We investigate the natural oscillations of sessile drops with a central trapped bubble on a plane using linear potential flow theory, considering both free and pinned contact lines. The system is governed by the contact angle $\alpha$ and the ratio $\tau$ of inner to outer contact line radii. For bubble-containing (BC) hemispherical drops with free contact lines (referred to as free BC semi-drops), the modes mirror half of those in concentric spherical BC drops due to plane symmetry. These modes are labelled ‘plus’ (with greater inner surface deformation) and ‘minus’ (with greater outer surface deformation). As $\tau \to 0$, minus modes converge to those of bubble-free drops. Results show that varying $\alpha$ from $90^\circ$ or pinning the contact line in free BC semi-drops alters the topology of spectral lines, turning original crossings of spectral lines between minus and plus modes into avoided crossings. This shift causes minus and plus modes to form spectral trends with avoided crossings, maintaining their original spectral shapes. In an avoided crossing, two coupled modes cannot be classified as plus or minus due to their comparable inner and outer surface deformations, resulting in mode beating when both are excited, as confirmed by our direct numerical simulations. This study on the impact of inner bubbles on the spectrum may help in predicting bubble size in opaque sessile drops.

In this experimental and numerical study, we revisit the question of the onset of the elastic regime in viscoelastic pinch-off. This is relevant to all modern filament thinning techniques, which aim to measure the extensional properties of low-viscosity polymer solutions. Examples are the slow retraction method (SRM) for capillary breakup extensional rheometry (CaBER), or the dripping method, in which a drop detaches from a nozzle. As part of these techniques, a stable liquid bridge is brought slowly to its stability threshold, where capillary-driven thinning starts. This thinning slows down dramatically at a critical radius $h_1$, marking the onset of the elasto-capillary regime, characterised by a filament of nearly uniform radius. While a theoretical scaling exists for this transition in the case of the classical step-strain CaBER protocol, where polymer chains stretch without relaxing during the fast plate separation, we show that this theory is not necessarily valid for a slow protocol such as the SRM. In that case, polymer chains start stretching (beyond their equilibrium coiled configuration) only when the bridge thinning rate becomes comparable to the inverse of their relaxation time. We derive a universal scaling for $h_1$, valid for both low- and high-viscosity polymer solutions. This scaling is validated by CaBER experiments with a slow plate separation protocol using different polymer solutions, plate diameters and sample volumes, as well as by numerical simulations using the FENE-P model.

In this paper, we simulate the process of two-dimensional axisymmetric fluid–structure coupling of a droplet impacting on a flexible disk. The effects of dimensionless disk stiffness (K = 0.1–1000), Weber number (We = 1–500) and contact angle (θ = 130° and 60°) on the dynamics of the droplet impacting on the flexible disk are analysed. The results indicate that there are five typical impact modes for a hydrophobic surface (θ = 130°) and four typical impact modes for a hydrophilic surface (θ = 60°) within the range of considered parameters. The analysis of spreading factor reveals that a part of the energy is transferred to the substrate, which is manifested as a weakening of the droplet spreading, when the substrate deforms downwards due to the droplet impact; the squeezing of the droplet causes a tendency to flow from the centre of the droplet to the edge, which is manifested as an enhancement of the droplet spreading, when the substrate recovers from the downward deformation. The effect of the substrate flexibility on the maximum spreading factor depends on the competition of the two mechanisms above. Based on this, a modified scaling law of βmax has been proposed by introducing the effective Weber number (Wem). The analysis of impact force demonstrates that the peak of the impact force is related to the deflection of the flexible substrate which is different from that of a rigid wall; and three typical processes of impact force variation have been summarised. In addition, unlike the rigid substrate scenario, there is an energy interaction between the droplet and the flexible substrate after impact occurs, which is classified as three typical energy transformation processes.

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.

The lubricated motion of an object near a deformable boundary presents striking subtleties arising from the coupling between the elasticity of the boundary and lubricated flow, including but not limited to the emergence of a lift force acting on the object despite the zero Reynolds number. In this study, we characterize the hydrodynamic forces and torques felt by a sphere translating in close proximity to a fluid interface, separating the viscous medium of the sphere's motion from an infinitely more viscous medium. We employ lubrication theory and perform a perturbation analysis in capillary compliance. The dominant response of the interface owing to surface tension results in a long-ranged interface deformation, which leads to a modification of the forces and torques with respect to the rigid reference case, that we characterize in detail with scaling arguments and numerical integrations.

Liquid plug formation in thin channels due to the Plateau–Rayleigh instability of a liquid film is observed in a variety of fields. In this paper, complementarity between theoretical solutions and direct numerical simulations (DNS) based on a front-tracking algorithm is explored to evaluate the importance of inertia for the case of a cylindrical capillary. A linear stability analysis is first performed and DNS results are then used to investigate the spatial distributions of inertial, convective and viscous terms of the Navier–Stokes equation. The existence of both viscous and inertial regimes is evidenced with a threshold given by the film thickness. The presence of the core fluid slows down the instability. In the viscous regime, predictions of the lubrication theory are verified. An example of liquid water as the outer fluid film and water vapour as the inner core fluid is simulated with application to the fuel cells.

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

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

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.

A model is formulated of a two-dimensional migrating, or swimming, inviscid bubble in a viscous fluid whose unsteady displacement is caused by the spreading over its surface of an initial distribution of insoluble surfactant. Assuming small capillary and Reynolds numbers, and a linear equation of state giving the surface tension as a function of surfactant concentration, the quasi-steady Stokes flow around the bubble is found analytically and explicit formulas are determined for the time-dependent bubble speed and its final overall displacement. At infinite surface Péclet number this is done using a complex version of the method of characteristics to solve a complex partial differential equation of Burgers type. For a finite non-zero surface Péclet number, the problem is shown to be linearizable by a complex variant of the classical Cole–Hopf transformation. The formulation allows general statements to be made on the bubble speed and its total net displacement in terms of the initial surfactant distribution. A weak finite-time singularity in the surface activity associated with an isolated clean point on the bubble surface is also identified and studied in detail.

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

Linear instability analysis of a viscous swirling liquid jet surrounded by ambient gas is carried out by considering the significant influence of axial shear effect. The jet azimuthal flow is assumed as a Rankine vortex, and the non-uniform velocity distribution in the jet axial direction is approximated by parabolic and error functions. The enhancement of jet rotation is found to promote the predominant mode with larger azimuthal wavenumbers, and the mode transition is decided by the competition between centrifugal force and axial shear stress. Subsequently, the influence of the axial shear effect is examined through changing the degree of shear stress and the thickness of the gas velocity boundary layer. It is found that an increase of jet average velocity or surface velocity in the axial direction leads to the predominant mode transition to smaller azimuthal wavenumbers, due to the combined effects of shear stress and gas pressure perturbation. A larger velocity difference between ambient gas and liquid jet also promotes the predominant modes with smaller azimuthal wavenumbers, and the physical mechanism is attributed to gas pressure perturbation. Phase diagrams of different azimuthal modes are given and compared with the study of Kubitschek & Weidman (J. Fluid Mech., vol. 572, 2007, pp. 261–286), where a static swirling column without axial shear stress was considered. The strengthened axial shear effect is found to delay the transition of predominant modes with the increase of angular velocity. Experimental studies considering the swirling jets with different axial velocities are further carried out, which validate the theoretical findings. Different instability mechanisms and their transition rules are also identified through energy budget analysis. This study is expected to give scientific guidance on understanding the instability mechanisms of the swirling jets that widely exist in natural phenomena and engineering applications.

We perform simulations of an impulsively started, axisymmetric viscoelastic jet exiting a nozzle and entering a stagnant gas phase using the open-source code Basilisk. This code allows for efficient computations through an adaptively refined volume-of-fluid technique that can accurately capture the deformation of the liquid–gas interface. We use the FENE-P constitutive equation to describe the viscoelasticity of the liquid, and employ the log-conformation transformation, which provides stable solutions for the evolution of the conformation tensor as the jet thins down under the action of interfacial tension. For the first time, the entire jetting and breakup process of a viscoelastic fluid is simulated, including the pre-shearing flow through the nozzle, which results in an inhomogeneous initial radial stress distribution in the fluid thread that affects the subsequent breakup dynamics. The evolution of the velocity field and the elastic stresses in the nozzle are validated against analytical solutions where possible, and the early-stage dynamics of the jet evolution are compared favourably to the predictions of linear stability theory. We study the effect of the flow inside the nozzle on the thinning dynamics of the viscoelastic jet (which develops distinctive ‘beads-on-a-string’ structures) and on the spatio-temporal evolution of the polymeric stresses in order to systematically explore the dependence of the filament thinning and breakup characteristics on the initial axial momentum of the jet and the extensibility of the dissolved polymer chains.

We numerically study the influence of a soluble surfactant on the microjetting mode of the liquid–liquid flow focusing configuration. The surfactant adsorbs on the interface next to the feeding capillary and accumulates in front of the emitted jet, significantly lowering the surface tension there. The resulting Marangoni stress substantially alters the balance of the tangential stresses at the interface but does not modify the interface velocity. The global stability analysis at the minimum flow rate stability limit shows that the Marangoni stress collaborates with soluto-capillarity to stabilize the microjetting mode. Our analysis unveils the noticeable effect of the Marangoni stress associated with the surface tension perturbation. Surfactant diffusion and desorption hardly affect the stability limit. Transient numerical simulations show how subcritical and supercritical base flows respond to a spatially localized initial perturbation. Our parametric study indicates that the minimum flow rate ratio depends on the adsorption constant and the surfactant concentration through the product of these two variables. The surfactant stabilizing effect increases with the outer stream flow rate. We show that surfactants not only stabilize the microemulsion resulting from the jet breakup in hydrodynamic focusing, but also allow for the reduction of droplet size. Our findings advance the fundamental understanding of the complex role of surfactants in tip streaming via hydrodynamic focusing. In particular, our results contradict the common assumption that adding surfactant favours tip streaming simply because it reduces the meniscus tip surface tension.

When subjected to external stress exceeding their yield stress ($\sigma _{Y}$), yield stress fluids (YSFs) undergo transition from solid to liquid. While the flow dynamics of capillary flows – including wetting, coating, spreading, and wicking – can be predicted accurately for fluids without yield stress, only a few asymptotic solutions have been validated against experiments for YSFs. In this study, we explore experimentally and theoretically the entire dynamics and solidifying process during capillary imbibition of YSFs, representing the fundamental aspect of capillarity-driven flow. We observe that during the gradual deceleration of capillary rise, YSFs exhibit apparent flow below $\sigma _{Y}$. This unexpected behaviour is linked to wall slip, manifested as plug flow in particle image velocimetry analyses. We introduce a model with numerical solutions, grounded in the rheological properties of YSFs and slip mechanisms, to accurately describe this liquid–solid transitional behaviour.

We explore the drawing of a shear-thinning or shear-thickening thread with an axisymmetric hole that evolves due to axial drawing, inertia and surface tension effects. The stress is assumed to be proportional to the shear rate raised to the $n$th power. The presence of non-Newtonian rheology and surface tension forces acting on the hole introduces radial pressure gradients that make the derivation of long-wavelength equations significantly more challenging than either a Newtonian thread with a hole or shear-thinning and shear-thickening threads without a hole. In the case of weak surface tension, we determine the steady-state profiles. Our results show that for negligible inertia the hole size at the exit becomes smaller as $n$ is decreased (i.e. strong shear-thinning effects) above a critical draw ratio, but surprisingly the opposite is true below this critical draw ratio. We determine an accurate estimate of the critical draw ratio and also discuss how inertia affects this process. We further show that the dynamics of hole closure is dominated by a different limit, and we determine the asymptotic forms of the hole closure process for shear-thinning and shear-thickening fluids with inertia. A linear instability analysis is conducted to predict the onset of draw resonance. We show that increased shear thinning, surface tension and inlet hole size all act to destabilise the flow. We also show that increasing shear-thinning effects reduce the critical Reynolds number required for unconditional stability. Our study provides valuable insights into the drawing process and its dependence on the physical effects.

Bistable states for a sufficiently large amount of liquid can appear in an eccentric capillary due to the eccentricity effect under zero gravity (J. Fluid Mech, vol. 863, 2019, pp. 364–385). A transverse body force, which can lead to rich physical phenomena of a droplet, may lead to multistable states (bistability, tristability and the likes) of a sufficiently large amount of liquid in a capillary. We theoretically investigate this situation in a circular or annular capillary tube under a transverse body force. The results show that there can be tristable (bistable) states in an annular (circular) capillary tube: an occluding configuration and two (one) non-occluding configurations. In the annular tube, for one of the non-occluding configurations, the gas–liquid interface in the middle cross-section of the droplet meets both the inner and outer walls of the tube (bridging configuration); for the other non-occluding configuration, the gas–liquid interface in the middle cross-section of the droplet does not meet the inner wall (non-bridging configuration). The multistability is dependent on the Bond number, the contact angle and the cross-sectional shape. The multistability cannot occur for a zero or very large Bond number. A hydrophilic condition (the contact angle smaller than 90°) contributes to the non-occluding non-bridging configuration, while the hydrophobic condition (the contact angle larger than 90°) contributes to the non-occluding bridging configuration (only for the annular capillary). For the annular capillary with a not-so-large contact angle, increasing the inner-to-outer radius ratio can lead to a larger range of Bond numbers, in which the multistability occurs.

An interesting resurrection phenomenon (including the initial complete submersion, subsequent resurfacing and final rebounding) of a superhydrophobic sphere impacting onto a liquid bath was observed in experiments and direct numerical simulations by Galeano-Rios et al. (J. Fluid Mech., vol. 912, 2021, A17). We investigate the mechanisms of the liquid entry for a superhydrophobic cylinder in this paper. The superhydrophobic cylinder, commonly employed as supporting legs for insects and robots at the liquid surface, can exhibit liquid-entry mechanisms different from those observed with the sphere. The direct numerical simulation method is applied to the impact of a two-dimensional (2-D) superhydrophobic cylinder (modelled as a pseudo-solid) onto a liquid bath. We find that for the impacting cylinder the resurrection phenomenon can also exist, and the cylinder can either rebound (get detached from the liquid surface) or stay afloat after resurfacing. The cylinder impact behaviour is classified into four regimes, i.e. floating, bouncing, resurrecting (resurrecting-floating and resurrecting-bouncing) and sinking, dependent on the Weber number and the density ratio of the cylinder to the liquid. For the regimes of floating and bouncing, the force analysis indicates that the form drag dominates the motion of the cylinder in the very beginning of the impact, while subsequently the surface tension force also plays a role with the contact line pinning on the horizontal midline of the cylinder. For the critical states of the highlighted resurrecting regime, our numerical results show that the rising height for the completely submerged cylinder of different density ratios remains nearly unchanged. Accordingly, a relation between the maximum ascending velocity and the density ratio is derived to predict whether the completely submerged cylinder can resurface.

Diffuse interface models are an important class of models used to describe multi-phase flows. In the case of incompressible viscous fluids there are a number of different diffuse interface models which have been known for many years. Nevertheless, a model based on mixture theory with a full set of conservation laws for the conservation of linear momentum of each constituent was not yet available. This gap was filled by ten Eikelder et al. (J. Fluid Mech., in press) recently and a first comparison with known models of Navier–Stokes/Cahn–Hilliard type is given. A detailed understanding of the relations between these models remains an important question.

Thin-film equations are utilised in many different areas of fluid dynamics when there exists a direction in which the aspect ratio can be considered small. We consider thin free films with Marangoni effects in the extensional flow regime, where velocity gradients occur predominantly along the film. In practice, because of the local deposition of surfactants or input of energy, asymmetric distributions of surfactants or surface tension more generally, are possible. Such examples include the surface of bubbles and the rupture of thin films. In this study, we consider the asymmetric thin-film equations for extensional flow with Marangoni effects. Concentrating on the case of small Reynolds number $ Re $, we study the deposition of insoluble surfactants on one side of a liquid sheet otherwise at rest and the resulting thinning and rupture of the sheet. The analogous problem with a uniformly thinning liquid sheet is also considered. In addition, the centreline deformation is discussed. In particular, we show analytically that if the surface tension isotherm $\sigma = \sigma (\varGamma )$ is nonlinear (surface tension $\sigma$ varies with surfactant concentration $\varGamma$), then accounting for top–bottom asymmetry leads to slower (faster) thinning and pinching if $\sigma = \sigma (\varGamma )$ is convex (concave). The analytical progress reported in this paper allows us to discuss the production of satellite drops from rupture via Marangoni effects, which, if relevant to surface bubbles, would be an aerosol production mechanism that is distinct from jet drops and film drops.