhancement regime.

The present study reports experiments of water droplets impacting on ice or on a cold metal substrate, with the aim of understanding the effects of liquid solidification or substrate melting on the impingement process. Both liquid and substrate temperatures are varied, as well as the height of fall of the droplet. The dimensionless maximum spreading diameter, $\beta _m$, is found to increase with both temperatures as well as with the impact velocity. Here $\beta _m$ is reduced when liquid solidification, which enhances dissipation, is present, whereas fusion, i.e. substrate melting, favours the spreading of the impacting droplet. These observations are rationalized by extending an existing model of effective viscosity, in which phase change alters the size and shape of the developing viscous boundary layer, thereby modifying the value of $\beta _m$. The use of this correction allows us to adapt a scaling recently developed in the context of isothermal drop impacts to propose a law giving the maximum diameter of an impacting water droplet in the presence of melting or solidification. Finally, additional experiments of dimethyl sulfoxide drop impacts onto a cold brass substrate have been performed and are also captured by the proposed modelling, generalizing our results to other fluids.

Direct numerical simulations of the droplet impact on a flat solid surface with an annular part are conducted. We investigate droplet impact on a superhydrophobic substrate with a superhydrophilic annulus to understand the formation conditions of droplets in different states. The location and size of superhydrophilic annulus are carried out through the phase diagram. We describe the formation process of droplets in three different states and the spreading radius with time to catch the rupture time of the film. Two different ruptures occur in the spreading stage or the retraction stage, respectively. The rupture times from these two mechanisms observed numerically are found to be a key factor resulting in partial rebound and lens-shaped/ring-shaped droplets. Finally, the influence of non-dimensional numbers on the formation of the ring-shaped droplet is demonstrated. The Weber number can alter the amplitude of the up and down oscillation on the droplet's upper surface, while the Froude number affects primarily the time to form the central penetrating hole. This gives the guidance and method to control the ring-shaped droplets formation time.

Bistable states for a sufficiently large amount of liquid can appear in an eccentric capillary due to the eccentricity effect under zero gravity (J. Fluid Mech, vol. 863, 2019, pp. 364–385). A transverse body force, which can lead to rich physical phenomena of a droplet, may lead to multistable states (bistability, tristability and the likes) of a sufficiently large amount of liquid in a capillary. We theoretically investigate this situation in a circular or annular capillary tube under a transverse body force. The results show that there can be tristable (bistable) states in an annular (circular) capillary tube: an occluding configuration and two (one) non-occluding configurations. In the annular tube, for one of the non-occluding configurations, the gas–liquid interface in the middle cross-section of the droplet meets both the inner and outer walls of the tube (bridging configuration); for the other non-occluding configuration, the gas–liquid interface in the middle cross-section of the droplet does not meet the inner wall (non-bridging configuration). The multistability is dependent on the Bond number, the contact angle and the cross-sectional shape. The multistability cannot occur for a zero or very large Bond number. A hydrophilic condition (the contact angle smaller than 90°) contributes to the non-occluding non-bridging configuration, while the hydrophobic condition (the contact angle larger than 90°) contributes to the non-occluding bridging configuration (only for the annular capillary). For the annular capillary with a not-so-large contact angle, increasing the inner-to-outer radius ratio can lead to a larger range of Bond numbers, in which the multistability occurs.

Drop collision with a solid particle is a ubiquitous phenomenon in a wide range of applications, including rain, spray coating, cooling or cleaning, particle encapsulation, inkjet printing, and additive manufacturing. Understanding the dynamics of drop collision is essential for optimizing these processes. In this study, we present a comprehensive experimental and analytical investigation of non-axisymmetric as well as axisymmetric drop impact on a solid particle. We use a high-speed video system to visualize the drop profile during the impact, and measure the drop height and spreading diameter for different liquid viscosities, ratios of the target to drop diameters, offsets, and various other impact parameters. We then develop a theoretical model for drop spreading on a solid spherical particle that relies on the formulation of a remote asymptotic solution for the inviscid flows, generated by non-axisymmetric drop impact. Next, the viscous effects in a thin viscous boundary layer are considered, which allows the formulation of an expression for the residual lamella thickness and maximum spreading. The theoretically predicted evolution of the lamella thickness, the residual film thickness, and the maximum spreading angle agree well with the experimental data presented in this work and the literature. Finally, we present a novel approach for in situ measurement of liquid viscosity, drop impact viscometry, at high shear rates via a single drop impact experiment, with potential application in industries where non-Newtonian drops play a major role, such as pesticide spraying, paint droplet spreading, blood drop impact and fuel injectors.

Waves are formed on the surface of a sessile drop driven through substrate vibrations oriented at a slanting angle from the normal. A mathematical model is derived, which leads to an infinite system of coupled Mathieu equations governing the wave dynamics that are solved using Floquet theory. The spatial structure of the waves is described by the mode number pair $[\ell,m]$, where $\ell$ and $m$ are the polar and azimuthal mode numbers, respectively. Limiting cases corresponding to horizontal and vertical vibrations are discussed with predictions agreeing well with prior literature. We focus our results on three drop motions – (1) harmonic $[1,1]$ rocking mode, (2) harmonic $[2,0]$ pumping mode, and (3) subharmonic rocking $[1,1]$ mode – as they depend upon the slanting angle, static contact angle, and contact-line conditions, which we assume to be either pinned or freely moving with fixed contact angle. New theoretical predictions are tested through experiments over a range of parameters, showing good agreement.

Whether colliding drops will merge with or bounce off each other is critical to numerous processes, and the physics involved is notoriously complex. In particular, experiments show that both sufficiently slow and fast head-on drop collisions lead to merging, but that there is often an intermediate regime in which bouncing is observed; these transitions in behaviour were recently discovered to be surprisingly sensitive to the radius of the drops and the ambient gas pressure. We show here that these transitions between bouncing and merging are governed by nanoscale phenomena; namely, gas-kinetic and disjoining pressure effects. To capture these crucial effects, a novel, open-source computational model is developed for the simulation of colliding drops. The model uses a hybrid approach, based on solving the Navier–Stokes equations in the drop with a lubrication approach for the unconventional physics of the gas film. Our simulations show remarkably good agreement with experiments of head-on collisions and also provide new experimentally verifiable predictions.

In this study, we investigate the stability of a film that is attached to a corner between a cylinder and a substrate, using a combination of theoretical and numerical approaches. Notably, we place our focus on flat and thin films where the contact line is almost perpendicular to the cylinder wall whereas a small angle forms between the contact line and the substrate, and the film size is smaller than the cylinder radius. The film stability, which depends on the film size and the wall wettability, is first predicted by a standard linear stability analysis (LSA) within the long-wave theoretical framework. We find that the film size plays the most important role in controlling the film stability. Specifically, the thicker the film is, the less sensitive it becomes to the large-wavenumber perturbation. The wall wettability mainly impacts the growth rates of perturbations and slightly influences the marginal stability and postinstability patterns of wrapping films. We compare the LSA predictions with numerical results obtained from a disjoining pressure model (DPM) and volume-of-fluid (VOF) simulations, which provide more insights into the film breakup process. At the early stage there is a strong agreement between the LSA predictions and the DPM results. Notably, as the perturbation grows, thin film regions connecting two neighbouring satellite droplets form which may eventually lead to a stable or temporary secondary droplet, an aspect which the LSA is incapable of capturing. In addition, the VOF simulations suggest that beyond a critical film size, merging between two neighbouring drops becomes involved during the breakup stage. Therefore, the LSA predictions are able to provide only an upper limit on the final number of satellite droplets.

We consider the dynamic wetting and dewetting processes of films and droplets of complex liquids on planar surfaces, focusing on the case of colloidal suspensions, where the particle interactions can be sufficiently attractive to cause agglomeration of the colloids within the film. This leads to an interesting array of dynamic behaviours within the liquid and of the liquid–air interface. Incorporating concepts from thermodynamics and using the thin-film approximation, we construct a model consisting of a pair of coupled partial differential equations that represent the evolution of the liquid film and the effective colloidal height profiles. We determine the relevant phase behaviour of the uniform system, including finding associated binodal and spinodal curves, helping to uncover how the emerging behaviour depends on the particle interactions. Performing a linear stability analysis of our system enables us to identify parameter regimes where agglomerates form, which we independently confirm through numerical simulations and continuation of steady states, to construct bifurcation diagrams. We obtain various dynamics such as uniform colloidal profiles in an unstable situation evolving into agglomerates and thus elucidate the interplay between dewetting and particle aggregation in complex liquids on surfaces.

Shock–droplet interaction in the early stage involves intricate wave structures. Investigating this phenomenon is inherently challenging due to the fine spatial and temporal scales involved. Past research has suggested that the occurrence of cavitation, marked by a negative peak pressure, is linked to the focus of the reflected expansion wave. In this study, a high-fidelity compressible numerical approach is utilized to replicate the initial phase of shock–droplet interactions. The location of the negative peak pressure is meticulously documented and compared with experimental measurement and numerical results. Results indicate a strong alignment between the negative peak pressure positions identified through numerical simulations and the focal points identified in theoretical models for low gas–liquid wave velocity ratios. However, this alignment is notably disrupted when dealing with higher gas–liquid wave velocity ratios. Further enhancements are made to the theoretical model, enabling a more precise depiction of internal wave structures and focus points, particularly under conditions of high gas–liquid wave velocity ratios. The study delves into the various factors influencing internal pressure fluctuations within the liquid droplet, categorizing them into four phases: the shock wave effect, relaxation effect, fluctuation effect, and expansion wave effect. Analysing the pressure decrease portion reveals that while the converging of the reflected expansion wave leads to a substantial pressure drop, it accounts for only a fraction of the total pressure variation. Consequently, any model predicting negative peak pressure positions must comprehensively consider all contributing factors.

We study the spreading of Newtonian viscous (aqueous glycerin solution) and viscoelastic (aqueous polymer solution) drops on solid substrates with different wettabilities. For drops of the same zero-shear viscosity, we find in the early stages of spreading that viscoelastic drops (i) spread faster and (ii) their contact radius shows a different power law vs time than Newtonian drops. We argue that the effect of viscoelasticity is only observable for experimental time scales of the order of or larger than the internal relaxation time of the viscoelastic polymer solution. We attribute this behaviour to the shear thinning of the viscoelastic polymer solution. When approaching the contact line, the shear rate increases and the steady-state viscosity of the viscoelastic drop is lower than that of the Newtonian drop. We support our experimental findings with a simple (first-order) perturbation model that qualitatively agrees with our findings.

A hybrid lattice Boltzmann and finite difference method is applied to study the influence of soluble surfactants on droplet deformation and breakup in simple shear flow. First, the influence of bulk surfactant parameters on droplet deformation in two-dimensional shear flow is investigated, and the surfactant solubility is found to influence droplet deformation by changing average interface surfactant concentration and non-uniform effects induced by non-uniform interfacial tension and Marangoni forces. In addition, the droplet deformation first increases and then decreases with Biot number, increases significantly with adsorption number $k$ and decreases with Péclet number or adsorption depth; and among the parameters, $k$ is the most influential one. Then, we consider three-dimensional shear flow and investigate the roles of surfactants on droplet deformation and breakup for different capillary numbers and viscosity ratios. Results show that in the soluble case with $k=0.429$, the droplet exhibits nearly the same deformation as in the insoluble case due to the balance between surfactant adsorption and desorption; upon increasing $k$ from 0.429 to 1, the average interface surfactant concentration is greatly enhanced, leading to significant increase in droplet deformation. The critical capillary number of droplet breakup $C{{a}_{cr}}$ is identified for varying viscosity ratios in clean, insoluble and soluble ($k=0.429$ and 1) systems. As the viscosity ratio increases, $C{{a}_{cr}}$ first decreases and then increases rapidly in all systems. The addition of surfactants always favours droplet breakup, and increasing solubility or $k$ could further reduce $C{{a}_{cr}}$ by increasing average interface surfactant concentration and local surfactant concentration near the neck during the necking stage.

Droplet impact on oscillating substrates is important for both natural and industrial processes. Recognizing the importance of the dynamics that arises from the interplay between droplet transport and substrate motion, in this work, we present an experimental investigation of the spreading of a droplet impacting a sinusoidally oscillating hydrophobic substrate. We focus particularly on the maximum spread of droplets as a function of various parameters of substrate oscillation. We first quantify the maximum spreading diameter attained by the droplets as a function of frequency, amplitude of vibration, and phase at the impact for various impact velocities. We highlight that there can be two stages of spreading. Stage I, which is observed at all impact conditions, is controlled by the droplet inertia and affected by the substrate oscillation. For certain conditions, a Stage II spreading is also observed, which occurs during the retraction process of Stage I due to additional energies imparted by the substrate oscillation. Subsequently, we derive scaling analyses to predict the maximum spreading diameters and the time for this maximum spread for both Stage I and Stage II. Furthermore, we identify the necessary condition for Stage II spreading to be greater than Stage I spreading. The results will enable optimization of the parameters in applications where substrate oscillation is used to control the droplet spread, and thus heat and mass transfer between the droplet and the substrate.

Spray formed by a myriad of secondary droplets generated by the impact of raindrops on a deep-water pool is studied with a laboratory rain facility. Experiments are performed with two rain rates and raindrops fall on the water surface at a nearly constant velocity. The secondary droplets at various heights above the pool's water surface are recorded with a cinematic digital in-line holographic technique that consists of a high-speed camera, a pulsed Nd:YLF laser and associated optics. The experimental results show that in the heat-map scatter plots of radius versus velocity near the water surface of the pool, the droplets are distributed into three regions, corresponding to distinct physical mechanisms of droplet generation. It is found that the diameter distribution of the droplets in the rain field changes with height above the pool's water surfaces. Both numerical simulation and experimental data reveal that the liquid water content, due to the presence of secondary droplets, in the atmospheric surface layer decreases exponentially with increasing height.