In this work, we revisit the Generalised Navier Boundary Condition (GNBC) introduced by Qian et al. in the sharp interface volume-of-fluid context. We replace the singular uncompensated Young stress by a smooth function with a characteristic width $\varepsilon \gt 0$ that is understood as a physical parameter of the model. Therefore, we call the model the ‘contact region GNBC’ (CR-GNBC). We show that the model is consistent with the fundamental kinematics of the contact angle transport described by Fricke, Köhne and Bothe. We implement the model in the geometrical volume-of-fluid solver Basilisk using a ‘free angle’ approach. This means that the dynamic contact angle is not prescribed, but reconstructed from the interface geometry and subsequently applied as an input parameter to compute the uncompensated Young stress. We couple this approach to the two-phase Navier–Stokes solver and study the withdrawing tape problem with a receding contact line. It is shown that the model allows for grid-independent solutions and leads to a full regularisation of the singularity at the moving contact line, which is in accordance with the thin film equation subject to this boundary condition. In particular, it is shown that the curvature at the moving contact line is finite and mesh converging. As predicted by the fundamental kinematics, the parallel shear stress component vanishes at the moving contact line for quasi-stationary states (i.e. for $\dot \theta _d=0$), and the dynamic contact angle is determined by a balance between the uncompensated Young stress and an effective contact line friction. Furthermore, a nonlinear generalisation of the model is proposed, which aims at reproducing the molecular kinetic theory of Blake and Haynes for quasi-stationary states.

The instability of a liquid film in a nanotube is significantly influenced by van der Waals forces. A theoretical framework based on the axisymmetric Stokes equations is developed to investigate their effects through linear stability analysis. The model reveals that van der Waals forces markedly enhance perturbation growth, reduce the dominant wavelength, and lower the critical film thickness that distinguishes collapse from non-collapse regimes. Direct numerical simulations of the Navier–Stokes equations both confirm these theoretical predictions and extend the analysis into the nonlinear regime. In this regime, van der Waals forces are found to alter the interfacial morphology and suppress the formation of satellite lobes. Both rupture and collapse follow a universal temporal scaling law with exponent $1/3$, and exhibit self-similar behaviour near the singularity.

This study presents an in-depth analysis of the energy dissipation and momentum balance during a laminar planar hydraulic jump in a viscous free surface flow, with shallow flow theory used to estimate the relevant jump parameters. The inclusion of momentum and kinetic energy correction factors incorporates the influence of the fluid nature. The fluid is described by the generalised Herschel–Bulkley model with Papanastasiou regularisation, which reduces to the Bingham plastic, power-law and Newtonian models under relevant limiting conditions. The analysis, extensively validated against experimental and simulated data, is explored to understand the physics of free surface flow during jump formation. Energy dissipation increases with an increase in the flow behaviour index n, flow consistency index k and yield stress τo since each of them increases the apparent viscosity. Interestingly, it is higher in the supercritical (upstream) compared with the subcritical (downstream) zone. For constant discharge rate and film thickness, the specific energy depends on the velocity profile and is thus a function of n and τo but not k, and the mechanism of influence of n and τo are also different. For a generalised approach, energy dissipation and jump parameters are discussed as a function of relevant non-dimensional numbers obtained from SFT. Energy dissipation during a hydraulic jump in non-Newtonian liquids is a hitherto unexplored aspect. In fact, energy dissipation during a planar jump in a viscous Newtonian liquid is also rare, although hydraulic jumps are primarily used as energy dissipators in free surface flows.

This study focuses on the modelling and dynamics of gravity-driven, axisymmetric thin liquid film flow along a conical surface. Spatial linear stability analysis is performed on the basis of a Benney-type equation derived for the present configuration. In particular, streamwise curvature of the free surface is found to exert a crucial influence on the stability threshold. For simulations of surface waves, a second-order low-dimensional model is developed under the long-wave assumption, achieving accuracy comparable to direct numerical simulations at far lower cost. With this model, the characteristics of both linear and nonlinear waves are examined. A key difference from the flow over a flat plate is the dependence of the wave dynamics on the radial distance from the cone apex. At relatively high flow rates, a transition from solitary to sinusoidal waves is observed, with the transition position correlating closely with the linear stability threshold. Within the parameter range investigated, quantitative results of the conical film flow are almost identical to those in the flat-plate case when local parameters are substituted, indicating that inertial effects of the conical geometry are negligible. The models and findings presented in this paper may aid the design and optimisation of industrial processes such as film coating and liquid-film-based heat and mass transfer on conical surfaces.

In this experimental work, a two-dimensional (wedge) and three-dimensional solids (conus, 4 and 6-sided pyramids) with different deadrise angles (1–$5^\circ$) impact a deep liquid pool (distilled water or 2.5 % butanol–water solution) at a speed varying from 0.50 to 19.75 cm s−1. Below a limit speed dependent on the deadrise angle, ‘exotic’ terminal forms of air entrapment are observed: a large central bubble, two parallel lines of bubbles for the two-dimensional solid, a trail of bubbles, necklace of bubbles, doughnut-shaped bubble and large central bubble for the three-dimensional solids. Above this limit speed, the collapse of the air film forms a line of bubbles near the central edge for the two-dimensional solid, and one/multiple bubbles near the vertex for the three-dimensional solids. The entrapment dynamic is observed using a high-speed camera with a total internal reflection set-up. The outer border of the wetted area expands linearly in time, with a speed that agrees with Wagner’s theory for wedge and conus, which provides the lower and upper limites for genuinely three-dimensional cases (pyramids). The decrease in the size of the air film over time is exponential. The measured initial characteristic size of the air film is proportional to the air dynamic viscosity and inversely proportional to the liquid density, impact velocity and squared deadrise angle, as expected from an air–water lubrication–inertia balance. The prefactor in the scaling law depends on the shape of the solid with a slight but detectable effect of liquid surface tension on its value.

A computational fluid dynamics simulation of subcooled flow boiling of water at 10.5 ${\rm bar}$, with an applied heat flux of $1\,{\rm MW}\,{\rm m}^{-2}$ and subcooling of 10 ${\rm K}$, was performed using an interface tracking method. The simulation replicated the conditions of an experiment conducted at MIT. The objectives are to elucidate heat-transfer mechanisms in moderate-pressure subcooled boiling and to validate the simulation method, with a focus on quantities that are difficult to measure experimentally, such as the distributions of velocity, temperature, bubble number density and heat-flux partitioning. Due to the small bubble size under high pressure, fine grids are required. Simulated bubble shapes, wall temperatures and vapour area fractions show good agreement with the experimental results. The simulations reveal that a very thin liquid layer (${\lt}4\,\unicode{x03BC}{\rm m}$) surrounding the bubbles is highly effective at removing heat from the surface. The local wall heat fluxes beneath medium and large bubbles, excluding the heat flux associated with seed-bubble generation, are approximately 0.9 and 0.4 ${\rm MW}\,{\rm m}^{-2}$, respectively; the latter is smaller because of the presence of thicker liquid films (14–70 $\unicode{x03BC}{\rm m}$) that thermally insulate the wall. In the single-phase liquid region, the heat transfer coefficient reaches $42\,{\rm kW}\,{\rm m}^{-2}\,{\rm K}^{-1}$ as a result of strong turbulent heat flux in the wall-normal direction; this turbulent heat flux is approximately eight times larger than in the equivalent single-phase liquid flow.

Thin liquid films play an instrumental role in the coating industry. In many cases, these films consist of multiple components and are applied in multiple layers. However, multilayer multicomponent coatings can readily develop thickness non-uniformities due to Marangoni flows driven by solute concentration gradients. Previous flow visualisation experiments have demonstrated that the addition of surfactant can suppress such non-uniformities, but the physical mechanisms underlying this suppression have not yet been definitively established. We investigate the growth of film-height non-uniformities in a two-layer multicomponent coating consisting of a solute-rich bottom layer, a solute-depleted top layer and surfactant. A lubrication-theory-based model that accounts for vertical and lateral gradients in solute and surfactant concentrations is developed. The resulting coupled nonlinear partial differential equations describing the film height, solute concentration and surfactant concentration are solved with a pseudospectral method. Our findings reveal that surfactant-induced Marangoni flows can significantly decrease film-height non-uniformities by competing with Marangoni flows due to solute concentration gradients. Several simplifications of the governing equations are explored to determine how well predictions from these simplified models compare with the full lubrication-theory-based model, thereby providing insight into dominant physical mechanisms in different parameter regimes. The role of surfactant solubility and sorption kinetics in controlling perturbation growth is also examined.

While studying soap film bursting to validate their opening velocity, i.e. the Taylor–Culick velocity, Mysels and co-workers discovered fifty years ago a compression region propagating in front of the hole that they called the aureole. In the wake of such a discovery, a series of papers ‘Bursting of soap films’ focused on the study of such peculiar Marangoni flow resulting from the rapid surfactant compression. Their pioneering theory postulates that surfactants remain insoluble at the interface, leading to a self-similar process that has been verified on small films. In the present study, by using films large enough to allow the surfactant to relax, we reveal a previously unexplored regime of aureole development. The surfactants forming the aureole initially behave as if they were insoluble, with an aureole front propagating at a constant speed. After a few milliseconds, however, the front slows down until it matches the hole-opening velocity, and the aureole length then becomes constant. In this steady regime, a model taking into account surfactant advection/diffusion in the film is developed. Our theory accurately captures the thickness and velocity exponential profiles observed in experiments, demonstrating that the observed deviations arise from a balance between the surfactant rapid compression and a desorption flux. Furthermore, measurements of the characteristic aureole lengths provide estimates of physico-chemical properties of the monolayer, which are discussed in the light of predictions based on adsorption laws. The present study highlights the transition from the insoluble limit to the soluble limit, and paves the way for measurement of out-of-equilibrium dynamics of surfactants.

We present a combined theoretical and numerical investigation of the inertial exit dynamics of a long horizontal circular cylinder vertically lifted out of a finite-size liquid bath at constant velocity. The various steps of the exit dynamics are studied in detail: from the formation of a bulge on the surface ahead of the cylinder to the coating of the cylinder by a liquid film while crossing the interface. We focus on inertial dynamics, a regime characteristic of large exit velocities, i.e. large Reynolds numbers ($500 \lt \textit{Re} \lt 10\,000$) and negligible interfacial effects. The dynamics is investigated through two-dimensional computations of the Navier–Stokes equations using a finite element method with moving boundaries. We describe in detail the exit dynamics while emphasising the effect of various parameters on surface deformation and resistive force. We identify subtle effects and interplay, such as initial free-surface response after impulsive start-up, the important role of the lateral bounding of the reservoir, and the close relationship between wake size and surge amplitudes as revealed by comparing with free-slip cylinder simulations. All these aspects are shown to be crucial to accurately predict the coated film thickness and the exit force.

The linear instability of liquid film with insoluble surfactants on a quasiperiodic oscillating plane for disturbances with arbitrary wavenumbers is investigated. The combined effects of insoluble surfactants and quasiperiodic oscillation on the instability are described using Floquet theory. For long-wavelength instability, the solution in the limit of long wave perturbations is obtained by the asymptotic expansion method. The results show that a new stable region emerges in the low-frequency domain of the neutral stability curve in the absence of gravity. As the imposed frequency increases, this newly formed stable region is progressively absorbed into a broader stable zone. The U-shaped neutral curves with separation bandwidth appear in the presence of gravity, and the presence of the surfactants will decrease the unstable frequency bandwidth and increase the critical Reynolds number. The finite-wavelength instability is solved numerically based on the Chebyshev spectral collocation method. Both travelling-wave and standing-wave modes are found due to the existence of surface surfactants. As the surfactant concentration increases, the finite-wavelength instability region expands significantly, and the intersection point marking the transition from travelling waves to standing waves shifts progressively towards lower frequencies. The physical mechanisms underlying perturbation growth are further elucidated through an energy budget analysis. Energy budget analysis demonstrates that long-wavelength instability is dominated mainly by surface shear stress, whereas finite-wavelength instability is primarily governed by the combined effects of Reynolds stress and surface shear stress.

We study the dynamic interaction of two viscous gravity currents in a confined porous layer using laboratory experiments in a vertically placed bead-packed Hele-Shaw cell. By varying the injection rate, along with the density and viscosity of the injecting and ambient fluids, these experiments cover three exact and eight approximate regimes of gravity current interaction, as identified based on the one-dimensional sharp-interface model. By superimposing the theoretically predicted profile shapes and time-dependent frontal locations, a verification is obtained in the different asymptotic regimes of viscous current interaction. Overall, fairly good agreement has been observed between the time-dependent numerical solutions and laboratory measurements on the profile shapes, particularly in the bulk region, where the aspect ratio of the interface shape is fairly large. Such an observation indicates the applicability of the sharp-interface model of viscous current interaction, including the very interesting dynamics of overriding and coflowing. However, the self-similar solutions in some of the exact regimes fail to make reasonable predictions in these experiments, presumably due to the influence of unfinished time transition. We have also observed some degree of disagreement in the frontal regions, which is likely due to the influence of fluid mixing, two-dimensional flow, local heterogeneity and the development of hydrodynamic instabilities for the viscously unstable experiments. The theoretical predictions of the model problem, along with the laboratory experimental observations, offer useful insights into the potential application of, e.g. the technology of co-flooding CO$_2$ and water in oil fields for the co-profits of geological CO$_2$ sequestration and enhanced oil recovery.

Recent work (Raufaste et al. 2022 Soft Matter, vol. 18, p. 4944) studied the dynamics of a soap film in the shape of an unstable minimal surface whose evolution is governed in part by the frictional forces associated with surface Plateau border (SPB) motion. In this note, we study a variant of this problem in which a half-catenoid bounded by a wire loop and a fluid bath axisymmetrically surrounds a cylindrical rod with a radius equal to the neck of the critical catenoid given by the wire loop. When the half-catenoid is brought just beyond the point of instability, the film touches the cylinder and separates from the bath, creating an SPB that is dragged upwards along the rod by the now unstable soap film, and asymptotically relaxes to a new stable annular minimal surface. For this free-boundary problem involving an unstable initial condition, we find the dynamics by balancing the capillary force of successive unstable minimal surfaces spanning the SPB and the wire loop with the frictional force associated with the moving SPB. We find good agreement between theory and experiment using the frictional force $f\sim \textit{Ca}^{2/3}$ given by Bretherton’s law, where $ \textit{Ca} $ is the capillary number.

Roll patterns on floating ice shelves have been suggested to arise from viscous buckling under compressive stresses. A model of this process is explored, allowing for a power-law fluid rheology for ice. Linear stability theory of uniformly compressing base flows confirms that buckling modes can be unstable over a range of intermediate wavelengths when gravity does not play a dominant role. The rate of compression of the base flow, however, ensures that linear perturbations have wavelengths that continually shorten with time. As a consequence, linear instability only ever arises over a certain window of time $t$, and its strength can be characterised by finding the net amplification factor a buckling mode acquires for $t\to \infty$, beginning from a given initial wavenumber. Bi-axial compression, in which sideways straining flow is introduced to prevent the thickening of the base flow, is found to be more unstable than purely two-dimensional (or uni-axial) compression. Shear-thinning enhances the degree of instability in both uni-axial and bi-axial flow. The implications of the theoretical results for the glaciological problem are discussed.

Sea surface films significantly influence air–sea interaction. While their damping effect on gravity–capillary waves is well recognised, the detailed mechanisms by which surface films alter small-scale wave dynamics – particularly energy dissipation and near-surface flow patterns – remain insufficiently understood. This paper presents experimental observations focusing on small-scale wave profiles and surface-flow dynamics in the presence of surfactants, providing direct experimental evidence of underlying mechanisms such as Marangoni effects. The experiments demonstrate enhanced energy dissipation and significant alterations in near-surface flow caused by surfactants, including the transformation of typical circular motion into elliptical-like trajectories and the emergence of reverse surface drift.

We investigate a short-wave instability mode recently identified via temporal stability analysis in weakly inclined falling liquid films sheared by a confined turbulent counter-current gas flow (Ishimura et al. J. Fluid Mech. vol. 971, 2023, p. A37). We perform spatio-temporal linear stability calculations based on the Navier–Stokes equations in the liquid film and the Reynolds-averaged Navier–Stokes equations in the gas, and compare these with our own experiments. We find that the short-wave instability mode is always upward-convective. The range of unstable group velocities is very wide and largely coincides with negative values of the wave velocity. Turbulence affects this mode both through the level of gas shear stress imparted and via the shape of the primary-flow gas velocity profile. Beyond a critical value of the counter-current gas flow rate, the short-wave mode merges with the long-wave Kapitza instability mode. The thus obtained merged mode is unstable for group velocities spanning from large negative to large positive values, i.e. it is absolute. The onset of the short-wave mode is precipitated by decreasing the channel height and inclination angle, and by increasing the liquid Reynolds number or the gas-to-liquid dynamic viscosity ratio. For vertically falling liquid films, merging occurs before the short-wave mode can become unstable on its own. Nonetheless, the ability to generate upward-travelling ripples is endowed to the merged mode. Preliminary calculations neglecting the linear perturbation of the turbulent viscosity suggest that three-dimensional perturbations could be more unstable than two-dimensional ones.

Contact between fluctuating, fluid-lubricated soft surfaces is prevalent in engineering and biological systems, a process starting with adhesive contact, which can give rise to complex coarsening dynamics. One representation of such a system, which is relevant to biological membrane adhesion, is a fluctuating elastic interface covered by adhesive molecules that bind and unbind to a solid substrate across a narrow gap filled with a viscous fluid. This flow is described by the stochastic elastohydrodynamic thin film equation, which incorporates thermal fluctuations into the description of viscous nanometric thin-film flow coupled to elastic membrane deformation. The average time it takes the fluctuating elastic membrane to adhere is predicted by the rare event theory, increasing exponentially with the square of the initial gap height. When the forces arising from spring-like adhesive molecules are included in the simulations, thermal fluctuations initiate phase separation of domains of bound and unbound molecules. The coarsening process of these unbound pockets displays close similarities to classical Ostwald ripening; however, the inclusion of hydrodynamics affects power-law growth. In particular, we identify a new bending-dominated coarsening regime, which is slower than the well-known tension-dominated case.

The stability of underwater bubbles is important to many natural phenomena and industrial applications. Since stability analyses are complex and influenced by numerous factors, they are often performed on a case-specific basis, with most being qualitative. In this work, we propose a unified and quantitative criterion for evaluating bubble stability by analysing its free energy. This criterion is broadly applicable across various bubble sizes (from nanometres to macroscale) and contact conditions (suspended, attached and trapped bubbles) on surfaces with diverse chemical (hydrophilic and hydrophobic) and morphological (flat and structured solid surfaces) features. This criterion not only applies to the classical stable bubble mode, which depends on contact line pinning at the tips of surface structures, but also predicts a new mode where the synergy between the geometry and wettability of the sidewalls maintains the bubble’s stable state. The contact line can spontaneously adjust its position on the solid surface to maintain pressure balance, which enhances bubble adaptability to environmental changes. A geometric standard for solid surfaces supporting this new stable state is raised, following which we realise the optimisation of solid surface geometries to control the stability of gas bubbles. This work not only provides a universal framework for understanding underwater bubble stability, but also opens avenues for applications.

Ice shelves that spread into the ocean can develop rifts that can trigger iceberg calving and enhance ocean-induced melting. Fluid mechanically, this system is analogous to an extensionally dominated radial spreading of a non-Newtonian fluid into a relatively inviscid and denser ambient fluid. Laboratory experiments have shown that rift patterns can emerge when the spreading fluid is shear thinning. Linear stability analysis supports these findings, predicting that while the instability mechanism is active in Newtonian fluids, it is suppressed by stabilising secondary-flow cellular vortices. Here, we explore the fully nonlinear evolution of a radially spreading Newtonian fluid, assessing whether large-amplitude perturbations could drive an instability. We use a quasi-three-dimensional numerical simulation that solves the full nonlinear shallow-shelf approximation, tracing the evolving fluid front, and validate it with known axisymmetric solutions and predictions from linear-stability theory. We find that large-amplitude perturbations induce nonlinear effects that give rise to non-axisymmetric patterns, including cusp-like patterns along the fluid front and complex secondary-flow eddies, which have neither been predicted theoretically nor observed experimentally. However, despite these nonlinear effects, large-amplitude perturbations alone are insufficient to induce rift-like patterns in Newtonian fluids. Strain-rate peaks at the troughs of the fluid front suggest that shear-thinning fluids may become more mobile in these regions, potentially leading to rift formation. This coincides with the likely weakening of stabilising forces as the fluid becomes more shear-thinning. These findings elucidate the critical role of nonlinear viscosity on the formation of rift-like patterns, which is the focus of Part 2 of this study.

Surface tension gradients of air–liquid–air films play a key role in governing the dynamics of systems such as bubble caps, foams, bubble coalescence and soap films. Furthermore, for common fluids such as water, the flow due to surface tension gradients, i.e. Marangoni flow, is often inertial, due to the low viscosity and high velocities. In this paper, we consider the localised deposition of insoluble surfactants onto a thin air–liquid–air film, where the resulting flow is inertial. As observed by Chomaz (2001 J. Fluid Mech. 442, 387–409), the resulting governing equations with only inertia and Marangoni stress are similar to the compressible gas equations. Thus, shocks are expected to form. We derive similarity solutions associated with the development of such shocks, where the mathematical structure is closely related to the Burgers equation. It is shown that the nonlinearity of the surface tension isotherm has an effect on the strength of the shock. When regularisation mechanisms are included, the shock front can propagate and late-time similarity solutions are derived. The late-time similarity solution due to regularisation by capillary pressure alone was found by Eshima et al. (2025 Phys. Rev. Lett. 134, 214002). Here, the regularisation mechanism is generalised to include viscous extensional stress.

We present a theoretical study, supported by simulations and experiments, on the spreading of a silicone oil drop under MHz-frequency surface acoustic wave (SAW) excitation in the underlying solid substrate. Our time-dependent theoretical model uses the long-wave approach and considers interactions between fluid dynamics and acoustic driving. While similar methods have analysed the micron-scale oil and water film dynamics under SAW excitation, acoustic forcing was linked to boundary layer flow, specifically Schlichting and Rayleigh streaming, and acoustic radiation pressure. For the macroscopic drops in this study, acoustic forcing arises from Reynolds stress variations in the liquid due to changes in the intensity of the acoustic field leaking from the SAW beneath the drop and the viscous dissipation of the leaked wave. Contributions from Schlichting and Rayleigh streaming are negligible in this case. Both experiments and simulations show that, after an initial phase where the oil drop deforms to accommodate acoustic stress, it accelerates, achieving nearly constant speed over time, leaving a thin wetting layer. Our model indicates that the steady speed of the drop results from the quasi-steady shape of its body. The drop speed depends on drop size and SAW intensity. Its steady shape and speed are further clarified by a simplified travelling-wave-type model that highlights various physical effects. Although the agreement between experiment and theory on drop speed is qualitative, the results’ trend regarding SAW amplitude variations suggests that the model realistically incorporates the primary physical effects driving drop dynamics.