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.

When a partially miscible binary mixture is quenched below its critical temperature, it transitions from its single-phase to a two-phase region, undergoing phase separation. The processes of formation and coalescence of droplets are driven by diffusive and convective phenomena, taking place isotropically in the system. The application of an external force field, which exerts a different contribution on the two species, breaks the symmetry of phase separation, leading to the segregation of two equilibrated phases separated by a single interface. This study investigates the dynamics of phase segregation under an external force. The effects of various force magnitudes, captured by the Bond number, in both high- and low-viscosity mixtures, distinguished by different fluidity numbers, are quantified via numerical simulations by using the phase field model. The intricate dynamics of formation, floating and coalescence of droplets towards complete segregation are described along with the quantification of the segregation time, revealing different patterns for high and low Bond numbers. Results show that in none of the cases, formation and floating can be regarded as strictly serial processes. A universal scaling between segregation time, Bond number, fluidity number and domain size is not possible, with a power-law dependence emerging only under the diffusion-dominated regime.

A Marangoni surfer is an object embedded in a gas–liquid interface, propelled by gradients in surface tension. We derive an analytical theorem for the lower bound on the viscous dissipation by a Marangoni surfer in the limit of small Reynolds and capillary numbers. The minimum dissipation can be expressed with the reciprocal difference between drag coefficients of two passive bodies of the same shape as the Marangoni surfer, one in a force-free interface and the other in an interface with surface incompressibility. The distribution of surface tension that gives the optimal propulsion is given by the surface tension of the solution for the incompressible surface and the flow is a superposition of both solutions. For a surfer taking the form of a thin circular disk, the minimum dissipation is $16\mu a V^2$, giving a Lighthill efficiency of $1/3$. This places the Marangoni surfers among the hydrodynamically most efficient microswimmers.

This paper investigates the effects of a light-actuated photosurfactant on the canonical problem of the linear stability of a viscous thread surrounded by a dynamically passive fluid. A model consisting of the Navier–Stokes equations and a set of molar concentration equations is presented that capture light-induced switching between two stable surfactant isomer states, trans and cis. These two states display significantly different interfacial properties, allowing for some external control of the stability behaviour of the thread via incident light. Normal modes are used to generate a generalized eigenvalue problem for the growth rate which is solved with a hybrid analytical and numerical method. The results are validated with appropriate analytical solutions of increasing complexity, beginning with a solution to a clean interface, then analytical solutions for one insoluble surfactant, one soluble surfactant and a special case of two photosurfactants with a spatially uniform undisturbed state. Presenting each of these cases allows for a holistic discussion of the effect of surfactants in general on the stability of a liquid thread. Finally, the numerical solutions in the presence of two photosurfactants that display radially non-uniform undisturbed states are presented, and details of the impact of the illumination on the linear stability of the thread are discussed.

We study a thin, laterally confined heated liquid layer subjected to mechanical parametric forcing without gravity. In the absence of parametric forcing, the liquid layer exhibits the Marangoni instability, provided the temperature difference across the layer exceeds a threshold. This threshold varies with the perturbation wavenumber, according to a curve with two minima, which correspond to long- and short-wave instability modes. The most unstable mode depends on the lateral confinement of the liquid layer. In wide containers, the long-wave mode is typically observed, and this can lead to the formation of dry spots. We focus on this mode, as the short-wave mode is found to be unaffected by parametric forcing. We use linear stability analysis and nonlinear computations based on a reduced-order model to investigate how parametric forcing can prevent the formation of dry spots. At low forcing frequencies, the liquid film can be rendered linearly stable within a finite range of forcing amplitudes, which decreases with increasing frequency and ultimately disappears at a cutoff frequency. Outside this range, the flow becomes unstable to either the Marangoni instability (for small amplitudes) or the Faraday instability (for large amplitudes). At high frequencies, beyond the cutoff frequency, linear stabilization through parametric forcing is not possible. However, a nonlinear saturation mechanism, occurring at forcing amplitudes below the Faraday instability threshold, can greatly reduce the film surface deformation and therefore prevent dry spots. Although dry spots can also be avoided at larger forcing amplitudes, this comes at the expense of generating large-amplitude Faraday waves.

The aim of the present study is to understand the role of steam/cold-air inhalation on the mucus layer dynamics in the proximal airways. We model these flows as a viscoelastic liquid (lining the inner side of a tube) sheared by a turbulent airflow and subjected to a radial heat flow. The linear stability analysis of the resulting flow is carried out using numerical and thin-film approaches. The air–mucus interface tension, coupled with the azimuthal curvature, leads to the previously predicted axisymmetric capillary mode responsible for airway closure in distal airways. The viscoelasticity of the mucus leads to axisymmetric and new non-axisymmetric elastic modes, with the former possessing a higher growth rate. The axisymmetric elastic and capillary modes merge to give rise to a new ‘elasto-capillary mode’ possessing a growth rate nearly equal to the combined growth rate of the elastic and capillary modes. The analysis predicts a strong stabilising effect of the thermocapillarity induced by steam inhalation on the elasto-capillary mode. Thus steam decreases airway resistance, which could explain the observed therapeutic effect of steam inhalation. The opposite is the case when the inhaled air is colder than the body temperature, in agreement with the observations. The theoretical predictions are corroborated by the physical mechanism explaining the suppression/amplification of the elasto-capillary mode due to thermocapillarity.

We report an experimental investigation of the heat transport and flow field in a rectangular Rayleigh–Bénard convection (RBC) cell with two immiscible fluids: silicone oil and glycerol. The global heat transport of the system is divided into three ranges corresponding to the different flow structures formed in the glycerol layer. In range I, the glycerol layer is dominated by conduction, and no plume is formed over the interface. In range II, cellular rolls are formed in the glycerol layer and the horizontal motion of rolls causes an oscillation of temperature in the interface. In range III, the cellular pattern is time-independent, and the interface forms a group of wavelets with wave numbers consistent with the mode of the cellular pattern. In lower-thin glycerol, the Nusselt (Nu) grows from conduction to convection through an oscillating subcritical bifurcation at critical Rayleigh number $Ra_c$. The value of $Ra_c$ in the present work is smaller than the theoretical prediction of both-rigid boundaries and greater than the prediction of one-rigid and one-free boundaries. In the upper-thick silicone oil layer, $Nu$ increases with increasing $Ra$, but it is smaller than that of traditional RBC. For the silicone oil layer in two-layer RBC, the hot plumes emitting over the liquid–liquid interface showed different shape and different velocity from cold plumes emitting from the top rigid plate. This implies that the velocity boundary condition strongly influences the flow structure in turbulent convection.

We study the effects of buoyancy, surface-tension gradients and phase boundary on the stability of a layer of water that is confined between air at the top and a layer of ice at the bottom. The temperature of the overlying air and flux condition at the free surface of the water layer are such that the layer is susceptible to both thermal and thermocapillary instabilities. We perform a linear stability analysis to identify these modes of instability and investigate the effects of the phase boundary on them. We find that with increasing thickness of the ice layer, the critical Rayleigh and Marangoni numbers for the instabilities are found to first decrease and then asymptote to constant values for ice thicknesses much larger than the thickness of the water layer. In the case of thermocapillary instability, we find that the thickness of the ice layer has negligible influence on the stability threshold for dimensionless wavenumber $k \gg 1$, and that the presence of an unstably stratified liquid layer significantly alters the stability threshold for $k = O (1)$. Furthermore, the inclusion of Marangoni stresses reduces the stability threshold of the thermal instability.

This study investigates the linear instability of a thin-film coating inside a rigid tube. The flow is assumed to be inertialess and driven by an axial body force (e.g. gravity), an interfacial shearing force, or their combinations. The interface and the bulk of the film are laden with soluble surfactant. The properties of the soluble surfactant, i.e. solubility, sorption kinetics and bulk diffusivity, modulate the interfacial dynamics of the film. The influence of these properties on the linear instability of the film is comprehensively investigated via long-wave approximation analysis and numerical calculation. Two modes, namely the interface mode and the surfactant mode, are identified to dominate the instability. For a quiescent film, it is found that solubility, sorption kinetics and bulk diffusivity act to improve the uniformity of the surface surfactant and mitigate the stabilizing effect of the Marangoni force. For the film driven by the axial body/interfacial shearing force, the results reveal that solubility plays contrasting roles in the interface mode and the surfactant mode. A window with intermediate solubility is detected where the film can be linearly stabilized. Moreover, sorption kinetics is found to destabilize the perturbations with long wavelength whereas it stabilizes the perturbations with finite wavelength. The bulk diffusivity of the surfactant has a non-monotonic influence on the flow instability, and the film can be relatively stable at both strong and weak diffusivity.

Axisymmetric numerical simulations of the hydrodynamics around rising bubbles are performed in order to investigate the impact of surfactants on the bubble dynamics. Surfactants are assumed to be insoluble. The transport of the adsorbed surfactants is computed along the deforming surface at large surface Péclet number, and Marangoni stresses are taken into account. This simulation model leads to the stagnant-cap regime, with partially immobile interfaces. A parametric study is performed on cases at given Archimedes number, by varying the degree of contamination (Marangoni number) but maintaining a nearly constant Eötvös number. The presence of surfactants affects the rise velocity for oblate bubbles less than for spherical bubbles: the increase of the drag coefficient, due to interface contamination, is mitigated by a lower bubble deformation. When the cap angle $\theta _{cap}$ belongs to the southern hemisphere, the aspect ratio $\chi$ is found to decrease with contamination: the dynamic pressure responsible for the bubble distortion is lowered, related to the decline of kinetic energy. As soon as $\theta _{cap}$ lies in the northern hemisphere, the pressure stress causing distortion becomes independent on $\theta _{cap}: \chi$ no longer evolves with contamination, and already matches the prediction for fully immobile interfaces. Mass transfer of a passive scalar across the contaminated interface is also analysed. Surprisingly, the Sherwood number $Sh$ is found to follow the same law as for spherical shapes (Kentheswaran et al., Intl J. Heat Mass Transfer, vol. 198, 2022, 123325), allowing us to predict the decrease in $Sh$ due to contamination. These results reveal the couplings between interface immobilisation, bubble deformation, rise velocity and interfacial mass transfer.

The presence of thermocapillary (Marangoni) convection in microgravity may help to enhance the heat transfer rate of phase change materials (PCMs) in space applications. We present a three-dimensional numerical investigation of the nonlinear dynamics of a melting PCM placed in a cylindrical container filled with n-octadecane and surrounded by passive air. The heat exchange between the PCM and ambient air is characterized in terms of the Biot number, when the air temperature has a linear profile. The effect of thermocapillary convection on heat transfer and the topology of the melting front is studied by varying the applied temperature difference between the circular supports and the heat transfer through the interface. The evolution of Marangoni convection during the PCM melting leads to the appearance of hydrothermal instabilities. A new mathematical approach for the nonlinear analysis of emerging hydrothermal waves (HTWs) is suggested. Being applied for the first time to the examination of PCMs, this procedure allows us to explore the nature of the coupling between HTWs and heat gain/loss through the interface, and how it changes over time. We observe a variety of dynamics, including standing and travelling waves, and determine their dominant and secondary azimuthal wavenumbers. Coexistence of multiple travelling waves with different wavenumbers, rotating in the same or opposite directions, is among the most fascinating observations.

When an immiscible oil drop is immersed in a stably stratified ethanol–water mixture, the Marangoni flow on the surface of the drop can experience an oscillatory instability, so that the drop undergoes a transition from levitating to bouncing. The onset of the instability and its mechanisms have been studied previously (Li et al., Phys. Rev. Lett., vol. 126, 2021, 124502; Li et al., J. Fluid Mech., vol. 932, 2022, A11), yet the bouncing motion of the drop itself, which is a completely different problem, has not yet been investigated. Here we study how the bouncing characteristics (jumping height, rising and sinking time) depend on the control parameters (drop radius, stratification strength, drop viscosity). We first record experimentally the bouncing trajectories of drops of different viscosities in different stratifications. Then a simplified dynamical analysis is performed to get the scaling relations of the jumping height and the rising and sinking times. The rising and sinking time scales are found to depend on the drag coefficient $C_D^S$ of the drop in the stratified liquid, which is determined empirically for the current parameter space (Zhang et al., J. Fluid Mech., vol. 875, 2019, 622–656). For low-viscosity (5 cSt) oil drops, the results on the drag coefficient match those from the literature (Yick et al., J. Fluid Mech., vol. 632, 2009, pp. 49–68; Candelier et al., J. Fluid Mech., vol. 749, 2014, pp. 184–200). For high-viscosity (100 cSt) oil drops, the parameter space had not been explored and the drag coefficients are not readily available. Numerical simulations are therefore performed to provide external verification for the drag coefficients, which well match with the experimental results.

Although superhydrophobic surfaces (SHSs) show promise for drag reduction applications, their performance can be compromised by traces of surfactant, which generate Marangoni stresses that increase drag. This question is addressed for soluble surfactant in a three-dimensional laminar channel flow, with periodic SHSs made of long finite-length longitudinal grooves located on both walls. We assume that bulk diffusion is sufficiently strong for cross-channel concentration gradients to be small. Exploiting long-wave theory and accounting for the difference between the rapid transverse and slower longitudinal Marangoni flows, we derive a one-dimensional model for surfactant transport from the full three-dimensional transport equations. Our one-dimensional model allows us to predict the drag reduction and surfactant distribution across the parameter space. The system exhibits multiple regimes, involving competition between Marangoni effects, bulk and interfacial diffusion, bulk and interfacial advection, shear dispersion and surfactant exchange between the bulk and the interface. We map out asymptotic regions in the high-dimensional parameter space, and derive explicit closed-form approximations of the drag reduction, without any fitting or empirical parameters. The physics underpinning the drag reduction effect and the negative effect of surfactant is discussed through analysis of the velocity field and surfactant concentrations, which show both uniform and non-uniform stress distributions. Our theoretical predictions of the drag reduction compare well with results from the literature solving numerically the full three-dimensional transport problem. Our atlas of maps provides a comprehensive analytical guide for designing surfactant-contaminated channels with SHSs, to maximise the drag reduction in applications.

To investigate the initial development of the Rayleigh–Bénard–Marangoni (RBM) instability in a relatively deep domain, direct numerical simulations for a large range of Marangoni and Rayleigh numbers were performed. In the simulations, the surface was assumed to be flat and surface cooling was modelled by a constant heat flux. The small-scale dynamics of the flow and temperature fields near the surface was fully resolved by using a non-uniform vertical grid distribution. A detailed investigation of the differences in physical mechanisms that drive the Rayleigh- and Marangoni-dominated instabilities is presented. To this end, various properties such as the maturation rate of convection cells, the fluctuating kinetic energy and the surface characteristic length scale were studied. It was confirmed that buoyancy forces and surface-temperature-gradient-driven Marangoni forces enhance one another in promoting the development of the RBM instability. When using a relevant measure of the effective thermal boundary layer thickness as length scale, both the critical Marangoni and Rayleigh numbers, obtained for the purely Marangoni- and purely Rayleigh-driven instabilities, were found to be in good agreement with the literature.

A colloidal motor driven by surface tension forces is theoretically designed by encapsulating an active Janus particle in a liquid drop which is immiscible in the suspending medium. The Janus particle produces an asymmetric flux of a solute species which induces surface tension gradients along the liquid–liquid interface between the drop and the surrounding fluid. The resulting Marangoni forces at the interface propel the compound drop/Janus particle system. The propulsion speeds of the motor are evaluated for a range of relative sizes and positions of the drop and the particle and across a range of transport properties of the drop and the suspending medium. It is demonstrated that the proposed design can produce higher propulsion velocities than the traditional Janus-particle-based colloidal motors propelled by neutral diffusiophoresis.

The present study investigates the linear spatio-temporal and weakly nonlinear stability of a pressure-driven two-layer channel flow subjected to a wall-normal temperature gradient commonly encountered in industrial applications. The liquid–liquid interface tension is assumed to be a linearly decreasing function of temperature. The study employs both numerical (pseudo-spectral method) and long-wave approaches. The general linear stability analysis (GLSA) predicts shear-flow and thermocapillary modes that arise due to the imposed pressure and temperature gradients, respectively. The previous stability analyses of the same problem predicted a negligible effect of the pressure-driven flow on the linear stability of the system. However, the GLSA reveals stabilising and destabilising effects of the pressure-driven flow depending on the viscosity ratio ($\mu _r$), thermal conductivity ratio ($\kappa _r$), interface position ($H$) and the sign of the imposed temperature gradient ($\beta _1$). The analysis predicts a range of $H$ for given $\mu _r$ and $\kappa _r$, which can not be stabilised by the thermocapillarity. The numerically predicted long-wave instability is then captured using the long-wave asymptotic approach. The arguments based on the physical mechanism further successfully explain the role of $\mu _r$, $\kappa _r$, $H$, the sign of $\beta _1$ and the interaction between the velocity and temperature perturbations in stabilising/destabilising the flow. The spatio-temporal analysis reveals the dominance of the spanwise mode in causing the absolutely unstable flow. The weakly nonlinear analysis reveals a subcritical pitchfork bifurcation without shear flow. However, with the shear flow, the streamwise mode undergoes a supercritical Hopf bifurcation.

At high surface Péclet numbers, it is common to associate the presence of surfactants with surface immobilization, where a free surface becomes indistinguishable from a no-slip surface. A different mechanism has recently been proposed for longitudinal shear flow along a unidirectional trench (Baier & Hardt, J. Fluid Mech., vol. 949, 2022, A34) wherein, at high Marangoni numbers, the meniscus spanning the finite-length trench becomes a constant-shear-stress surface due to contamination by incompressible surfactant. That model predicts recirculating interfacial flows on the meniscus, a phenomenon that has been observed experimentally (Song et al., Phys. Rev. Fluids, vol. 3, issue 3, 2018, 033303). By finding an explicit solution to the constant-shear-stress model at all protrusion angles and calculating the effective slip length for a dilute mattress of such surfactant-laden trenches, we show that those effective slip lengths are almost indistinguishable from those for a surface whose menisci have the same deflection but have been completely immobilized (i.e. they are no-slip surfaces). This means that, despite the presence of non-trivial recirculating vortical flows on the menisci, the aggregate slip characteristics of such surfaces are that they have been effectively immobilized. This surprising result underscores the need for caution in comparing theory with experiments based on effective slip properties alone.

A recent experiment by Wang et al. (Soft Matt., vol. 17, 2021, pp. 2985–2993) shows that a self-propelled compound drop in a surfactant-laden solution can autonomously change its motion from a straight line to a spiraling trajectory, enhancing its capability for chemical detection, catalytic reaction and pollutant removal in a large fluid region. To understand the underlying physics of this peculiar motion, we develop a two-dimensional minimal model to study the swimming dynamics of a compound droplet driven by a self-generated Marangoni stress. We find that, depending on the Péclet number ($Pe$) and the viscosity and volume ratios of the two compound phases, the drop can swim in a variety of trajectories, including straight lines, circles, zigzag curves and chaotic trajectories. The drop moves in circles when its two components have comparable volumes. Otherwise, it shows other types of motions depending on $Pe$. Our simulation results for the circular motion at small $Pe$ are qualitatively comparable to the experiment. The transition between zigzag and circular trajectories is mainly determined by the orientation of high-order modes with respect to the drop's swimming direction. For most compound drops, the speed decays as $Pe^{-1/3}$ at high Péclet numbers as it does for a single-phase drop. A drop with two equal components undergoes a run-and-reorient motion due to the competition between the even and odd modes.

The effects of insoluble surfactants, satisfying a Langmuir equation of state, on transverse Stokes flow over a superhydrophobic surface of unidirectional grooves with flat menisci are examined. The phenomenon of surface immobilisation, whereby surfactants cause part or all of the Cassie-state menisci to become effectively no-slip zones thereby degrading the slip properties of the surface, is of primary interest. The study employs a combined analytical and numerical approach allowing an exploration of surfactant effects over the full range of surface Péclet numbers, Marangoni numbers and surfactant loads. For small surface Péclet and Marangoni numbers, perturbation theory is used to gain basic insights into the physical mechanisms at work. That analysis also provides checks on a robust numerical scheme, built around a complex variable formulation of the problem, used to compute solutions across a wide range of non-dimensional parameter values. Two distinct mechanisms are found to be responsible for surface immobilisation. In the first, most commonly seen at higher surface Péclet numbers, a stagnant cap forms because swept surfactant immobilises a section of the meniscus before any portion of the meniscus reaches maximum packing. Such a cap exists even for a linear equation of state and grows in length with increasing Marangoni number. A second immobilisation mechanism is associated with the nonlinear equation of state: an immobilised region forms because the surfactant concentration reaches its maximum value near the downstream edge of the meniscus. Immobilisation of the latter form can occur at much lower surface Péclet numbers, even if the Marangoni number is small, as long as there is sufficient surfactant in the system. The study enhances understanding of how insoluble surfactants can degrade slip over superhydrophobic surfaces.

Phase segregation triggered by selective evaporation can emerge in multicomponent systems, leading to complex physicochemical hydrodynamics. Recently, Li et al. (Phys. Rev. Lett., vol. 120, 2018, 224501) and Kim & Stone (J. Fluid Mech., vol. 850, 2018, pp. 769–783) reported a segregative behaviour (i.e. demixing) in an evaporating binary droplet. In this work, by means of experiments and theoretical analysis, we investigate the flow dynamics after the occurrence of the phase segregation. As example, we take the 1,2-hexanediol–water binary droplet system. First, we reveal experimentally the overall physicochemical hydrodynamics of the evaporation process, including the segregative behaviour and the resulting flow structure close to the substrate. By quantifying the evolution of the radial flow, we identify three successive life stages of the evaporation process. At Stage I, a radially outward flow is observed, driven by the Marangoni effect. At the transition to Stage II, the radial flow reverses partially, starting from the contact line. This flow breaks the axial symmetry and remarkably is driven by the segregation itself. Finally at Stage III, the flow decays as the evaporation ceases gradually. At this stage, the segregation has grown to the entire droplet, and the flow is again controlled by the Marangoni effect. The resulting Marangoni flow homogenizes the distribution of the entrapped volatile water over the whole droplet.