Surfactants – molecules and particles that preferentially adsorb to fluid interfaces – play a ubiquitous role in the fluids of industry, of nature and of life. Since most surfactants cannot be seen directly, their behaviour must be inferred from their impact on observed flows, like the buoyant rise of a bubble, or the thickness of a coating film. In so doing, however, a difficulty arises: physically distinct surfactant processes can affect measurable flows in qualitatively identical ways, raising the spectre of confusion or even misinterpretation. This Perspective describes, in one coherent piece, both the equilibrium properties and dynamic processes of surfactants, to better enable the fluid mechanics community to understand, interpret and design surfactant/fluid systems. Specifically, we treat the equilibrium thermodynamics of surfactants at interfaces, including surface pressure, isotherms of soluble and insoluble surfactants and surface dilatational moduli (Gibbs and Marangoni). We describe surfactant dynamics in fluid systems, including surfactant transport and interfacial stress boundary conditions, the competition between surface diffusion, advection and adsorption/desorption, Marangoni stresses and flows and surface-excess rheology. We discuss paradigmatic problems from fluid mechanics that are impacted by surfactants, including translating drops and bubbles, surfactant adsorption to clean and oscillating interfaces; capillary wave damping, thin-film dynamics, foam drainage and the dynamics of particles and probes at surfactant-laden interfaces. Finally, we discuss the additional richness and complexity that frequently arise in ‘real’ surfactants, including phase transitions, phase coexistence and polycrystalline phases within surfactant monolayers, and their impact on non-Newtonian surface rheology.

Classical and symmetrical horizontal convection is studied by means of direct numerical simulations for Rayleigh numbers $Ra$ up to 3 × 1012 and Prandtl numbers $Pr=0.1$, 1 and 10. For both set-ups, a very good agreement in global quantities with respect to heat and momentum transport is attained. Similar to Shishkina & Wagner (Phys. Rev. Lett., vol. 116, 2016, 024302), we find Nusselt number $Nu$ versus $Ra$ scaling transitions in a region $10^{8}\leqslant Ra\leqslant 10^{11}$. Above a critical $Ra$, the flow undergoes either a steady–oscillatory transition (small $Pr$) or a transition from steady state to a transient state with detaching plumes (large $Pr$). The onset of the oscillations takes place at $Ra\,Pr^{-1}\approx 5\times 10^{9}$ and the onset of detaching plumes at $Ra\,Pr^{5/4}\approx 9\times 10^{10}$. These onsets coincide with the onsets of scaling transitions.

Liquid drop impact dynamics depends on the liquid–substrate interaction. In particular, when liquid–solid friction is decreased, the spreading of the impacting drop lasts longer. We characterise this effect by using two types of superhydrophobic surfaces, with similar wetting properties but different friction coefficients. It is found that, for large enough impact velocities, a reduced friction delays the buildup of a viscous boundary layer, and leads to an increase of the time required to reach the maximal radius of the impacting drop. An asymptotic analysis is carried out to quantify this effect, and agrees well with the experimental findings. Interestingly, this novel description complements the general picture of drop impact on solid surfaces, and more generally addresses the issue of drag reduction in the presence of slippage for non-stationary flows.

Shear bands are common in dense quasi-static granular flows. They can appear in the interior of the flowing material or at confining boundaries and are typically of the order of ten particle diameters in thickness. Deformation tends to be localized in shear bands separating non-deforming or weakly deforming regions. Dilatancy and sharp velocity variation are typical in these shear layers. Much work has been reported in the literature concerning the development of non-local quasi-static rheological models to predict the flow behaviour in shear layers. In a recent article, Dsouza & Nott (J. Fluid Mech., vol. 888, 2020, R3) derive a non-local extension to a classical plasticity model by postulating that some local quantities appearing in the yield function, which stipulates the relationship between different components of the stress for the material to undergo sustained yielding, and the flow rule which provides information on the rate of deformation tensor to within an arbitrary multiplicative constant, should be replaced by their local averages. They then obtain an explicit non-local model which does not involve new microstructural variables and they show that the model captures velocity and volume fraction fields in simple shear flows, although some model parameters must be fitted to achieve quantitative agreement. This article discusses the work of Dsouza & Nott (2020) and comments on work ahead for further testing and developing the model.

Dynamic wetting failure in shear-thinning and shear-thickening liquids is examined in this paper. Flow visualization experiments using a curtain-coating geometry suggest that shear thinning postpones the onset of wetting failure and the resulting air entrainment. To advance the fundamental understanding of the underlying physical mechanisms, a hydrodynamic model consisting of liquid displacing air in a rectangular channel in the absence of inertia is developed. Both shear thinning and shear thickening are considered by using Carreau-type models to describe the liquid rheology. Steady-state solutions are calculated using the Galerkin finite-element method and the critical capillary number where wetting failure occurs is identified. Shear thinning is found to postpone the onset of wetting failure whereas shear thickening is found to promote it. The underlying mechanism involves thickening/thinning of the air film as a consequence of shear thinning/thickening of the liquid and the tangential stress balance. The results can be interpreted in terms of an effective viscosity, and demonstrate that similar physical mechanisms govern dynamic wetting failure in Newtonian, shear-thinning and shear-thickening liquids.

When a drop impacts a dry surface at high velocity, it atomises into secondary droplets. These small droplets are generated by one of two types of splashes: either by a prompt splash from the spreading rim at the surface or by a thin corona splash, which levitates from the surface. This study investigates the splashing mechanisms experimentally using multiple high-resolution cameras and characterises the outcome of both splashing types at high Weber and Reynolds numbers. We demonstrate that the prompt splash is well described by the Rayleigh–Taylor instability of the rapidly advancing liquid lamella and determine the boundaries defining this splashing regime, which allows us to distinguish the prompt from the corona splash. Furthermore, we provide an expression to estimate the elapsed time during which the secondary droplets are generated, which is then implemented in the theory of Riboux & Gordillo (Phys. Rev. Lett., vol. 113 (2), 2014, 024507). This theoretical approach together with detailed quantification of the splashing outcome allows us to completely predict the outcome of both splashing types, which includes the mean size, velocity and total ejected volume of the secondary droplets. The detailed model proposed here can be indeed used to understand, characterise and predict more accurately the underlying physics in several applications.

Time-resolved particle image velocimetry measurements were performed in smooth-walled open channels to investigate the contributions of very large-scale motions (VLSMs) to the turbulence characteristics in open channel flows. The focal point is to clarify the free surface effects on the characteristics of VLSMs and the contributions of VLSMs to the unique statistical features in open channel flows (i.e., the turbulent kinetic energy (TKE) redistribution and smaller wake strength of the mean velocity profile). The resulting wavelength of VLSMs in present smooth-walled open channels is approximately $20h$ ($h$ is water depth), which is comparable to that in pipe and closed channels while smaller than that in rough-walled open channels, and they are shown to make a great contribution to turbulence statistics with over 50 % of streamwise turbulence intensity, Reynolds shear stress and negative net force coming from VLSMs in the outer layer. Compared with other wall-bounded flows, VLSMs maintain higher strength in the outer layer of open channel flows with non-negligible strength even in the near surface region ($y\sim >0.8h$), indicating that the free surface seems to sustain/promote VLSMs. This strength difference of VLSMs closely relates to the TKE redistribution and smaller wake strength of the mean velocity in the outer layer of open channel flows. The higher streamwise turbulence intensity is mainly contributed from the higher strength of VLSMs therein. The decelerating role of VLSMs combining with their higher strength is vital for shaping the mean velocity profile, which therefore is speculated to make a great contribution to the smaller wake strength phenomenon.

The formation of caustics by inertial particles is distinctive of dispersed flows. Their pressureless nature allows crossing trajectories resulting in singularities that cannot be captured accurately by standard Lagrangian approaches due to their fine spatial scale. A promising method for the investigation of caustics is the Osiptsov method or fully Lagrangian approach (FLA). The FLA has the advantage of identifying caustics, but its applicability is hindered by the occurrence of singularities. We present an original robust framework based on the FLA that provides an explicit expression of the dispersed phase structure that does not degenerate in the vicinity of caustics, using a single representative particle. The FLA is extended to account for the Hessian of the dispersed continuum (DC). It demonstrates the integrability of the FLA number density and allows for the calculation of the number density on a given length scale, retaining the functionality of the FLA. Number density models based on the second-order representation of the DC and on the one-dimensional structure of the particle distribution, that account for the anisotropy of the DC on caustics, are derived and applied for analytical flows. The number density is linked to a finite length scale, needed for the introduction of the FLA to spatially filtered flow fields. Finally, the method is used for the calculation of the interparticle separation on caustics. The identification of the structure of caustics presented in this work paves the way to a robust understanding of the mechanisms of particle accumulation.

The discovery of governing equations from data is revolutionizing the development of some research fields, where the scientific data are abundant but the well-characterized quantitative descriptions are probably scarce. In this work, we propose to combine the direct simulation Monte Carlo (DSMC) method, which is a popular molecular simulation tool for gas flows, and machine learning to discover the governing equations for fluid dynamics. The DSMC method does not assume any macroscopic governing equations a priori but just relies on the model of molecular interactions at the microscopic level. The data generated by DSMC are utilized to derive the underlying governing equations using a sparse regression method proposed recently. We demonstrate that this strategy is capable of deriving a variety of equations in fluid dynamics, such as the momentum equation, diffusion equation, Fokker–Planck equation and vorticity transport equation. The data-driven discovery not only provides the right forms of the governing equations, but also determines accurate values of the transport coefficients such as viscosity and diffusivity. This work proves that data-driven discovery combined with molecular simulations is a promising and alternative method to derive governing equations in fluid dynamics, and it is expected to pave a new way to establish the governing equations of non-equilibrium flows and complex fluids.

Electro- and diffusio-phoresis of particles correspond respectively to the transport of particles under electric field and solute concentration gradients. Such interfacial transport phenomena take their origin in a diffuse layer close to the particle surface, and the motion of the particle is force free. In the case of electrophoresis, it is further expected that the stress acting on the moving particle vanishes locally as a consequence of local electroneutrality. But the argument does not apply to diffusiophoresis, which takes its origin in solute concentration gradients. In this paper we investigate further the local and global force balance on a particle undergoing diffusiophoresis. We calculate the local tension applied on the particle surface and show that, counter-intuitively, the local force on the particle does not vanish for diffusiophoresis, in spite of the global force being zero, as expected. Incidentally, our description allows us to clarify the osmotic balance in diffusiophoresis, which has been a source of debate in recent years. We explore various cases, including hard and soft interactions, as well as porous particles, and provide analytic predictions for the local force balance in these various systems. The existence of local stresses may induce deformation of soft particles undergoing diffusiophoresis, hence suggesting applications in terms of particle separation based on capillary diffusiophoresis.

We use matched asymptotics to derive analytical formulae for the acoustic impedance of a subwavelength orifice consisting of a cylindrical perforation in a rigid plate. In the inviscid case, an end correction to the length of the orifice due to Rayleigh is shown to constitute an exponentially accurate approximation in the limit where the aspect ratio of the orifice is large; in the opposite limit, we derive an algebraically accurate correction, depending upon the logarithm of the aspect ratio, to the impedance of a circular aperture in a zero-thickness screen. Viscous effects are considered in the limit of thin Stokes boundary layers, where a boundary-layer analysis in conjunction with a reciprocity argument provides the perturbation to the impedance as a quadrature of the basic inviscid flow. We show that for large aspect ratios the latter perturbation can be captured with exponential accuracy by introducing a second end correction whose value is calculated to be in between two guesses commonly used in the literature; we also derive an algebraically accurate approximation in the small-aspect-ratio limit. The viscous theory reveals that the resistance exhibits a minimum as a function of aspect ratio, with the orifice radius held fixed. It is evident that the resistance grows in the long-aspect-ratio limit; in the opposite limit, resistance is amplified owing to the large velocities close to the sharp edge of the orifice. The latter amplification arrests only when the plate is as thin as the Stokes boundary layer. The analytical approximations derived in this paper could be used to improve circuit modelling of resonating acoustic devices.

We study the contact line dynamics of a viscous droplet deposited at the centre of a substrate subject to an axial thermal gradient. The temperature of the substrate decreases with distance from the centre, so the Marangoni stress induced at the liquid–air interface displaces the liquid radially outward. The flow experiences two stages. In the first stage, the droplet evolves towards an axially symmetric ring whose radius increases with time as $t^{1/3}$. Using the lubrication approximation, we perform numerical simulations that confirm this law for the motion of the front and show that the maximum thickness of the profile decreases as $t^{-0.374}$. We explain the evolution law of the contact line by balancing Marangoni and viscous stresses. In the second stage, the contact line becomes unstable and develops smooth corrugations whose amplitude increases with time and that eventually become long fingers. The temporal evolution of the Fourier spectra of the contour shows a shift of the most unstable mode from smaller to larger azimuthal wavenumbers.

A macroscopic condition to simulate the interaction between an incompressible fluid flow and a permeable micro-structured rigid surface (i.e. a thin membrane) has been developed using multiscale homogenization and matching asymptotic expansions between the near membrane and the far region. The condition allows us to write the fluid velocity across the membrane, seen macroscopically as a smooth equivalent surface, as an effective jump between the stresses computed on the two faces of this surface. The coefficients appearing in the jump condition are the entries of tensors which solve Stokes problems within the pores, enforced by boundary conditions depending on the flow outside the membrane. These problems, found via homogenization, definitely characterize the microscopic geometrical properties of thin permeable micro-structured sheets. The new macroscopic model is validated by comparisons with direct numerical simulations of the fluid flow across membranes in different configurations, proving that the formalism adopted to write the jump conditions is valid. As a result, a rational tool able to join a microscopic and a macroscopic analysis of fluid flows across membranes is delivered, showing some potentialities to provide advancement in membrane design. It suggests that the concept of permeability has to be substituted by a more general tensor, called here the Navier tensor, which plays the role of permeability only in some particular situations.

Coupled free-flow and porous-medium systems appear in a variety of industrial and environmental applications. Fluid flow in the free-flow domain is typically described by the (Navier–)Stokes equations while Darcy’s law is applied in the porous medium. The correct choice of coupling conditions on the fluid–porous interface is crucial for accurate numerical simulations of coupled problems. We found out that the Beavers–Joseph interface condition, which is widely used not only for fluid flow parallel to the porous layer but also for filtration problems, is unsuitable for arbitrary flow directions. To validate our statement, we provide several examples and compare numerical simulation results for the coupled Stokes–Darcy problems to the pore-scale resolved models. We show also that the Beavers–Joseph parameter cannot be fitted for arbitrary flow directions.

In this paper, we consider the dynamics of a helicoidal object, which can be either a passive particle or an active swimmer, with an arbitrary shape in a linear background flow at low Reynolds number, and derive a generalized version of the Jeffery equations for the angular dynamics of the object, including a new constant from the chirality of the object as well as the Bretherton constant. The new constant appears from the inhomogeneous chirality distribution along the axis of the helicoidal symmetry, whereas the overall chirality of the object contributes to the drift velocity. Further investigations are made for an object in a simple shear flow, and it is found that the chirality effects generate non-closed trajectories of the director vector which will be stably directed parallel or anti-parallel to the background vorticity vector depending on the sign of the chirality. A bacterial swimmer is considered as an example of a helicoidal object, and we calculate the values of the constants in the generalized Jeffery equations for a typical morphology of Escherichia coli. Our results provide useful expressions for the studies of microparticles and biological fluids, and emphasize the significance of the symmetry of an object on its motion at low Reynolds number.

The transition to turbulence in Taylor–Couette flow often occurs via a sequence of supercritical bifurcations to progressively more complex, yet stable, flows. We describe a subcritical laminar–turbulent transition in the counter-rotating regime mediated by a transient intermediate state in a system with an axial aspect ratio of $\unicode[STIX]{x1D6E4}=5.26$ and a radius ratio of $\unicode[STIX]{x1D702}=0.905$. In this regime, flow visualization experiments and numerical simulations indicate the intermediate state corresponds to an aperiodic flow featuring interpenetrating spirals. Furthermore, the reverse transition out of turbulence leads first to the same intermediate state, which is now stable, before returning to an azimuthally symmetric laminar flow. Time-resolved tomographic particle image velocimetry is used to characterize the experimental flows; these measurements compare favourably to direct numerical simulations with axial boundary conditions matching those of the experiments.

The interface between two miscible solutions in porous media and Hele-Shaw cells (two glass plates separated by a thin gap) in a gravity field can destabilise due to buoyancy-driven and double-diffusive effects. In this paper the conditions for instability to arise are presented within an analytical framework by considering the eigenvalue problem based on the tools used extensively by Chandrasekhar. The model considered here is Darcy’s law coupled to evolution equations for the concentrations of different solutes. We have shown that, when there is an interval in the spatial domain where the first derivative of the base-state density profile is negative, the flows are unstable to stationary or oscillatory modes. Whereas for base-state density profiles that are strictly monotonically increasing downwards such that the first derivative of the base-state density profile is positive throughout the domain (for instance, when a lighter solution containing a species A overlies a denser solution containing another species B), a necessary and sufficient condition for instability is the presence of a point on either side of the initial interface where the second derivative of the base-state density profile is zero such that it changes sign. In such regimes the instability arises as non-oscillatory modes (real eigenvalues). The neutral stability curve, which delimits the stable from the unstable regime, that follows from the discussion presented here along with the other results are in agreement with earlier observations made using numerical computations. The analytical approach adopted in this work could be extended to other instabilities arising in porous media.

Nonlinear interactions between surface and internal gravity waves in a two-layer system are studied using explicit second-order nonlinear evolution equations in Hamiltonian form. Motivated by the detailed experiment of Lewis, Lake & Ko (J. Fluid Mech., vol. 63, 1974, pp. 773–800), our focus is on surface wave modulation by the group resonance mechanism that corresponds to near-resonant triad interactions between a long internal wave and short surface waves. Our numerical solutions show good agreement with laboratory measurements of the local wave amplitude and slope, and confirm that the surface modulation becomes significant when the group velocity of the surface waves matches the phase speed of the internal wave, as the linear modulation theory predicts. It is shown, however, that, after the envelope amplitude is increased sufficiently, the surface and internal waves start to exchange energy through near-resonant triad interactions, which is found to be crucial to accurately describe the long-term surface wave modulation by an internal wave. The reduced amplitude equations are also adopted to validate this observation. For oceanic applications, numerical solutions are presented for a density ratio close to one and it is found that significant energy exchanges occur through primary and successive resonant triad interactions.

The behaviours of micro-organisms at interfaces play important roles in various biological, medical and engineering phenomena. Despite its widely recognized importance, our understanding of swimming micro-organisms at interfaces is limited. Ferracci et al. (PLoS One, vol. 8, 2013, e75238) reported that the ciliate, Tetrahymena, was entrapped at a water–air interface, while it escaped from a solid wall. Although the entrapment was speculated to be induced by physical processes, the mechanism is still unclear. To clarify the entrapment phenomenon, we focus on cell shape and numerically investigate the behaviour of a swimming micro-organism at interfaces from a hydrodynamic point of view. The model cell is assumed to propel itself by generating homogeneous tangential stress above the cell body. The results reveal that two major shape parameters, i.e. fore-and-aft asymmetry and a constriction, are dominant in the entrapment phenomenon. The mechanism can be explained by the balance of two opposite rotational velocities: repelling velocity due to the ciliary beat and attracting velocity due to the collision at the interface. In other words, the mechanism can be understood by hydrodynamic and steric effects. Moreover, cells tend to be entrapped more by the water–air interface than by the solid wall, which agrees with experimental observations reported previously (Ferracci et al. 2013). Finally, we experimentally observe Tetrahymena thermophila entrapped on the surface of an air bubble, and qualitatively discuss the shape of entrapped cells. The knowledge obtained provides a basis for understanding the behaviours of swimming micro-organisms at various interfaces, both in nature and in industrial applications.

The equations governing the mean fluid motions within a turbulent boundary layer adjoining a stationary plane beneath an axisymmetric circumferential flow $v_{\infty }(r)$, where $r$ is cylindrical radius, are solved by assuming the eddy diffusivity is proportional to $v_{\infty }$ times a diffusivity function $L(r,z)$, where $z$ is axial distance from the plane. The boundary-layer shape and structure depend on the dimensionless vorticity $\unicode[STIX]{x1D703}=\text{d}(rv_{\infty })/2v_{\infty }\,\text{d}r$, but are independent of the strength of the circumferential flow. This problem has been solved using a spectral method in the case of rigid-body motion ($\unicode[STIX]{x1D703}=1$ and $v_{\infty }\sim r$) for two models of $L$: $L$ constant (model A) and $L$ constant within a rough layer of thickness $z_{0}$ adjoining the boundary and increasing linearly with $z$ outside that layer (model B). The influence of the rough layer is quantified by the dimensionless radial coordinate $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D716}r/z_{0}$, where $\unicode[STIX]{x1D716}\ll 1$. The boundary-layer thickness varies parabolically with $r$ for model A and nearly linearly with $r$ for model B. Inertial stability of the outer flow causes the velocity components to decay with axial distance as exponentially damped oscillations, with the radial flow consisting of a sequence of jets. Axial flow is positive (flowing out of the boundary layer). Outflow from the layer, velocity gradients at the bounding plane, meridional-plane circulation and oscillations all increase as radius decreases.