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.

We investigate the motion of a thin liquid drop on a pre-stretched, highly bendable elastic sheet. Under the lubrication approximation, we derive a system of fourth-order partial differential equations, along with appropriate boundary and contact line conditions, to describe the evolution of the fluid interface and the elastic sheet. Extending the classical analysis of Cox and Voinov, we perform a four-region matched asymptotic analysis of the model in the limit of small slip length. The central result is an asymptotic relation for the contact line speed in terms of the apparent contact angles. We validate the relation through numerical simulations. A key implication of this result is that a soft substrate retards drop spreading but enhances receding, compared to the dynamics on a rigid substrate. The relation remains valid across a wide range of bending modulus, despite the distinguished limit assumed in the analysis.

We derive a mathematical model for the overflow fusion glass manufacturing process. In the limit of zero wedge angle, the model leads to a canonical fluid mechanics problem in which, under the effects of gravity and surface tension, a free-surface viscous flow transitions from lubrication flow to extensional flow. We explore the leading-order behaviour of this problem in the limit of small capillary number, and find that there are four distinct regions where different physical effects control the flow. We obtain leading-order governing equations, and determine the solution in each region using asymptotic matching. The downstream behaviour reveals appropriate far-field conditions to impose on the full problem, resulting in a simple governing equation for the film thickness that holds at leading order across the entire domain.

An asymptotic model for the flow of a highly viscous film coating the interior of a slippery, flexible tube is developed and studied. The model is valid for the axisymmetric flow of moderately thick films, and accounts for tube flexibility, wall damping, longitudinal tension, slip length and strength of base flow due either to gravity or airflow. In the absence of base flow, linear stability analysis shows the existence of one unstable mode; the presence of base flow allows for multiple unstable modes arising due to the Plateau–Rayleigh instability and elastic instability, with stronger base flow reducing the maximum growth rate. Numerical solutions in the absence of base flow show that slip decreases the amplitude of wall deformations and can significantly decrease the time to plug formation in weakly flexible or strongly damped tubes. For falling films, the impact of model parameters on the critical thickness required for plug formation was analysed by studying turning points in families of travelling-wave solutions; this thickness decreases with slip, flexibility and tension, while damping had a non-monotonic impact on critical thickness. In contrast to model solutions in rigid tubes, for flexible tubes the critical thickness cannot be made arbitrarily large through simply increasing the strength of the base flow. For air-driven films, both slip and flexibility increase the rate of film transport along the tube.

We study the transport and deposition of inhaled aerosols in a mid-generation, mucus-lined lung airway, with the aim of understanding if and how airborne particles can avoid the mucus and deposit on the airway wall – an outcome that is harmful in case of allergens and pathogens, but beneficial in case of aerosolised drugs. We adopt the weighted-residual integral boundary-layer model of Dietze and Ruyer-Quil (J. Fluid Mech. 762, 2015, 68–109, to describe the dynamics of the mucus–air interface, as well as the flow in both phases. The transport of mucus induced by wall-attached cilia is also considered, via a coarse-grained boundary condition at the base of the mucus. We show that the capillary-driven Rayleigh–Plateau instability plays an important role in particle deposition by drawing the mucus into large annular humps and leaving substantial areas of the wall exposed to particles. We find, counter-intuitively, that these mucus-depleted zones enlarge on increasing the mucus volume fraction. Our simulations are eased by the fact that the effects of cilia and air turn out to be rather simple: the long-term interface profile is slowly translated by cilia and is unaffected by the laminar airflow. The streamlines of the airflow, though, are strongly modified by the non-uniform mucus film, and this has important implications for aerosol entrapment. Particles spanning a range of sizes (0.1–50 microns) are modelled using the Maxey–Riley equation, augmented with Brownian forces. We find a non-monotonic dependence of deposition on size. Small particles diffuse across streamlines due to Brownian motion, while large particles are thrown off streamlines by inertial forces – particularly when air flows past mucus humps. Intermediate-sized particles are tracer-like and deposit the least. Remarkably, increasing the mucus volume need not increase entrapment: the effect depends on particle size, because more mucus produces not only deeper humps that intercept inertial particles, but also larger depleted zones that enable diffusive particles to deposit on the wall.

Thixotropic fluids with a non-monotonic flow curve display viscosity bifurcations at certain stresses. It has been proposed that these transitions can introduce interfaces (or shear bands) into thin films that can destabilize inertialess flows over inclined planes. This proposition is confirmed in the present paper by formulating a thin-film model, then using this model to construct sheet-like base flows and test their linear stability. It is also found that viscosity bifurcations, and the associated interfaces, are not necessary for instability, but that the time-dependent relaxation of the microstructure responsible for thixotropy within the bulk of the film can promote instability instead. Computations with the thin-film model demonstrate that instabilities saturate supercritically into steadily propagating nonlinear waves that travel faster than the mean flow.

The dynamics of thin viscous liquid films flowing down an inclined wall under gravity in the presence of an upward flowing high-speed air stream is considered. The air stream induces nonlinear waves on the interface and asymptotic solutions are developed to derive a non-local evolution equation forced by the air pressure which is obtained analytically, and incorporating a constant tangential stress. Benney equations in the capillary (strong surface tension) and inertio-capillary regimes are derived and studied. The air stream produces Turing-type short wave instabilities in sub-critical Reynolds number regimes that would be stable in the absence of the outer flow. Extensive numerical experiments are carried out to elucidate the rich dynamics in the above-mentioned short-wave regime. The stability of different branches of solutions of non-uniform steady states is carried out, along with time-dependent nonlinear computations that are used to track the large-time behaviour of attractors. A fairly complete picture of different solution types are categorised in parameter space. The effect of the Reynolds number on the wave characteristics in the inertio-capillary regime is also investigated. It is observed that, for each value of the slenderness parameter $\delta$, there exists a critical Reynolds number $R_c$ above which the solutions become unbounded by encountering finite-time singularities. Increasing the air speed significantly decreases $R_c$, making the system more prone to large amplitude singular events even at low Reynolds numbers when the system would have been stable in the absence of the air stream.

When a liquid film on a horizontal plate is driven in motion by a shear stress, surface waves are easily generated. This paper studies such flow at moderate Reynolds numbers, where the surface tension and inertial force are equally important. The governing equations for two-dimensional flows are derived using the long-wave approximation along with the integral boundary-layer theory. For small disturbances, the dispersion relation and neutral curves are determined by the linear stability analysis. For finite-amplitude perturbations, the numerical simulation suggests that the oscillations generated by the perturbation in a certain place continuously spread to the surrounding areas. When the effects of surface tension and gravity reach equilibrium, steady-state solutions will emerge, which include two cases: solitary waves and periodic waves. The former have heteroclinic trajectories between two stationary points, while the latter include five patterns at different parameters. In addition, there are also periodic waves that do not converge after a long period of time. During these evolution processes, strange attractors appear in the phase space. By examining the Poincaré section and the sensitivity to initial values, we demonstrate that these waves can be divided into two types: quasi-periodic and chaotic solutions. The specific type depends on parameters and initial conditions.

This study investigates the formation and evolution of fishbone patterns in oblique impinging liquid microjets through high-speed imaging experiments and numerical simulations. The results identify periodic oscillations in the upper region of the liquid sheet as the primary mechanism driving fishbone instabilities, which induce rim disturbances and lead to bifurcations into diverse fishbone morphologies. Transitions between stable and unstable flow patterns are systematically mapped across varying Weber numbers and impingement angles, providing a comprehensive framework for understanding this interfacial dynamics. Two critical transitions – marking the onset and disappearance of fishbone patterns – are characterised, offering insights into the underlying physics governing the stability and instability of these flow structures.

A new arbitrary Lagrangian–Eulerian (ALE) formulation for Navier–Stokes flow on self-evolving surfaces is presented. It is based on a general curvilinear surface parameterisation that describes the motion of the ALE frame. Its in-plane part becomes fully arbitrary, while its out-of-plane part follows the material motion of the surface. This allows for the description of flows on deforming surfaces using only surface meshes. The unknown fields are the fluid density or pressure, the fluid velocity and the surface motion, where the latter two share the same normal velocity. The corresponding field equations are the continuity equation or area-incompressibility constraint, the surface Navier–Stokes equations and suitable surface mesh equations. Particularly advantageous are mesh equations based on membrane elasticity. The presentation focuses on the coupled set of strong and weak form equations, and presents several manufactured steady and transient solutions. These solutions are used together with numerical simulations to illustrate and discuss the properties of the proposed new ALE formulation. They also serve as basis for the development and verification of corresponding computational methods. The new formulation allows for a detailed study of fluidic membranes such as soap films, capillary menisci and lipid bilayers.

A liquid film flowing down a fibre becomes unstable, leading to the formation of droplets that travel downstream. The droplet spacing and speed depend on the flow rate for a given nozzle and fibre radii. We show that fibre morphology further modifies the droplet spacing. In particular, we study the effect of the size of the beads in a granular chain on the evolution of the film thickness. We show that, when the size of the bead exceeds a critical value, the selection mechanism for instability modes is modified from regularly spaced droplets to coarsening by droplet merging. Droplet formation for flow over a single bead on the fibre is modified successively over subsequent beads in the downstream. Further, we show that if the perturbation in the flow produced by the bead is introduced as a velocity perturbation at the nozzle inlet, the formation of droplets on the fibre is qualitatively similar to that for the bead.