We consider the dynamics of a liquid film with a pinned contact line (for example, a drop), as described by the one-dimensional, surface-tension-driven thin-film equation h_t + (h^n h_{xxx})_x = 0$h_t + (h^n h_{xxx})_x = 0$, where h(x,t)$h(x,t)$ is the thickness of the film. The case n=3$n=3$ corresponds to a film on a solid substrate. We derive an evolution equation for the contact angle \theta (t)$\theta (t)$, which couples to the shape of the film. Starting from a regular initial condition h_0(x)$h_0(x)$, we investigate the dynamics of the drop both analytically and numerically, focusing on the contact angle. For short times t\ll 1$t\ll 1$, and if n\ne 3$n\ne 3$, the contact angle changes according to a power law \displaystyle t^{\frac {n-2}{4-n}}$\displaystyle t^{\frac {n-2}{4-n}}$. In the critical case n=3$n=3$, the dynamics become non-local, and \dot {\theta }$\dot {\theta }$ is now of order \displaystyle {\rm{e}}^{-3/(2t^{1/3})}$\displaystyle {\rm{e}}^{-3/(2t^{1/3})}$. This implies that, for n=3$n=3$, the standard contact line problem with prescribed contact angle is ill posed. In the long time limit, the solution relaxes exponentially towards equilibrium.

We study the dynamics of a thin liquid sheet that flows upwards along the sides of a vertically aligned, impacting plate. Upon impact of the vertical solid plate onto a liquid pool, the liquid film is ejected and subsequently continues to flow over the solid surface while the plate enters the water. With increasing impact velocity, the liquid film is observed to rise up faster and higher. We focus on the time evolution of the liquid film height and the thickness of its upper rim and discuss their dynamics in detail. Similar to findings in previous literature on sheet fragmentation during drop impact, we find the rim thickness to be governed by the local instantaneous capillary number based on gravity and the deceleration of the liquid sheet, showing that the retraction of the rim is primarily due to capillarity. In contrast, for the liquid film height, we demonstrate that the viscous dissipation in the thin boundary layer is the dominant factor for the vertical deceleration of the liquid sheet, by modelling the time evolution of the film height and showing that the influences of capillarity, gravity and deceleration due to the air phase are all negligible compared with the viscous term. Finally, we introduce characteristic viscous time and length scales based on the initial rim thickness and show that the maximum height of the film and the corresponding time can be determined from these viscous scales.

Spin coating is the process of generating a uniform coating film on a substrate by centrifugal forces during rotation. In the framework of lubrication theory, we investigate the axisymmetric film evolution and contact-line dynamics in spin coating on a partially wetting substrate. The contact-line singularity is regularized by imposing a Navier slip model. The interface morphology and the contact-line movement are obtained by numerical solution and asymptotic analysis of the lubrication equation. The results show that the evolution of the liquid film can be classified into two modes, depending on the rotational speed. At lower speeds, the film eventually reaches an equilibrium state, and we provide a theoretical description of how the equilibrium state can be approached through matched asymptotic expansions. At higher speeds, the film exhibits two or three distinct regions: a uniform thinning film in the central region, an annular ridge near the contact line, and a possible Landau–Levich–Derjaguin-type (LLD-type) film in between that has not been reported previously. In particular, the LLD-type film occurs only at speeds slightly higher than the critical value for the existence of the equilibrium state, and leads to the decoupling of the uniform film and the ridge. It is found that the evolution of the ridge can be well described by a two-dimensional quasi-steady analysis. As a result, the ridge volume approaches a constant and cannot be neglected to predict the variation of the contact-line radius. The long-time behaviours of the film thickness and the contact radius agree with derived asymptotic solutions.

A model of imbibition dynamics in a channel of flattened triangular cross-section is presented, taking into account the liquid film flow in the corners of the channel. The quasi-analytical solutions are derived on the basis of a lubrication approximation. The analysis encompasses two imbibition scenarios corresponding to a constant flow rate or constant pressure imposed in the wetting fluid at the inlet of the channel. In the former case, the process starts with a liquid film flow regime in the corners that is followed by a bulk and corner film flow regime characterised by a triple point advancing (far) ahead of the bulk meniscus after its entrance in the channel. In the latter case, the occurrence of the bulk and corner film flow regime is conditioned by an imposed pressure yielding a capillary pressure at the inlet smaller than a threshold capillary pressure. Above this threshold, the liquid film regime remains. For both imbibition scenarios under concern, important features are highlighted, including (i) the time scalings of the dynamics of both the triple point and apex of the bulk meniscus (when it exists), (ii) the contrast in the positions of these two points showing that the classical Washburn approach, which neglects the effect of the corner films, overpredicts the dynamics of the bulk meniscus. The important consequence is an early wetting fluid breakthrough at the channel outlet much before the bulk meniscus arrival. Comparisons with experimental data available in the literature are provided, validating the approach proposed in this work.

The flow near a moving contact line depends on the dynamic contact angle, viscosity ratio and capillary number. We report experiments involving immersing a plate into a liquid bath, concurrently measuring the interface shape, interfacial velocity and fluid flow using digital image processing and particle image velocimetry. All experiments were performed at low plate speeds to maintain small Reynolds and capillary numbers for comparison with viscous theories. The dynamic contact angle, measured in the viscous phase, was kept below 90^{\circ }$90^{\circ }$ and the viscosity ratio, \lambda < 1$\lambda < 1$. This region of parameter space is largely unexplored for advancing contact lines. An important aim of the present study is to provide new experimental data against which new contact line models can be developed. The flow field is directly compared against the prediction from the viscous theory of Huh & Scriven (J. Colloid Interface Sci., vol. 35, issue 1, 1971, pp. 85–101) but with a slight modification involving the curved interface. Remarkable agreement is found between experiments and theory across a wide parameter range. The prediction for interfacial speed from Huh & Scriven is also in excellent agreement with experiments except in the vicinity of the contact line. Material points along the interface were found to rapidly slow down near the contact line, thus alleviating the singularity at the moving contact line. To the best of our knowledge, such a detailed test of theoretical models has not been performed before and we hope the present study will spur new modelling efforts in the field.

In spray cleaning, a multitude of small drops, violently accelerated by a high-speed gas stream, strike a dirty surface. This process is extremely effective: very little dirt can resist it. This is true even for dirt particles whose characteristic size is less than 100 nm. Spray cleaning is classically modelled by balancing particle adhesion with either inertial stress or viscous shear near the surface, the latter being calculated using droplet size and velocity as the characteristic length and velocity. This results in dimensionless numbers that are often well below one, suggesting that the mechanical stress exerted on the surface by the drop impact that detaches the particle is not well captured. Using quantitative nanoscale measurements, we show that the remarkable efficiency of spray cleaning results from the forced spreading of each droplet on the surface, which generates an unsteady and inhomogeneous shear confined to a boundary layer entrained in the wake of the liquid–solid contact line. In the very first moments of impact, the boundary layer is extremely thin, yielding a gigantic stress: the contact line of the spreading droplets sweeps all the surface particles away. We propose a quantitative model of spray cleaning based on this unsteady surface stress, which agrees well with (i) experimental data obtained with spray droplets of 34\ \mathrm {\mu }$34\ \mathrm {\mu }$m mean radius impacting the surface to be cleaned at a mean velocity ranging between 30 and 70 m s^{-1}$^{-1}$ and contamination by nanoparticles of varying nature and shape and (ii) data in the literature on spray cleaning.

During oscillatory wetting, a phase retardation emerges between contact angle variation and contact line velocity, presenting as a hysteresis loop in their correlation – an effect we term dynamic hysteresis. This phenomenon is found to be tunable by modifying the surface with different molecular layers. A comparative analysis of dynamic hysteresis, static hysteresis and contact line friction coefficients across diverse substrates reveals that dynamic hysteresis is not a result of dissipative effects but is instead proportionally linked to the static hysteresis of the surface. In the quest for appropriate conditions to model oscillatory contact line motion, we identify the generalized Hocking's linear law and modified generalized Navier boundary condition as alternative options for predicting realistic dynamic hysteresis.

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.

Superhydrophobic (SHPo) surfaces can capture a thin layer of air called a plastron under water to reduce skin friction. Although a ~30 % drag reduction has been recently reported with longitudinal micro-trench SHPo surfaces under a boat and in a towing tank, the results lacked the consistency to establish a clear trend. Designed based on Yu et al. (J. Fluid Mech, vol. 962, 2023, A9), this work develops and tests a series of high-performance SHPo surface coupons that can sustain a pinned plastron underneath a passenger motorboat revamped to reach 14 knots. Importantly, plastrons in a pinned state, not just their existence, are confirmed during flow experiments for the first time. All the drag-reduction data measured on different longitudinal micro-trenches are found to collapse if plotted against slip length in wall units. In comparison, aligned posts and transverse trenches show less and little drag reduction, respectively, confirming the adverse effect of the spanwise slip in turbulent flows. This report not only verifies SHPo surfaces can provide a consistent drag reduction at high speeds in open sea but also shows that one may predict the amount of drag reduction in turbulent flows using the physical slip length obtained for Stokes flows.

When placed at the surface of a volatile liquid, a sphere of hot dense non-volatile material remains suspended until it cools sufficiently. The duration of this ‘inverse Leidenfrost’ phenomenon depends on the Nusselt number Nu$Nu$ of the sphere, itself determined by flow in the film of vapour separating particle and liquid. It is shown that provided the Nusselt number is large, it can be calculated numerically using only the Laplace relation and the equations governing the thin film; patching to a solution for the outer thick film is not necessary. This method is demonstrated by using it to determine Nu$Nu$ for a sphere sufficiently small that in the governing equations, the acceleration due to gravity is negligible except where multiplied by the density of the sphere. Numerical results giving Nu$Nu$ as a function of a dimensionless measure of sphere weight are supplemented with analysis showing that, when the weight is of the order of the maximum supportable by surface tension alone, the film consists of a spherical bubble cap bounded by its contact rim. The solutions for these regions are coupled: although the apparent contact angle \chi$\chi$ for the cap is determined within the rim, its value depends on the flow rate arriving from the cap as well as on the additional evaporation from the rim. The latter acts to reduce \chi$\chi$ from the value it would otherwise have, thereby reducing the thickness of the entire cap. For the example treated here, the value of Nu$Nu$ is doubled by this mechanism.

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]$[\ell,m]$, where \ell$\ell$ and m$m$ 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]$[1,1]$ rocking mode, (2) harmonic [2,0]$[2,0]$ pumping mode, and (3) subharmonic rocking [1,1]$[1,1]$ 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.

Depinning of liquid droplets on substrates by flow of a surrounding immiscible fluid is central to applications such as cross-flow microemulsification, oil recovery and waste cleanup. Surface roughness, either natural or engineered, can cause droplet pinning, so it is of both fundamental and practical interest to determine the flow strength of the surrounding fluid required for droplet depinning on rough substrates. Here, we develop a lubrication-theory-based model for droplet depinning on a substrate with topographical defects by flow of a surrounding immiscible fluid. The droplet and surrounding fluid are in a rectangular channel, a pressure gradient is imposed to drive flow and the defects are modelled as Gaussian-shaped bumps. Using a precursor-film/disjoining-pressure approach to capture contact-line motion, a nonlinear evolution equation is derived describing the droplet thickness as a function of distance along the channel and time. Numerical solutions of the evolution equation are used to investigate how the critical pressure gradient for droplet depinning depends on the viscosity ratio, surface wettability and droplet volume. Simple analytical models are able to account for many of the features observed in the numerical simulations. The influence of defect height is also investigated, and it is found that, when the maximum defect slope is larger than the receding contact angle of the droplet, smaller residual droplets are left behind at the defect after the original droplet depins and slides away. The model presented here yields considerably more information than commonly used models based on simple force balances, and provides a framework that can readily be extended to study more complicated situations involving chemical heterogeneity and three-dimensional effects.

Both experimental and theoretical studies of fast and microscale physical phenomena occurring during the growth of vapour bubbles in nucleate pool boiling are reported. The focus is on the liquid film of micrometric thickness (a ‘microlayer’) that can form between the heater and the liquid–vapour interface of a bubble. The microlayer strongly affects the macroscale heat transfer and is thus important to be understood. The microlayer appears as a result of the inertial forces that cause the hemispherical bubble shape. It is shown that the microlayer can be seen as the Landau–Levich film deposited by the bubble foot edge during its receding. Paradoxically, the deposition is controlled by viscosity and surface tension. The microlayer profile measured with white-light interferometry, the temperature distribution over the heater, and the bubble shape are observed with synchronised high-speed cameras. According to the numerical simulations, the microlayer consists of two regions: a dewetting ridge near the contact line, and a longer and flatter bumped part. It is shown that the ridge cannot be measured by interferometry because of its intrinsic limitation on the interface slope. The ridge growth is linked to the contact line receding. The simulated dynamics of both the bumped part and the contact line agrees with the experiment. The physical origin of the bump in the flatter part of microlayer is explained.

Surface tension and wetting are dominating physical effects in microscale and nanoscale flows. We present an efficient and reliable model of surface tension and equilibrium contact angles in smoothed particle hydrodynamics for free-surface problems. We demonstrate its robustness and accuracy by simulating several three-dimensional free-surface flow problems driven by interfacial tension.

Droplet spreading is ubiquitous and plays a significant role in liquid-based energy systems, thermal management devices and microfluidics. While the spreading of non-volatile droplets is quantitatively understood, the spreading and flow transition in volatile droplets remains elusive due to the complexity added by interfacial phase change and non-equilibrium thermal transport. Here we show, using both mathematical modelling and experiments, that the wetting dynamics of volatile droplets can be scaled by the spatial–temporal interplay between capillary, evaporation and thermal Marangoni effects. We elucidate and quantify these complex interactions using phase diagrams based on systematic theoretical and experimental investigations. A spreading law of evaporative droplets is derived by extending Tanner's law (valid for non-volatile liquids) to a full range of liquids with saturation vapour pressure spanning from 10^1$10^1$ to 10^4$10^4$ Pa and on substrates with thermal conductivity from 10^{-1}$10^{-1}$ to 10^3\ {\rm W}\ {\rm m}^{-1}\ {\rm K}^{-1}$10^3\ {\rm W}\ {\rm m}^{-1}\ {\rm K}^{-1}$. In addition to its importance in fluid-based industries, the conclusions also enable a unifying explanation to a series of individual works including the criterion of flow reversal and the state of dynamic wetting, making it possible to control liquid transport in diverse application scenarios.

Using as a starting point conservation of momentum, a multiphase mechanical energy balance equation is derived that accounts for multiple material phases and interfaces present within a moving control volume. This balance is applied to a control volume that is anchored to a three-phase contact line as it advances continuously over the surface of a rough and chemically homogeneous and inert solid. Using semi-quantitative models for the material behaviour occurring within the control volume, an order of magnitude analysis is performed to neglect insignificant terms, producing an equation for predicting contact-angle hysteresis from a knowledge of the interface dynamics occurring around the three-phase contact line. It is shown that the viscous energy dissipation that occurs during the ‘stick–slip’ motion of the three-phase contact line, being the cause of contact-angle hysteresis on rough surfaces, can be calculated from changes in intermediate equilibrium interface states. The balance is applied to the Wenzel, Cassie–Baxter and Fakir (super-hydrophobic) wetting states, showing for the Fakir case that significant dissipation occurs during both interface advance and recede, and relating these dissipations to interfacial area changes that occur around the ‘stick–slip’ events.

The classical Cox–Voinov theory of contact line motion provides a relation between the macroscopically observable contact angle, and the microscopic wetting angle as a function of contact-line velocity. Here, we investigate how viscoelasticity, specifically the normal stress effect, modifies the wetting dynamics. Using the thin film equation for the second-order fluid, it is found that the normal stress effect is dominant at small scales yet can significantly affect macroscopic motion. We show that the effect can be incorporated in the Cox–Voinov theory through an apparent microscopic angle, which differs from the true microscopic angle. The theory is applied to the classical problems of drop spreading and dip coating, which shows how normal stress facilitates (inhibits) the motion of advancing (receding) contact lines. For rapid advancing motion, the apparent microscopic angle can tend to zero, in which case the dynamics is described by a regime that was already anticipated in Boudaoud (Eur. Phys. J. E, vol. 22, 2007, pp. 107–109).

Capillary folding is the process of folding planar objects into three-dimensional (3-D) structures using capillary force. It has many important industrial applications, e.g. the fabrication of microelectromechanical systems. In this work, we propose a 3-D model for the capillary folding of thin elastic sheets with pinned contact lines. The energy of the system consists of interfacial energies between the different phases and the elastic energy of the sheet. The latter is described by the nonlinear Koiter's model, which can accommodate large deformations of the sheet. From the energy, we derive the governing equations for the static system using a variational approach. We then develop a numerical method to find equilibrium solutions via a relaxation dynamics. At each time step, we evolve the sheet by using the subdivision element method, and update the droplet surface by minimizing a squared area functional using linear finite elements. Qualitatively, numerical solutions for the equilibrium configurations of the sheet–droplet system agree well with those observed in experiments. Furthermore, we identify the critical bendabilities and present bifurcation diagrams for the folding of a triangular sheet. The results exhibit rich and fully 3-D behaviours that were not captured by previous two-dimensional models. Our results provide new insights into the nonlinear process of capillary folding, and may contribute to the advancement of microfabrication techniques.

From industrial-scale production to small-scale fabrication of functional films, the blade coating method is used widely to apply a uniform thin liquid film on a moving substrate. However, conventional hydrodynamic models are inadequate for laboratory-scale low-speed blade coating, where capillary forces dominate. In this study, the low-speed blade coating of non-evaporative Newtonian fluids was investigated in experimental, computational and analytical approaches. The transient free boundary problem was solved utilizing a two-dimensional finite element method, and a simple viscocapillary model was developed to describe the viscous stress and capillary forces within the puddle, and predict the thickness of the wet film as a function of the speed of the substrate. Comparing the predicted film thickness with the computational results, and a flow visualization experiment of blade coating with silicone oil, respectively, confirmed the model's validity. The study indicates that the proposed model may be a useful tool for optimizing laboratory coating processes, as it provides a greater understanding of the low-speed blade coating system on a laboratory scale.

We investigate jumping of sessile droplets from a solid surface in ambient oil using modulated electrowetting actuation. We focus on the case in which the electrowetting effect is activated to cause droplet spreading and then deactivated exactly at the moment the droplet reaches its maximum deformation. By systematically varying the control parameters such as the droplet radius, liquid viscosity and applied voltage, we provide detailed characterisation of the resulting behaviours including a comprehensive phase diagram separating detachment from non-detachment behaviours, as well as how the detach velocity and detach time, i.e. duration leading to detachment, depend on the control parameters. We then construct a theoretical model predicting the detachment condition using energy conservation principles. We finally validate our theoretical analysis by experimental data obtained in the explored ranges of the control parameters.