In this study, we present the discovery of an inverse-square force between evaporating sessile droplets. We find that one evaporating sessile droplet exerts a driving force on another, with the magnitude of the force decaying inversely proportional to the square of distance between their centres. The inverse-distance decay of the vapour concentration generated by the source droplet is a key ingredient leading to the inverse-square form of the force. A theoretical framework combining continuum hydrodynamics with an effective capillary line-force representation is developed to rationalize this force. The inverse-square form is robust over ranges of droplet size, composition and ambient humidity. Its broader applicability is further supported by our analysis of previously published kinematic data for pure-liquid sessile droplets and thermocapillary floating droplets.

The coffee-ring effect is a universal feature of evaporating sessile droplets with a pinned contact line, wherein solutes or particles are advected to the droplet’s edge due to evaporation-driven flows. Here, we investigate model respiratory droplets primarily composed of water, salt and a type of the glycoprotein mucin. Using fluorescence microscopy, we observe the formation of a well-defined protein ring at the droplet edge as water evaporates. The growth and morphology of this ring exhibit a strong dependence on ambient relative humidity ($H_r$), revealing a dynamics that existing models cannot capture. While existing models have successfully described the formation of the coffee ring in particle-laden droplets, they often assume that the evaporative flux, and thus the hydrodynamics, is decoupled from solute transport. This assumption breaks down in complex fluids, such as protein or polymeric solutions, where the solute can influence evaporation through changes in water activity. Specifically, we find that protein accumulation at the edge is governed by the feedback between local solute concentration and evaporation rate. To account for this, we develop a minimal theoretical model based on the lubrication approximation, incorporating the coupling between the hydrodynamics and solute transport through the evaporation rate. Our framework reproduces key features of the experimental observations and suggests a physical basis for the $H_r$-dependent stability and infectivity of respiratory droplets containing viruses.

We develop a three-dimensional numerical scheme for investigating interfacial flows coupled with frictional solid particles. Our approach combines the lattice Boltzmann method (LBM) to model the dynamics of a two-component fluid, and the discrete element method (DEM) to model normal reaction, sliding friction and rolling friction between solid particles and between particles and solid surfaces. Key to the coupling between the fluid and particle dynamics are the momentum exchange method to transfer hydrodynamic forces between the fluids and particles, a geometric boundary condition to tune particle wettability, and a capillary force model describing surface tension forces between particles and liquid–fluid interfaces. We rigorously validate the contact forces by investigating the dynamics of a particle bouncing off a solid surface and rolling down an inclined plane, the hydrodynamic force by the Segrè–Silberberg effect and the capillary force by particle detachment from a liquid–fluid interface. Motivated by the self-cleaning properties of lotus leaves, we apply the method to investigate how drops remove contaminant particles from surfaces. We successfully reproduce scenarios reported experimentally by Naga et al. (2021a Soft Matt. 17(7), 1746–1755) by tuning the particle friction. Furthermore, the LBM–DEM approach allows us to systematically explore the effects of particle friction coefficients, drop size and speed. Our method opens opportunities to study numerous phenomena involving particle dynamics interacting with interfacial flows, including soil erosion, capillary-driven colloidal self-assembly and how raindrops transport microplastics in the environment. It also makes it possible to control parameters that are difficult to tune independently in experiments, including contact angles, surface tension and friction coefficients.

This theoretical report explores a double emulsion droplet in a two-directional shear creeping flow. The droplet consists of a spherical inner drop (fluid 3) enclosed by a spherical shell (fluid 2), all immersed in external fluid 1. Six parameters govern: the capillary number (Ca), two viscosity ratios (λ21, λ31), the radii ratio (κ), the surface tensions ratio (Ω) and the flow-type (G). Using Lamb’s solution, an analytical solution for the spherical case (Ca = 0, Ω = ∞) was found. We explored small deformations with: λ21 = O(1), λ31 = O(1), G = O(1), $\textit{Ca} \ll 1$ and $ \textit{Ca}/\Omega \ll 1$. If $ G \gt -1$, the outer surface is oriented at $\phi$ = +45°, while the inner surface at $\phi$ = −45°. The reverse occurs for $ G \lt -1$. If G = −1, no deformation is possible. For a highly viscous shell ($ \lambda_{21} \gg 1$), the deformation of the outer surface is independent of Ca and oriented at $\phi$ = 0° or $\phi$ = 90°, while the deformation of the inner surface depends on Ca and at $\phi$ = ±45°. The opposite occurs for highly viscous inner drop ($ \lambda_{31} \gg 1$). Two breakup mechanisms were identified: contact between inner and outer surfaces, and inner drop rupture. Finally, we considered the effective viscosity of a dilute emulsion, containing spherical double emulsion droplets. In a single drop (κ → 0), Taylor’s expression is recovered. For a highly viscous or thin shell (λ21 → ∞, κ → 1), Einstein’s formula is obtained.

In laser-produced plasma (LPP) extreme ultraviolet (EUV) sources, deformation of a tin droplet into an optimal target shape is determined by its interaction with a pre-pulse laser-generated plasma. This interaction is mediated by a transient ablation pressure, whose complex spatio-temporal evolution remains experimentally inaccessible. Existing modelling approaches are limited: empirical pressure–impulse models neglect dynamic plasma feedback, while advanced radiation hydrodynamics codes often fail to resolve late-time droplet hydrodynamics. To bridge this gap, we propose a radiation two-phase flow model based on a diffuse interface approach. The model integrates radiation hydrodynamics for the plasma with the Euler equations for a weakly compressible liquid, extending a five-equation diffuse interface formulation to incorporate radiation transport, thermal conduction, ionisation and surface tension. This formulation enforces pressure and velocity equilibrium across the diffuse interface region, with closure models constructed to ensure correct jump conditions at interfaces and asymptotically recover the pure-phase equations in bulk regions. Then we apply the model to simulate a benchmark pre-pulse scenario, where a $50\ \unicode{x03BC} \mathrm{m}$ tin droplet is irradiated by a $10\ \mathrm{ns}$ laser pulse. Our axisymmetric simulations capture the rapid plasma expansion and subsequent inertial flattening of the droplet into a thin, curved sheet over microsecond time scales. Notably, the model reproduces experimentally observed features such as an axial jet – rarely replicated in prior simulations. Quantitative agreement with experimental data for sheet dimensions and velocity validates the approach. The proposed model self-consistently couples laser–plasma physics with compressible droplet dynamics, providing a powerful tool for studying plasma–liquid interactions in LPP-EUV source optimisation.

The impact of droplets, such as rain and sea sprays, on wind turbine blade surfaces can lead to significant damage due to leading-edge erosion (LEE). Traditionally, LEE on wind turbine blades has been solely attributed to material fatigue from repeated impacts; however, studies in similar engineering applications, such as steam turbines, suggest that cavitation may also play a critical role. This study develops a novel fluid–structure interaction (FSI) model that, for the first time, explicitly incorporates phase change in the liquid flow inside the droplet. Droplet impingement simulations cover a broad range of impact speeds relevant to most engineering applications, considering both cavitation and non-cavitation scenarios. The results present temporal evolutions of the pressure wave inside the droplet during its impact on the solid surface and demonstrates the triggering mechanisms of both homogeneous and heterogeneous cavitation. The developed FSI model suggests that the significant impact of cavitation, which can subject the material to two separate stress events during a single droplet impact, potentially reduces the material’s fatigue lifetime by half. Furthermore, the study explores the ability of heterogeneous cavitation to cause localised damage on the coating surface upon a single raindrop impact at speeds below $100$ $\mathrm{m\,s^{-1}}$. The comparison of two-dimensional and axisymmetric simulations shows that the intensified shock waves in the latter produce more focused cavitation near the surface centre, with minimal spreading along the surface. These findings highlight the need to incorporate heterogeneous cavitation effects in future studies, particularly as the turbine blade size and impact velocities increase.

We develop a thermodynamically consistent phase-field model for soluble surfactants in two-phase flows, incorporating both interfacial and solid surface adsorption. The model is derived via variational principles consistent with the second law of thermodynamics, resulting in modified free energies and boundary conditions that capture surfactant transport, adsorption and wetting dynamics. A key contribution of this work is the inclusion of surfactant adsorption on solid walls, which leads to qualitative agreement with experimental observations: unlike prior numerical studies that predicted hydrophilic surfaces becoming more hydrophilic and hydrophobic surfaces more hydrophobic, our model shows a shift towards increased hydrophilicity across all contact angles – consistent with experimental trends. Our results establish that solid adsorption provides the missing mechanism required for predictive modelling of surfactant-laden contact line dynamics.

Drop deformation and motion driven by shear flows are widespread phenomena in nature and underpin a variety of technological applications. We investigate the deformation, motion and contact-line dynamics of two-dimensional drops in steady shear flows within the framework of lubrication theory. For pinned drops with $180^\circ$ hysteresis, we identify a critical shear rate, beyond which the upstream side becomes unstable and a thinning film develops. At late stage, the thinning film connects to the bulk drop through a self-similar capillary transition structure. For moving drops with $0^\circ$ hysteresis, the dynamical behaviours can be classified into three regions: in the low-shear region, the steady-state drop velocity increases linearly with the shear number $S$; in the intermediate-shear region, the drop velocity exhibits a non-monotonic dependence on $S$; in the high-shear region, the steady-state drop profile can be clearly divided into the entrained film, dimple and capillary ridge. Moreover, during the drop evolution under high shear, a Landau–Levich–Derjaguin-like (LLD-like) film emerges and exhibits a scaling law that differs from the classical LLD law. These results provide a unified theoretical and numerical framework for understanding shear-driven drop dynamics and may offer insights into analogous three-dimensional systems.

Droplet impingement is one of the most common phenomena in nature. However, the impact dynamics of droplets on structured heterogeneous wettability surfaces remain little explored. In this work, an investigation is conducted into the droplet impact on heterogeneous wettability surfaces composed of superhydrophobic micropillars on a hydrophilic substrate, by using an improved phase-field lattice Boltzmann model. In particular, the effects of surface geometry and impact conditions on droplet bouncing and wetting behaviours are scrutinised. Four distinct impact outcomes are identified: complete bouncing, pancake bouncing, partial wetting and complete wetting. Based on the competition among capillary, inertial and viscous forces, an analytical model is proposed to predict the maximum droplet penetration depth within the pillar gaps. A transition diagram is constructed to distinguish these different impact outcomes, with regime boundaries derived from the proposed analytical model. Finally, from the perspective of energy analysis, both the evolution of the energy budget and the surface energy distribution during the droplet impact process are revealed. These findings provide guidance for the design of heterogeneous wettability surfaces, enabling predictable control over droplet bouncing and wetting behaviours.

When a fast droplet impacts a pool, the resulting ejecta sheet dynamics determine the final impact outcome. At low capillary numbers, the ejecta sheet remains separate from a deep static pool, while at higher values, it develops into a lamella. Here, we show that the common natural scenario of a slowly moving deep pool can change the upstream impact outcome, creating highly three-dimensional dynamics no longer characterised by a single descriptor. By considering how pool movement constrains the evolution of the ejecta sheet angle, we reach a length-scale invariant parametrisation for the upstream transition that holds for a wide range of fluids and impact conditions. Direct numerical simulations show similar dynamics for an equivalent oblique impact, indicating that the air boundary layer above a moving pool does not play a decisive role for low pool–droplet speed ratios. Our results also provide insight into the physical mechanisms that underpin pool impact outcomes more generally.

We study experimentally the detachment of a large pendant drop of non-Brownian monodisperse solid spheres in a Newtonian liquid, for high solid volume fractions ($0.55\leq \phi \leq 0.605$) compared with the critical volume fraction $\phi _{{c}}$ of the suspension. This set-up allows us to monitor both the bulk-effective deformation rate and the mean stress at the neck of the drop throughout detachment. The neck effective viscosity is found to decrease close to exponentially with the current neck radius, from a large initial value, $\eta _{{i}}$, down to the suspending liquid one. Experiments with different drop-to-particle size ratios indicate that, above $\phi _{{c}}$, the initial value $\eta _{{i}}$ is set by the viscosity-limited dilation of the particle phase. Observations on a transparent, doubly index-matched system reveal increasing deviation from axial symmetry and the collapse of capillary menisci at the drop interface during the exponential decay. A simplified model, based on the observed rheological evolution and the close-to-conical neck shape, captures the steeply accelerated detachment dynamics and shows how $\eta _{{i}}$ sets the breakup duration.

We present the results of a combined experimental and theoretical study of the spreading of viscous droplets over rigid substrates. First, we experimentally investigate the wetting of a roughened glass surface by a viscous droplet of silicone oil, wide and shallow relative to the capillary length $\ell _c$. The horizontal radius of the droplet grows according to an $R_{\textit{drop}}\sim t^{1/8}$ scaling, reminiscent of viscous gravity currents (Lopez et al. 1976 J. Colloid Interface Sci., vol. 56, pp. 460–468). The droplet is preceded by a mesoscopic fluid film that percolates through the rough substrate, its radius increasing according to $R_{\textit{film}}\sim t^{3/8}/(\log t)^{1/2}$. To rationalise these observed scalings, we develop a new ‘capillary current’ model for the spreading of shallow droplets with arbitrary radius on rough surfaces. Furthermore, on the basis of established similarities between droplet spreading over wetted rough and smooth substrates (Cazabat & Cohen Stuart, 1986 J. Phys. Chem., vol. 90, pp. 5845–5845), we argue its relevance to a broader class of spreading problems. We propose that, throughout their evolution, shallow droplets maintain a quasi-equilibrium balance between hydrostatic and curvature pressure, perturbed only by unbalanced contact line forces arising along the droplet’s edge. For drops with horizontal radii small with respect to $\ell _c$, our model converges to the original description of Hervet & de Gennes (1984 C. R. Acad. Sci. II, vol. 299, pp. 499–503) and thereby recovers the classic spreading laws of Hoffman (1975 J. Colloid Interface Sci., vol. 50, pp. 228–241), Voinov (1976 Fluid Dyn., vol. 11, pp. 714–721) and Tanner (1979 J. Phys. D: Appl. Phys. vol. 12, p. 1473). For drops wide with respect to $\ell _c$, it rationalises why millimetric, surface-tension-driven capillary currents exhibit the same spreading behaviour as relatively large-scale viscous gravity currents.

We present new experiments about the transport of a passive constituent around a bubble rising at large Reynolds number in a vertical thin-gap cell filled with a reactive liquid. A chemical reaction occurs at the interface and the constituents are transported in the liquid. First, 4D-Lagrangian particle tracking (4D-LPT) measurements are performed to scrutinise the three-dimensional structure of the flow in the vicinity of a large bubble rising steadily. They reveal that two horseshoe vortices, superposed in the direction perpendicular to the walls, precede the bubble. The attached wake also consists of two superposed recirculations and two transverse vortices stacked to the rear interface. The horseshoe vortices are also observed for oscillating bubbles, and their radius scales with the bubble diameter, as explained by inviscid arguments. Then, time-resolved concentration fields of a non-diffusive constituent are obtained around the bubbles by a colorimetry method. A comparative discussion with 4D-LPT results or bubble kinematics provides new insights into the mode of transport of this constituent transferred at the interface. Regardless of the size of the bubble, mass transfer occurs predominantly in the liquid films located between the bubble and the walls, resulting in a direct deposit of mass near the walls. On another hand, the high-Reynolds number flow surrounding the bubble combines transport in the horseshoe vortices, intermittent detachment of mass from the equator of the bubble synchronised with vortex shedding and bubble shape contraction and finally transport in a strongly stretched and dampened von Kármán array leading to final deposits of mass.

The droplet–microfluidic system under acoustic-field actuation represents a classical multiphase flow problem with broad relevance to engineering and biomedical applications. When an acoustic wave interacts with a viscous droplet suspended in another liquid, it induces acoustic streaming both inside and outside the droplet and exerts acoustic radiation stresses that deform its shape. In this work, we present a comprehensive analytical and numerical framework to investigate the coupled dynamics of droplet shape evolution, acoustic fields and acoustic streaming. The study considers droplets stably positioned within a standing wave at either a pressure node or a velocity node, undergoing deformations analogous to the classical Taylor regime. Our analytical model, supported by numerical simulations, reveals that the acoustic field is predominantly governed by low-order modes, up to the quadrupole. Furthermore, the acoustic-streaming pattern is dictated by the density contrast between the droplet and the surrounding fluid, whereas both density and compressibility contrasts control the droplet deformation. We also demonstrate that impedance mismatch is not the fundamental criterion for acoustic scattering. Instead, mismatches in density, or compressibility, or both, serve as the primary mechanisms. In addition, we develop a phase diagram illustrating droplet shape-deformation and streaming-pattern regimes as functions of the material properties (density ratio and compressibility ratio). These results provide deep mechanistic insights for the design and control of efficient droplet–acoustofluidic platforms and lay the groundwork for next-generation ultrasound-driven strategies for precise droplet manipulation in soft-matter and microfluidic applications.

Deciphering the incipient stages of phase change in superheated fluids – and their subsequent evolution – remains a major challenge, as it requires bridging microscopic nucleation with macroscopic bubble dynamics and heat transfer. This gap has long hindered predictive boiling models and limited progress in emerging thermal technologies. Here, we leverage large-scale simulations of fluctuating hydrodynamics, combined with a rare-event technique, to rationalise liquid–vapour transformation from nucleation to bubble growth. We quantify the statistics of nucleated bubbles and hydrodynamic fields, onset temperatures, and non-trivial wettability effects on the nucleation pathways. Despite operating at the mesoscale, the simulations recover microscopic nucleation signatures observed in atomistic studies, including the dependence of both the boiling onset temperature and the crossover between homogeneous and heterogeneous nucleation pathways on surface wettability. At larger scales, they reproduce the experimentally observed transition from an early exponential regime of bubble-contact areas to a post-critical power-law scaling. These findings establish a quantitative bridge between fluctuation-driven nucleation and emergent boiling dynamics across scales.

We investigate the asymmetric freezing of a liquid droplet sliding on an inclined cold surface using numerical simulations based on the lubrication approximation. The combined effects of gravity, capillarity and solidification kinetics on droplet motion, interfacial deformation and the resulting frozen morphology are examined through systematic variations in substrate inclination, wettability, effective Bond number and Stefan number. Our results show that sliding prior to and during the early stages of freezing plays a dominant role in governing the asymmetry of the frozen droplet. A tilted ice cusp forms at the droplet tip due to the competition between gravitational forces and capillary resistance, with its orientation and magnitude strongly dependent on substrate wettability and inclination. Greater inclination and increased wettability enhance asymmetry in droplet morphology. Further, highly wetting substrates favour capillary-driven retraction and induce transient liquid motion opposite to gravity during freezing. The evolution of contact-angle hysteresis at both the solid surface and the liquid–ice interface underscores the importance of early-time dynamics, when the unfrozen liquid remains mobile and gravitational effects are most pronounced. Decomposition of the liquid motion into capillary- and gravity-driven contributions provides physical insight into contact-line pinning, receding-edge thinning and the development of asymmetric liquid–ice contact angles. Increasing the Stefan number accelerates freezing, limits sliding-induced deformation, and reduces both the cusp angle and the post-freezing contact-angle contrast. Overall, this study establishes a physical framework for understanding the morphology of frozen droplets on inclined substrates.

This study presents a combined analytical and experimental investigation into the rising dynamics of Newtonian droplets in viscoelastic fluids, focusing on discontinuous transitions in droplet velocity. Although velocity jumps have been observed in gas bubbles, this work is the first systematic investigation of the phenomenon in Newtonian droplets. Analytically, a triple-perturbation approach is developed using Reynolds (Re), Deborah (De) and capillary (Ca) numbers as expansion parameters. The surrounding viscoelastic medium is modelled using the nonlinear-Giesekus constitutive equation. The analysis reveals that increasing Deborah number intensifies and reorients normal stresses – particularly the polar ($\theta$-direction) and hoop components – along the droplet interface. These stresses migrate rearward with increasing elasticity, resulting in localised elongational flows and concentrated radial stresses near the rear stagnation point, which indicates the onset of elastic instabilities. Experimentally, Newtonian droplets are introduced into a variety of exterior fluids: (i) quasilinear Boger fluids, which are constant-viscosity viscoelastic solutions that isolate elastic effects; and (ii) a strongly nonlinear viscoelastic solution, which exhibits both high elasticity and significant shear-thinning behaviour. Equivalent Newtonian fluids of matched viscosity are also used as controls. Viscoelastic fluids are solutions of polyacrylamide in the water–glycerine mixture. Across all experiments, the dimensionless ranges were $\textit{Re}\approx 6\times 10^{-4}{-}4.59$, $\textit{De}\approx 0.057{-}0.802$ and $\textit{Ca}\approx 0.0026{-}0.638$, framing the scope of applicability. The experimental observations challenge the classical expectation of smooth droplet motion and demonstrate that Newtonian droplets in viscoelastic environments can exhibit sharp, weak or no velocity jumps, depending on the balance between elasticity, viscosity and deformability.

Boiling and bubble injection are effective strategies for enhancing heat transfer between solid boundaries and a working fluid in numerous industrial applications, including nuclear reactors and molten metal processing. Motivated by this, we conduct direct numerical simulations of a vertical, turbulent, differentially heated, bubble-laden channel flow. The Prandtl number $\textit{Pr}$, kept identical in both phases, is varied across three representative values – $0.07$ (liquid metals), $0.7$ (vapour) and $7$ (water) – to span thermal transport regimes across three orders of magnitude. The simulations are conducted at a friction Reynolds number $\textit{Re}_\tau =150$, void fraction $\alpha =5.4\,\%$ and a density ratio $\rho _r=0.1$ (defined as the bubble-to-carrier density). The bubbles substantially alter the hydrodynamic structure of the flow, amplifying turbulent fluctuations and mixing. Their interaction with the thermal boundary layers disrupts the characteristic streaky structures near the heated walls, fragmenting them into smaller and more chaotic patterns. To elucidate this mechanism, we examine the bubble-induced modifications to the temperature field and show that temperature becomes decorrelated from velocity. Consequently, the heat-transfer enhancement arises primarily from an increase in convective heat flux driven by intensified wall-normal velocity fluctuations. The thermal boundary layer is markedly thinned, and the Nusselt number nearly doubles across all examined cases.

A mathematical model for the deposition of particles from a thin sessile droplet undergoing diffusion-limited evaporation in four different modes of evaporation, namely the constant contact radius (CR), constant contact angle (CA), stick–slide (SS), and stick–jump (SJ) modes, is formulated and analysed. Explicit expressions are obtained for the flow and concentration of particles within the droplet, as well as the evolutions of the mass of particles in the bulk of the droplet and in a distributed deposit and/or in one or more ring deposits on the substrate. It is shown that the nature of the deposit depends on both the local evaporative flux and the motion of the contact line. In particular, for a droplet undergoing diffusion-limited evaporation, the flow is outwards towards the contact line in both the CR and CA modes, however, the receding contact line in the CA mode results in a qualitatively different deposit from that in the CR mode, specifically a switch from a ring deposit in the CR mode to a near-uniform deposit in the CA mode. This contrasts with the behaviour of a droplet undergoing spatially uniform evaporation in the CA mode, in which the flow is radially inwards resulting in a peak deposit. For a droplet evaporating in the SS or SJ modes, the final deposit is a combination of the deposit types associated with the CR and CA modes. The present model is validated by finding good agreement between the theoretical predictions for the deposit and previous experimental results.

We investigate droplet deformation following laser-pulse impact at low Weber numbers (${\textit{We}}\sim 0.1{-}100$). Droplet dynamics can be characterised by two key parameters: the impact Weber number and the width, W, of the distribution of the impact force over the droplet surface. By varying laser-pulse energy, our experiments traverse a phase space comprising (i) droplet oscillation, (ii) breakup or (iii) sheet formation. Numerical simulations complement the experiments by determining the pressure width and by allowing We and W to be varied independently, despite their correlation in the experiments. A single phase diagram, integrating observations from both experiments and simulations, demonstrates that all phenomena can be explained by a single parameter: the deformation Weber number ${\textit{We}}_{{d}}=f({\textit{We}},{W})$ that is based on the initial radial expansion speed of the droplet, following impact. The resulting phase diagram separates (i) droplet oscillation for ${\textit{We}}_{{d}}\lt 5$, from (ii) breakup for $5\lt {\textit{We}}_{{d}}\lt 60$ and (iii) sheet formation for ${\textit{We}}_{{d}}\gt 60$.