We develop a time-dependent conformal method to study the effect of viscosity on steep surface waves. When the effect of surface tension is included, numerical solutions are found that contain highly oscillatory parasitic capillary ripples. These small-amplitude ripples are associated with the high curvature at the crest of the underlying viscous-gravity wave, and display asymmetry about the wave crest. Previous inviscid studies of steep surface waves have calculated intricate bifurcation structures that appear for small surface tension. We show numerically that viscosity suppresses these. While the discrete solution branches still appear, they collapse to form a single smooth branch in the limit of small surface tension. These solutions are shown to be temporally stable, both to small superharmonic perturbations in a linear stability analysis, and to some larger amplitude perturbations in different initial-value problems. Our work provides a convenient method for the numerical computation and analysis of water waves with viscosity, without evaluating the free-boundary problem for the full Navier–Stokes equations, which becomes increasingly challenging at larger Reynolds numbers.

Gravito–capillary waves at free surfaces are ubiquitous in several natural and industrial processes involving quiescent liquid pools bounded by cylindrical walls. These waves emanate from the relaxation of initial interface distortions, which often take the form of a cavity (depression) centred on the symmetry axis of the container. The surface waves reflect from the container walls leading to a radially inward propagating wavetrain converging (focussing) onto the symmetry axis. Under the inviscid approximation and for sufficiently shallow cavities, the relaxation is well-described by the linearised potential-flow equations. Naturally, adding viscosity to such a system introduces viscous dissipation that enervates energy and dampens the oscillations at the symmetry axis. However, for viscous liquids and deeper cavities, these equations are qualitatively inaccurate. In this study, we decompose the initial localised interface distortion into several Bessel functions and study their time evolution governing the propagation of concentric gravito–capillary waves on a free surface. This is carried out for inviscid as well as viscous liquids. For a sufficiently deep cavity, the inward focussing of waves results in large interfacial oscillations at the axis, necessitating a second-order nonlinear theory. We demonstrate that this theory effectively models the interfacial behaviour and highlights the crucial role of nonlinearity near the symmetry axis. This is rationalised via demonstration of the contribution of bound wave components to the interface displacement at the symmetry axis Contrary to expectations, the addition of slight viscosity further intensifies the oscillations at the symmetry axis although the mechanism of wavetrain generation here is quite different compared with bubble bursting where such behaviour is well known (Duchemin et al., Phys. Fluids, vol. 14, issue 9, 2002, pp. 3000–3008). This finding underscores the limitations of the potential flow model and suggests avenues for more accurate modelling of such complex free-surface flows.

The stability and dynamics of solitary waves propagating along the surface of an inviscid ferrofluid jet in the absence of gravity are investigated analytically and numerically. For the axisymmetric geometry, the problem is shown to be a conservative system with total energy as the Hamiltonian; however, one of the canonical variables differs from those in the classic water-wave problem in the Cartesian coordinate system. The Dirichlet–Neumann operator appearing in the kinetic energy is then expanded as a Taylor series, described in homogeneous powers of the surface displacement. Based on the further analysis of the Dirichlet–Neumann operator, a systematic procedure is proposed to derive reduced model equations of multiple scales in various asymptotic limits from the full Euler equations in the Hamiltonian/Lagrangian framework. Particularly, a fully dispersive model arising from retaining terms valid up to the quartic order in the series expansion of the kinetic energy, which results in quadratic and cubic algebraic nonlinearities in Hamilton's equations and henceforth is abbreviated as the cubic full-dispersion model, is proposed. By comparing bifurcation curves and wave profiles of various types of axisymmetric solitary waves among different model equations, the cubic full-dispersion model is found to agree well with the full Euler equations, even for waves of considerably large amplitudes. The stability properties of axisymmetric solitary waves subjected to longitudinal disturbances are verified with the newly proposed model. Our analytical results, consistent with Saffman's theory, indicate that in the axisymmetric cylindrical system, the stability exchange subjected to superharmonic perturbations also occurs at the stationary point of the speed-energy bifurcation curve. A series of numerical experiments for the stability and dynamics of solitary waves are performed via the numerical time integration of the model equation, and collision interactions between stable solitary waves show non-elastic features.

We present herein the derivation of a lubrication-mediated (LM) quasi-potential model for droplet rebounds off deep liquid baths, assuming the presence of a persistent dynamic air layer which acts as a lubricating pressure transfer. We then present numerical simulations of the LM model for axisymmetric rebounds of solid spheres and compare quantitatively to current results in the literature, including experimental data in the low-speed impact regime. In this regime the LM model has the advantage of being far more computationally tractable than direct numerical simulation (DNS) and is also able to provide detailed behaviour within the micro-metric thin lubrication region. The LM system has an interesting mathematical structure, with the lubrication layer providing a free-boundary elliptic problem mediating the drop and bath free-boundary evolutionary equations.

A theory is presented for wave-driven propulsion of floating bodies driven into oscillation at the fluid interface. By coupling the equations of motion of the body to a quasipotential flow model of the fluid, we derive expressions for the drift speed and propulsive thrust of the body which in turn are shown to be consistent with global momentum conservation. We explore the efficacy of our model in describing the motion of SurferBot (Rhee et al., Bioinspir. Biomim., vol. 17, issue 5, 2022), demonstrating close agreement with the experimentally determined drift speed and oscillatory dynamics. The efficiency of wave-driven propulsion is then computed as a function of driving oscillation frequency and the forcing location, revealing optimal values for both of these parameters which await confirmation in experiments. A comparison with other modes of locomotion and applications of our model with competitive water sports is discussed in conclusion.

The impact of a liquid droplet with another droplet or onto a solid surface are important basic processes that occur in many applications such as agricultural sprays and inkjet printing, and in nature such as pathogens transport by raindrops. We investigated the head-on collision of unequal-size droplets of the same liquid on wetting surfaces using the direct numerical simulations technique at different size ratios. The unsteady Navier–Stokes equations are solved and the liquid–gas interface is tracked using the geometric volume-of-fluid method. The numerical model is validated by comparing simulation results of two extreme cases of droplets bouncing with the experimental data from previous studies and the agreement is quite accurate. The validated model is employed to simulate droplets bouncing at several size ratios at different Weber numbers and Ohnesorge number. Two distinct regimes are identified, namely, the inertial regime, where the restitution coefficient is a constant value close to 0.3, the viscous regime, where the restitution coefficient declines. To understand the bouncing behaviour, the velocity field is analysed and an energy budget calculation is performed. The distribution of the sessile droplet energy is found to be important and the sessile droplet surface energy is calculated by its deformation characteristics such as crater depth. Finally, a scaling analysis is performed to rationalize the insensitivity of the coefficient of restitution in the inertial regime, and its decline in the viscous regime, at large size ratios.

The rupture of the thin film at the top of a bubble at a liquid–gas interface leads to an axisymmetric collapse of the bubble cavity. We present scaling laws for such a cavity collapse, established from experiments conducted with bubbles spanning a wide range of Bond (${10^{-3}< Bo\leq 1}$) and Ohnesorge numbers (${10^{-3}< Oh<10^{-1}}$), defined with the bubble radius $R$. The cavity collapse is a capillary-driven process, with a dependency on viscosity and gravity, affecting respectively, precursory capillary waves on the cavity boundary and the static bubble shape. The collapse is characterised by the normal interface velocity ($U_n$) and by the tangential wave propagation velocity of the kink ($U_t$), defined by the intersection of the concave cavity boundary formed after the rupture of the thin film with the convex boundary of the bubble cavity. During the collapse, $U_t$ remains constant and is shown to be $U_t=4.5U_c{\mathcal {W}}_R$, where $U_c$ is the capillary velocity and ${\mathcal {W}}_R(Oh,Bo)={(1-\sqrt {Oh {\mathscr {L}}} )^{-1/2}}$ is the wave resistance factor due to the precursory capillary waves, with $\mathscr {L}(Bo)$ being the path correction of the kink motion. The movement of the kink in the normal direction is part of the inward shrinkage of the whole cavity due to the sudden reduction of gas pressure inside the bubble cavity after the thin film rupture. This normal velocity is shown to scale as $U_c$ in the equatorial plane, while at the bottom of the cavity $\bar {U}_{nb}=U_c(Z_c/R)({\mathcal {W}_R}/ {\mathscr {L}})$, where $Z_c(Bo)$ is the static cavity depth. The filling rate of the cavity, which remains a constant throughout the collapse, is shown to be entirely determined by the shrinking velocity and scales as ${Q_T\simeq 2{\rm \pi} R Z_c U_c}$. From $Q_T$ we recover the jet velocity scaling, thereby relating the cavity collapse with the jet velocity scaling.

We present new numerical solutions for nonlinear standing water waves when the effects of both gravity and surface tension are considered. For small values of the surface tension parameter, solutions are shown to exhibit highly oscillatory capillary waves (parasitic ripples), which are both time- and space-periodic, and which lie on the surface of an underlying gravity-driven standing wave. Our numerical scheme combines a time-dependent conformal mapping together with a shooting method, for which the residual is minimised by Newton iteration. Previous numerical investigations typically clustered gridpoints near the wave crest, and thus lacked the fine detail across the domain required to capture this phenomenon of small-scale parasitic ripples. The amplitude of these ripples is shown to be exponentially small in the zero surface tension limit, and their behaviour is linked to (or explains) the generation of an elaborate bifurcation structure.

Gravity–capillary waves are waves influenced by both the effects of surface tension and gravity; these waves are at small scales with wavelength range from approximately 10 cm to less than 1 cm. Gravity–capillary waves play a significant role in air–sea interactions, and they exhibit much different features compared with gravity waves. They can be observed widely on the sea surface. Parasitic waves (capillary waves generated by and that ride on gravity and/or gravity–capillary waves) and micro-breaking can be observed on the water surface with winds; however, the presence of wind makes it difficult to analyse the mechanisms of the wave itself. In this paper, parasitic waves and micro-breaking on gravity–capillary waves are examined experimentally, both in the absence of wind. Parasitic waves and axisymmetric micro-breaking waves are generated mechanically in a convergent channel, where energy density increases due to spatial convergence. Three experimental techniques are used to measure wave properties: planar laser-induced fluorescence, particle image velocimetry and shadowgraphs. The wave profile evolution and vortices beneath the parasitic waves are studied. The micro-breaking of gravity–capillary waves is observed on a surface with added surfactant. The surfactant increases the Bond number, and makes breaking possible in these small-scale waves. Energy dissipation of parasitic waves and micro-breaking is quantified, and the enhanced dissipation caused by parasitic waves is identified through the experiments. In this study, mechanically generated breaking waves with wavelengths less than 10 cm are studied for the first time, without the effect of wind. The results yield insight into wave characteristics and energy dissipation on the air–sea interface at small scales.

Natural oscillations of sessile drops with a free or pinned contact line in different gravity environments are studied based on a linear inviscid irrotational theory. The inviscid Navier–Stokes equations and boundary conditions are reduced to a functional eigenvalue problem by the normal-mode decomposition. We develop a boundary element method model to numerically solve the eigenvalue problem for predicting the natural frequencies. Emphasis is placed on the frequency shifts of modes due to gravity for a wide range of contact angles $\alpha$ and Bond numbers $Bo$. Three types of $\alpha$–$Bo$ diagrams reflecting how gravity shifts the frequency are identified. Specifically, the frequency of zonal modes shifts downwards (upwards) when $\alpha$ is smaller (larger) than a critical value, while the frequencies of most sectoral modes are shifted downwards regardless of $\alpha$. As a result, gravity can transform the lowest mode from a zonal mode to a sectoral mode. The spectral degeneracy of hemispherical drops inherited from the Rayleigh–Lamb spectrum is also broken by gravity. However, we discover that gravity has no effect on the mode associated with the horizontal motion of the centre of mass, whose frequency is always zero regardless of $\alpha$ and $Bo$. This implies that the ‘walking’ drop instability reported in previous literature does not exist.

We study the dynamics of capillary waves at the interface of a two-layer stratified turbulent channel flow. We use a combined pseudo-spectral/phase field method to solve for the turbulent flow in the two liquid layers and to track the dynamics of the liquid–liquid interface. The two liquid layers have same thickness and same density, but different viscosity. We vary the viscosity of the upper layer (two different values) to mimic a stratified oil–water flow. This allows us to study the interplay between inertial, viscous and surface tension forces in the absence of gravity. In the present set-up, waves are naturally forced by turbulence over a broad range of scales, from the larger scales, whose size is of order of the system scale, down to the smaller dissipative scales. After an initial transient, we observe the emergence of a stationary capillary wave regime, which we study by means of temporal and spatial spectra. The computed frequency and wavenumber power spectra of wave elevation are in line with previous experimental findings and can be explained in the frame of the weak wave turbulence theory. Finally, we show that the dispersion relation, which gives the frequency ($\omega$) as a function of the wavenumber ($k$), is in good agreement with the well-established theoretical prediction, $\omega (k) \sim k^{3/2}$.

The rebound of droplets impacting a deep fluid bath is studied both experimentally and theoretically. Millimetric drops are generated using a piezoelectric droplet-on-demand generator and normally impact a bath of the same fluid. Measurements of the droplet trajectory and other rebound metrics are compared directly with the predictions of a linear quasipotential model, as well as fully resolved direct numerical simulations of the unsteady Navier–Stokes equations. Both models resolve the time-dependent bath and droplet shapes in addition to the droplet trajectory. In the quasipotential model, the droplet and bath shape are decomposed using orthogonal function decompositions leading to two sets of coupled damped linear harmonic oscillator equations solved using an implicit numerical method. The underdamped dynamics of the drop are directly coupled to the response of the bath through a single-point kinematic match condition which we demonstrate to be an effective and efficient model in our parameter regime of interest. Starting from the inertio-capillary limit in which both gravitational and viscous effects are negligible, increases in gravity or viscosity lead to a decrease in the coefficient of restitution and an increase in the contact time. The inertio-capillary limit defines an upper bound on the possible coefficient of restitution for droplet–bath impact, depending only on the Weber number. The quasipotential model is able to rationalize historical experimental measurements for the coefficient of restitution, first presented by Jayaratne & Mason (Proc. R. Soc. Lond. A, vol. 280, issue 1383, 1964, pp. 545–565).

We calculate the net energy per unit time exchanged between two sets of modes in a generic system governed by a three-wave kinetic equation. Our calculation is based on the property of detailed energy conservation of the triadic resonant interactions. In a first application to isotropic systems, we re-derive the previously used formula for the energy flux as a particular case for adjacent sets. We then exploit the new formalism to quantify the level of locality of the energy transfers in the example of surface capillary waves. A second application to anisotropic wave systems expands the currently available set of tools to investigate magnitude and direction of the energy fluxes in these systems. We illustrate the use of the formalism by characterizing the energy pathways in the oceanic internal wavefield. Our proposed approach, unlike traditional approaches, is not limited to stationarity, scale invariance and strict locality. In addition, we define a number $w$ that quantifies the scale separation necessary for two sets of modes to having negligible mutual energy exchange, with potential consequences in the interpretation of wave turbulence experiments. The methodology presented here provides a general, simple and systematic approach to energy fluxes in wave turbulence.

Progressive capillary waves on the interface between two homogeneous fluids confined in a channel with rigid walls parallel to the undisturbed interface are investigated. This problem is formulated as a system of integrodifferential equations that can be solved numerically via a boundary integral equation method coupled with series expansions of the unknown functions. With this highly accurate scheme and numerical continuation, we explore the global bifurcation of periodic travelling waves. It is found that there are two types of limiting profile, self-intersecting and boundary-touching, which appear either along a primary branch bifurcating from infinitesimal periodic waves or on an isolated branch existing above a certain finite amplitude. For particular sets of parameters, these two types of bifurcation curves can intersect, which can be viewed as a secondary bifurcation phenomenon occurring on the primary branch. Based on asymptotic and numerical analyses of the almost limiting waves, it is found that the boundary-touching solutions feature a circular geometry, i.e. the interface is pieced together by circular arcs of the same radius. A theoretical investigation yields the necessary conditions for the existence of these extreme waves, whereby we can predict the limiting configurations for most parameter sets. The comparisons between theoretical predictions and numerical results show good agreement.

Numerical studies of dimple and jet formation from a collapsing cavity often model the initial cavity shape as a truncated sphere, mimicking a bursting bubble. In this study, we present a minimal model containing only nonlinear inertial and capillary forces, which produces dimples and jets from a collapsing capillary wave trough. The trough in our simulation develops from a smooth initial perturbation, chosen to be an eigenmode to the linearised ${O}(\epsilon )$ problem ($\epsilon$ is the non-dimensional amplitude). We explain the physical mechanism of dimple formation and demonstrate that, for moderate $\epsilon$, the sharp dimple seen in simulations is well captured by the weakly nonlinear ${O}(\epsilon ^3)$ theory developed here. For $\epsilon \gg 1$ the regime is strongly nonlinear, spreading surface energy into many modes, and the precursor dimple now develops into a sharply rising jet. Here, simulations reveal a novel localised window (in space and time) where the jet evolves self-similarly following inviscid (Keller & Miksis, SIAM J. Appl. Maths, vol. 43, issue 2, 1983, pp. 268–277) scales. We develop an analogy of this regime to a self-similar solution of the first kind, for linearised capillary waves. Our first-principles study demonstrates that, at sufficiently small scales, dimples and jets can form from radial inward focusing of capillary waves, and the formation of this may be described by a relatively simple model employing (nonlinear) inertial and capillary effects. Viscosity and gravity can, however, significantly influence the focusing process, either intensifying the singularity or weakening it (Walls et al., Phys. Rev. E, vol. 92, issue 2, 2015, 021002; Gordillo & Rodríguez-Rodríguez, J. Fluid Mech., vol. 867, 2019, pp. 556–571). This leads, in particular, to critical values of Ohnesorge and Bond numbers, which cannot be obtained from our minimal model.

The influence of a magnetic field on the pattern of stationary waves formed on the surface of a magnetic fluid (ferrofluid) when an obstacle moves has been studied both theoretically and experimentally. It is found that a vertical magnetic field narrows the cone of stationary waves and increases their amplitude. In the wake region, the peaks of the Rosensweig instability appear in a magnetic field that is smaller than the critical field that determines this instability occurrence. A horizontal magnetic field parallel to the obstacle velocity expands the cone of waves but reduces their amplitude up to the suppression of stationary waves. A horizontal field perpendicular to the obstacle velocity also expands the cone of waves and stabilizes their amplitude.

We demonstrate dynamic stabilisation of axisymmetric Fourier modes susceptible to the classical Rayleigh–Plateau (RP) instability on a liquid cylinder by subjecting it to a radial oscillatory body force. Viscosity is found to play a crucial role in this stabilisation. Linear stability predictions are obtained via Floquet analysis demonstrating that RP unstable modes can be stabilised using radial forcing. We also solve the linearised, viscous initial-value problem for free-surface deformation obtaining an equation governing the amplitude of a three-dimensional Fourier mode. This equation generalizes the Mathieu equation governing Faraday waves on a cylinder derived earlier in Patankar et al. (J. Fluid Mech., vol. 857, 2018, pp. 80–110), is non-local in time and represents the cylindrical analogue of its Cartesian counterpart (Beyer & Friedrich, Phys. Rev. E, vol. 51, issue 2, 1995, p. 1162). The memory term in this equation is physically interpreted and it is shown that, for highly viscous fluids, its contribution can be sizeable. Predictions from the numerical solution to this equation demonstrate the predicted RP mode stabilisation and are in excellent agreement with simulations of the incompressible Navier–Stokes equations (up to the simulation time of several hundred forcing cycles).

The propagation speed, shape and stability of the rim generated by a liquid-curtain breakup are studied. In the experiment, a liquid curtain surrounded by a slot die, edge guides and the surface of a roller breaks at the contact point between the edge guide and roller in a low-Weber-number range, and the rim propagates in the horizontal direction. Except for the initial time, the rim is almost straight and has a nearly constant propagation speed. For an Ohnesorge number much smaller than 1, unevenness occurs on the rim and the droplets separate from it. When the Ohnesorge number is of the order of unity, the rim becomes convex vertically downward, and the liquid lump flows down. The shape, propagation speed and surface stability of the rim are discussed by analysing the equation proposed by Entov & Yarin (J. Fluid Mech., vol. 140, 1984, pp. 91–111). It is shown that the volume flow rate condition at the slot die exit is important to explain the propagation of the rim. Additionally, in the initial stage of the curtain breakup, the Plateau–Rayleigh instability causes unevenness on the rim surface, and after the rim reaches the slot die exit, the Rayleigh–Taylor instability generates a liquid lump on the rim, which grows into droplets when the Ohnesorge number is much less than 1.

Interfacial waves between two superimposed dielectric fluid layers under a horizontal electric field are investigated from asymptotic and numerical aspects. The fluid is taken to be inviscid, incompressible and non-conducting in each layer. The competing forces resulting from gravity, surface tension and electric field are all considered. A systematic procedure is proposed to derive model equations of multiple scales in various possible limits from the electrified Euler equations in the framework of the Zakharov–Craig–Sulem formulation. Based on thorough analyses of the Dirichlet–Neumann operators in the long-wave approximation, classic weakly nonlinear models – including the Boussinesq-type system, the Korteweg–de Vries (KdV) equation and its variants, and the Benjamin-type equation – are obtained under different scaling assumptions. In addition, strongly nonlinear models (without the smallness assumption on the wave amplitude) in the Benjamin and Barannyk–Papageorgiou–Petropoulos regimes are derived. In these models, the electric effects are shown to produce dispersive regularisations of long and short waves. A modified boundary integral equation method is developed to compute solitary waves in the original electrified Euler equations. Through comparisons with solitary-wave solutions in the Euler equations, it is found that in various asymptotic regimes, weakly nonlinear models are in overall good agreement when wave amplitudes are small. In contrast, the range of validity of the strongly nonlinear model is much broader. It is shown that the horizontal electric field plays a significant role in the physical system: it expands the range of parameters for the existence of progressive waves, changes the qualitative characteristics of solitary waves and leads to a new type of solitary wave, namely the KdV–wavepacket mixed type.

We consider a spherical particle levitating above a liquid bath owing to the Leidenfrost effect, where the vapour of either the bath or sphere forms an insulating film whose pressure supports the sphere’s weight. Starting from a reduced formulation based on a lubrication-type approximation, we use matched asymptotics to describe the morphology of the vapour film assuming that the sphere is small relative to the capillary length (small Bond number) and that the densities of the bath and sphere are comparable. We find that this regime is comprised of two formally infinite sequences of distinguished limits which meet at an accumulation point, the limits being defined by the smallness of an intrinsic evaporation number relative to the Bond number. These sequences of limits reveal a surprisingly intricate evolution of the film morphology with increasing sphere size. Initially, the vapour film transitions from a featureless morphology, where the thickness profile is parabolic, to a neck–bubble morphology, which consists of a uniform pressure bubble bounded by a narrow and much thinner annular neck. Gravity effects then become important in the bubble leading to sequential formation of increasingly smaller neck–bubble pairs near the symmetry axis. This process terminates when the pairs closest to the symmetry axis become indistinguishable and merge. Subsequently, the inner section of that merger transitions into a uniform-thickness film that expands radially, gradually squishing increasingly larger neck–bubble pairs into a region of localised oscillations sandwiched between the uniform film and what remains of the bubble whose radial extent is presently comparable to the uniform film; the neck–bubble pairs farther from the axis remain essentially intact. Ultimately, the uniform film gobbles up the largest outermost bubble, whereby the morphology simplifies to a uniform film bounded by localised oscillations. Overall, the asymptotic analysis describes the continuous evolution of the vapour film from a neck–bubble morphology typical of a Leidenfrost drop levitating above a flat solid substrate to a uniform-film morphology which resembles that in the case of a large liquid drop levitating above a liquid bath.