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.

We investigate the sliding dynamics of a millimetre-sized particle trapped in a horizontal soap film. Once released, the particle moves toward the centre of the film in damped oscillations. We study experimentally and model the forces acting on the particle, and evidence the key role of the mass of the film on the shape of the film and particle dynamics. Not only is the gravitational distortion of the film measurable, it completely determines the force responsible for the motion of the particle – the catenoid-like deformation induced by the particle has negligible effect on the dynamics. Surprisingly, this is expected for all film sizes as long as the particle radius remains much smaller than the film width. We also measure the friction force, and show that ambient air and the film contribute almost equally to the friction. The theoretical model that we propose predicts exactly the friction coefficient as long as inertial effects can be neglected in air (for the smallest and slowest particles). The fit between theory and experiments sets an upper boundary $\eta _s \leqslant 10^{-8}$ Pa s m for the surface viscosity, in excellent agreement with recent interfacial microrheology measurements.

We study the dynamics of fracture deflation following hydraulic fracturing of an infinite elastic solid, with fluid removal from a narrow conduit at the centre. This process involves coupled lubricating flow and elastic deformation, now subject to appropriate descriptions of fluid removal through the conduit towards the ambient, driven by elastic stresses and extraction/suction. When the influence of material toughness is negligible, the dynamics is found to be governed by two dimensionless parameters that describe the relative influence of elasticity-driven backflow ($\Pi _c$) and ambient-pressure-driven backflow ($\Pi _e$), respectively. We also found that the fracture’s thickness eventually approaches zero at the centre, while the fracture evolves into a self-similar shape of the dipole type that conserves the dipole moment $M$. The fracture’s front continues to elongate according to $x_f \propto t^{1/9}$, while the total fluid volume within the fracture decreases according to $V \propto t^{-1/9}$. The model and solutions might find use in practical problems to estimate the rate of backflow and effective permeability of a fractured reservoir once pressure is released.

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.

Gravity-driven film flow in circular pipes with isolated topography was examined with fluorescence imaging for three flow rates, two angles of inclination, and four topography shapes. The time-averaged free surface response in the vicinity of the topography depended on flow rate, inclination angle and topography shape. For some flow conditions, the time-averaged free surface included a capillary ridge, and for a subset of those conditions, a series of capillary waves developed upstream with a spacing often approximated by half the capillary length. In contrast to film flow over planar topography, the capillary ridge often formed downstream of the topography, and for the lowest flow rate over rectangular step down topography, the free surface developed a steady overhang along the downstream face of the topography. Possible dynamic causes of the unique film flow behaviour in circular pipes are discussed. Transient free surface fluctuations were observed at half the magnitude reported in film flow over corrugated circular pipes, and local maxima in transient magnitude corresponded to axial locations of inflection points in the time-averaged free surface. Local maxima are related to low surface pressure regions that vary in location and amplitude. Rectangular step down topography generated an extra ridge of fluid that formed on top of the capillary ridge for flow conditions, resulting in a capillary ridge downstream of the step. The extra ridge varied in temporal duration and spatial extent, and finds no comparison in planar film flow. No evidence of periodic behaviour was detected in the transient film response.

Experiments are presented to explore the non-axisymmetric instabilities of spreading films of aqueous suspensions of Carbopol and Xanthan gum floating on a bath of perfluoropolyether oil. The experimental observations are compared against theoretical predictions exploiting a shallow-film model in which the viscoplastic rheology is captured by the Herschel–Bulkley constitutive law. With this model, we construct axisymmetric base states that evolve from the moment that the film floats onto the bath, out towards long times at which spreading becomes self-similar, and then test their linear stability towards non-axisymmetric perturbations. In the geometry of a thinning expanding film, we find that shear thinning does not drive a loss of axisymmetry at early times (when the degree of expansion is small), but when the film has expanded in radius by a factor of two or so, shear-thinning hoop stresses drive non-axisymmetric instabilities. Unstable modes possess relatively low angular wavenumber, and the loss of symmetry is not particularly dramatic. When the oil in the bath is replaced by salty water, the experiments are completely different, with dramatic non-axisymmetric patterns emerging from interfacial effects.

In binary mixtures, the lifetimes of surface bubbles can be five orders of magnitude longer than those in pure liquids because of slightly different compositions of the bulk and the surfaces, leading to a thickness-dependent surface tension of thin films. Taking advantage of the resulting simple surface rheology, we derive the equations describing the thickness, flow velocity and surface tension of a single liquid film, using thermodynamics of ideal solutions and thin film mechanics. Numerical resolution shows that, after a first step of tension equilibration, the Laplace-pressure-driven flow is associated with a flow at the interface driven by an induced Marangoni stress. The resulting parabolic flow with mobile interfaces in the film further leads to its pinching, eventually causing its rupture. Our model paves the way for a better understanding of the rupture dynamics of liquid films of binary mixtures.

We derive leading-order governing equations and boundary conditions for a sheet of viscous fluid retracting freely under surface tension. We show that small thickness perturbations about a flat base state can lead to regions of compression, where one or both of the principal tensions in the sheet becomes negative, and thus drive transient buckling of the sheet centre-surface. The general theory is applied to the simple model problem of a retracting viscous disc with small axisymmetric thickness variations. Transient growth in the centre-surface is found to be possible generically, with the dominant mode selected depending on the imposed initial thickness and centre-surface perturbations. An asymptotic reduction of the boundary conditions at the edge of the disc, valid in the limit of large normalised thickness perturbations, reduces the centre-surface evolution equation to an ordinary differentional equation (ODE) eigenvalue problem. Analysis of this eigenvalue problem leads to insights such as how the degree of transient buckling depends on the imposed thickness perturbation and which thickness perturbation gives rise to the largest transient buckling.

The dynamics of evolving fluid films in the viscous Stokes limit is relevant to various applications, such as the modelling of lipid bilayers in cells. While the governing equations were formulated by Scriven (1960), solving for the flow of a deformable viscous surface with arbitrary shape and topology has remained a challenge. In this study, we present a straightforward discrete model based on variational principles to address this long-standing problem. We replace the classical equations, which are expressed with tensor calculus in local coordinates, with a simple coordinate-free, differential-geometric formulation. The formulation provides a fundamental understanding of the underlying mechanics and translates directly to discretization. We construct a discrete analogue of the system using Onsager's variational principle, which, in a smooth context, governs the flow of a viscous medium. In the discrete setting, instead of term-wise discretizing the coordinate-based Stokes equations, we construct a discrete Rayleighian for the system and derive the discrete Stokes equations via the variational principle. This approach results in a stable, structure-preserving variational integrator that solves the system on general manifolds.

Even though liquid foams are ubiquitous in everyday life and industrial processes, their ageing and eventual destruction remain a puzzling problem. Soap films are known to drain through marginal regeneration, which depends upon periodic patterns of film thickness along the rim of the film. The origin of these patterns in horizontal films (i.e. neglecting gravity) still resists theoretical modelling. In this work, we theoretically address the case of a flat horizontal film with a thickness perturbation, either positive (a bump) or negative (a groove), which is initially invariant under translation along one direction. This pattern relaxes towards a flat film by capillarity. By performing a linear stability analysis on this evolving pattern, we demonstrate that the invariance is spontaneously broken, causing the elongated thickness perturbation pattern to destabilise into a necklace of circular spots. The unstable and stable modes are derived analytically in well-defined limits, and the full evolution of the thickness profile is characterised. The original destabilisation process we identify may be relevant to explain the appearance of the marginal regeneration patterns near a meniscus and thus shed new light on soap-film drainage.

Aqueous suspensions of cornstarch abruptly increase their viscosity on raising either shear rate or stress, and display the formation of large-amplitude waves when flowing down inclined channels. The two features have been recently connected using constitutive models designed to describe discontinuous shear thickening. By including time-dependent relaxation and spatial diffusion of the frictional contact density responsible for shear thickening, an analysis of steady sheet flow and its linear stability is presented. The inclusion of such effects is motivated by the need to avoid an ill-posed mathematical problem in thin-film theory and the resulting failure to select any preferred wavelength for unstable linear waves. Relaxation, in particular, eliminates an ultraviolet catastrophe in the spectrum of unstable waves and furnishes a preferred wavelength at which growth is maximized. The nonlinear dynamics of the unstable waves is briefly explored. It is found that the linear instability saturates once disturbances reach finite amplitude, creating steadily propagating nonlinear waves. These waves take the form of a series of steep, shear-thickened steps that translate relatively slowly in comparison with the mean flow.

Liquid flowing down a fibre readily destabilises into a train of beads, commonly called a bead-on-fibre pattern. Bead formation results from capillary-driven instability and gives rise to patterns with constant velocity and time-invariant bead frequency $f$ whenever the instability is absolute. In this study, we develop a scaling law for $f$ that relates the Strouhal number $St$ and capillary number $Ca$ for Ostwaldian power-law liquids with Newtonian liquids recovered as a limiting case. We validate our proposed scaling law by comparing it with prior experimental data and new experimental data using xanthan gum solutions to produce a low capillary number $Ca \leq 10^{-2}$ regime. The experimental data encompasses both Ostwaldian and Newtonian flow, as well as symmetric and asymmetric patterns, and we find the data collapses along the predicted trend across seven orders of magnitude in $Ca$. Our proposed scaling law is a powerful tool for studying and applying bead-on-fibre flows where $f$ is critical, such as heat and mass transfer systems.

Surface roughness significantly modifies the liquid film thickness entrained when dip coating a solid surface, particularly at low coating velocity. Using a homogenization approach, we present a predictive model for determining the liquid film thickness coated on a rough plate. A homogenized boundary condition at an equivalent flat surface is used to model the rough boundary, accounting for both flow through the rough texture layer, through an interface permeability term, and slip at the equivalent surface. While the slip term accounts for tangential velocity induced by viscous shear stress, accurately predicting the film thickness requires the interface permeability term to account for additional tangential flow driven by pressure gradients along the interface. We find that a greater degree of slip and interface permeability signifies less viscous stress that would promote deposition, thus reducing the amount of free film coated above the textures. The model is found to be in good agreement with experimental measurements, and requires no fitting parameters. Furthermore, our model may be applied to arbitrary periodic roughness patterns, opening the door to flexible characterization of surfaces found in natural and industrial coating processes.

We examine the gravity-driven flow of a thin film of viscous fluid spreading over a rigid plate that is lubricated by another viscous fluid. We model the flow over such a ‘soft’ substrate by applying the principles of lubrication theory, assuming that vertical shear provides the dominant resistance to the flow. We do so in axisymmetric and two-dimensional geometries in settings in which the flow is self-similar. Different flow regimes arise, depending on the values of four key dimensionless parameters. As the viscosity ratio varies, the behaviour of the intruding layer ranges from that of a thin coating film, which exerts negligible traction on the underlying layer, to a very viscous gravity current spreading over a low-viscosity, near-rigid layer. As the density difference between the two layers approaches zero, the nose of the intruding layer steepens, approaching a shock front in the equal-density limit. We characterise a frontal stress singularity, which forms near the nose of the intruding layer, by performing an asymptotic analysis in a small neighbourhood of the front. We find from our asymptotic analysis that unlike single-layer viscous gravity currents, which exhibit a cube-root frontal singularity, the nose of a viscous gravity current propagating over another viscous fluid instead exhibits a square-root singularity, to leading order. We also find that large differences in the densities between the two fluids give rise to flows similar to that of thin films of a single viscous fluid spreading over a rigid, yet mobile, substrate.

A long-wave asymptotic model is developed for a viscoelastic falling film along the inside of a tube; viscoelasticity is incorporated using an upper convected Maxwell model. The dynamics of the resulting model in the inertialess limit is determined by three parameters: Bond number Bo, Weissenberg number We and a film thickness parameter $a$. The free surface is unstable to long waves due to the Plateau–Rayleigh instability; linear stability analysis of the model equation quantifies the degree to which viscoelasticity increases both the rate and wavenumber of maximum growth of instability. Elasticity also affects the classification of instabilities as absolute or convective, with elasticity promoting absolute instability. Numerical solutions of the nonlinear evolution equation demonstrate that elasticity promotes plug formation by reducing the critical film thickness required for plugs to form. Turning points in travelling wave solution families may be used as a proxy for this critical thickness in the model. By continuation of these turning points, it is demonstrated that in contrast to Newtonian films in the inertialess limit, in which plug formation may be suppressed for a film of any thickness so long as the base flow is strong enough relative to surface tension, elasticity introduces a maximum critical thickness past which plug formation occurs regardless of the base flow strength. Attention is also paid to the trade-off of the competing effects introduced by increasing We (which increases growth rate and promotes plug formation) and increasing Bo (which decreases growth rate and inhibits plug formation) simultaneously.

A cylindrical liquid thread readily destabilizes into a series of drops due to capillary instability, which is also responsible for undesirable bead-on-fibre structures observed when coating a thin fibre. In this experimental study, we show how a falling liquid thread can be stabilized by internally distorting the cross-sectional shape using two vertically hung fibres. Below a critical flow rate $Q_c$, the dual-fibre system deforms the falling thread into a smooth column with a non-circular cross-section, thereby suppressing instability. Above $Q_{{c}}$, the cylindrical thread is left undeformed by the fibres and destabilizes into beads connected by a stable, non-cylindrical film. An empirical stability threshold is identified showing that flow delays the onset of capillary instability when compared with a marginally stable quasi-static coating. When the flow is unstable $Q>Q_c$, the bead velocity $v$ obeys a simple scaling law that is well supported by our experiments over a large parameter range. This suppression technique can be extended to other slender geometries, such as a ribbon, which shows similar qualitative results but exhibits a different stability threshold due to spontaneous dewetting about its short edge.

The stability of liquid-film flows is essential in many industrial applications. In the dip-coating process, a liquid film forms over a substrate extracted at a constant speed from a bath. We studied the linear stability of this film considering different thicknesses $\hat {h}$ for four liquids, spanning an extensive range of Kapitza numbers ($Ka$). By solving the Orr–Sommerfeld eigenvalue problem with the Chebyshev–Tau spectral method, we calculated the threshold between growing and decaying perturbations, investigated the instability mechanism, and computed the absolute/convective threshold. The instability mechanism was studied by analysing the perturbations’ vorticity distribution and the kinetic energy balance. It was found that liquids with low $Ka$ (e.g. corn oil, $Ka = 4$) are stable for a smaller range of wavenumbers compared with liquid with high $Ka$ (e.g. liquid zinc, $Ka = 11\,525$). Surface tension has a stabilising and a destabilising effect. For long waves, it curves the vorticity lines near the substrate, reducing the flow under the crests. For short waves, it fosters vorticity production at the interface and creates a region of intense vorticity near the substrate. In addition, we discovered that the surface tension contributes to both the production and dissipation of perturbation's energy depending on the $Ka$ number. Regarding the absolute/convective threshold, we identified a window in the parameter space where unstable waves propagate throughout the entire domain (indicating absolute instability). Perturbations affecting Derjaguin's solution ($\hat {h}=1$) for $Ka<17$ and the Landau–Levich–Derjaguin solution ($\hat {h}=0.945 Re^{1/9}Ka^{-1/6}$), are advected by the flow (indicating convective instability).

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.

The motion of a bubble of negligible viscosity, such as air, forced down a tube filled with a viscous fluid which wets the walls of the tube has become a classic of the fluid dynamical literature. The differential motion of the bubble and the fluid are determined by the thin film which surrounds the bubble, whose shape and thickness are set by the interplay between gradients in surface tension and viscous shear stresses. Bretherton (J. Fluid Mech., vol. 10, issue 2, 1961, p. 166) provided a first, clear mathematical analysis in the lubrication limit coupled with carefully constructed experimental confirmation of the thin films deposited by a bubble moving in the confining geometry of the capillary tube. Its lasting impact has been not only in the migration of bubbles, but in a host of related fluid dynamical, industrial, biological and environmental processes for which thin lubricating films on the sometimes convoluted geometries of complex microstructures, such as porous media, determine the large-scale behaviour.