Mixing describes the process by which solutes evolve from an initial heterogeneous state to uniformity under the stirring action of a fluid flow. Fluid stretching forms thin scalar lamellae that coalesce due to molecular diffusion. Owing to the linearity of the advection–diffusion equation, coalescence can be envisioned as an aggregation process. Here, we demonstrate that in smooth two-dimensional chaotic flows, mixing obeys a correlated aggregation process, where the spatial distribution of the number of lamellae in aggregates is highly correlated with their elongation, and is set by the fractal properties of the advected material lines. We show that the presence of correlations makes mixing less efficient than a completely random aggregation process because lamellae with similar elongations and scalar levels tend to remain isolated from each other. We show that correlated aggregation is uniquely determined by a single exponent that quantifies the effective number of random aggregation events. These findings expand aggregation theories to a larger class of systems, which have relevance to various fundamental and applied mixing problems.

The flow inside a rotating annulus tilted with respect to gravity is characterized experimentally and theoretically. As in the case of a tilted rotating cylinder the flow is forced by the free surface, maintained flat by gravity. It leads to resonances of global inertial modes (Kelvin modes) when the height of fluid is a multiple of half the wavelength of the mode. The divergence of the mode is saturated by viscous effects at the resonance. The maximum amplitude scales as the Ekman number to the power $-1/2$ when surface Ekman pumping is dominant, and to the power $-1$ when volumic damping is dominant. An analytical prediction is given with no fitting parameter, in excellent agreement with experimental results. At lower Ekman numbers, the flow destabilizes with respect to a triadic resonance instability, as already observed by Xu & Harlander (Phys. Rev. Fluids, 2020). We provide here a linear stability analysis leading to the viscous threshold of the instability for small tilt angles. For large tilt angles, a centrifugal instability is observed due to the acceleration of the flow by the inner cylinder. Finally, the features of the turbulent flow and its mixing efficiency are characterized experimentally. We underline the potential interest of this configuration for bioreactors.

The composition of the Earth's mantle, built by impacts of metallic core planetesimals, depends on the time offered to metal–silicate equilibration (concentration equalization by mixing) as the impactor fragments settle down. Smaller fragments equilibrate while bigger ones remain segregated and accumulate in the – therefore iron rich – Earth's core. To understand the primary instability setting the interpenetration depth and wavelength between the impactor and the mantle, Lherm et al. (J. Fluid Mech., vol. 937, 2022, A20) analyse the intermingled phenomena occurring when a drop impacts a substrate of a pool of another liquid, digging a crater. The density difference between the phases triggers a Rayleigh–Taylor instability when the drop is denser than the pool. The overall planet chemical composition relies on a tradeoff between fragmentation and mixing; this study provides elements to decipher their relative importance.

The interplay between chemical reaction and substrate deformation is discussed by adapting Ranz's formulation for scalar mixing to the case of a reactive mixture between segregated reactants, initially separated by an interface whose thickness may not be vanishingly small. Experiments in a simple shear flow demonstrate the existence of three regimes depending on the Damköhler number $Da=t_s/t_c$ where $t_s$ is the mixing time of the interface width and $t_c$ is the chemical time. Instead of treating explicitly the chemical cross-term, we rationalize these different regimes by globalizing it as a production term involving a flux which depends on the rate at which the reaction zone is fed by the reactants, a formulation valid for $Da>1$. For $Da<1$, the reactants interpenetrate before they react, giving rise to a ‘diffusio-chemical’ regime where chemical production occurs within a substrate whose width is controlled by molecular diffusion.

Fines are smaller droplets produced from an auxiliary mechanism besides the formation of the standard drops in a fragmentation process. We report their formation in a controlled experiment which isolates an individual fragmentation protocol: the collision of two rims bordering growing adjacent holes on a liquid sheet. The standard drops come from the capillary breakup of the fused rims. Occasionally, the rims collision is strong enough to trigger a new, splash-like mechanism, producing an expanding lamellae perpendicular to the main sheet, which destabilizes into finer drops. We quantify the threshold condition for the onset of this mechanism first discovered by Lhuissier & Villermaux (J. Fluid Mech., vol. 714, 2013, pp. 361–392), we document the resulting lamellae dynamics and explain why it affects the mean drop size in the spray, broadening substantially the overall drop size distribution, which we determine. Possible applications of these findings are mentioned.

Upon burst, surface bubbles transfer biological and chemical material from water bodies to the air we breathe via the production of droplets. An understanding of what shapes the size and payload of such droplets starts by understanding the fundamental physics of bubble birth, drainage and burst. Our combined experimental and theoretical investigation focuses on film-drop-producing surface bubbles. Controlling fluid properties such as temperature, salinity and volatility, coupled with changes of ambient air saturation, we elucidate the ageing and lifetime of bubbles. We derive and validate a generalized bubble cap drainage model accounting for both curvature-pressure-induced drainage and Marangoni flows induced by the coupling between the bubble and its surrounding air. We show that this deterministic drainage is coupled with stochastic local perturbations, both intrinsic and extrinsic, from impacts by mist droplets to microbubbles. We derive the conditions for such perturbations to be lethal to the cap film, involving the competition of mixing and drainage time scales on the bubble, the film thickness, the size of the perturbation and the local Marangoni stresses introduced. We explain how the mixing dynamics on the cap ensures that bursts mostly occur at the foot of bubbles rather than on their cap. Our study sheds light on the coupling between the deterministic cap thinning and the stochastic events leading to bubble death. We conclude that ubiquitous water contaminants enable the birth of a bubble, sustain it through its ageing, but ultimately also kill it.

We call thick those films for which the disjoining pressure and thermal fluctuations are ineffective. Water films with thickness $h$ in the $1{-}100~\unicode[STIX]{x03BC}\text{m}$ range are thick, but are also known, paradoxically, to nucleate holes spontaneously. We have uncovered a mechanism solving the paradox, relying on the extreme sensitivity of the film to surface tension inhomogeneities. The surface tension of a free liquid film is lowered by an amount $\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}$ over a size $a$ by chemical or thermal contamination. At the same time this spot diffuses (within a time $a^{2}/D$, with $D$ the diffusion coefficient of the pollutant in the substrate), the Marangoni stress $\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}/a$ induces an inhomogeneous outward interstitial flow which digs the film within a time $\unicode[STIX]{x1D70F}_{0}\sim \sqrt{\unicode[STIX]{x1D70C}ha^{2}/\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}}$, with $\unicode[STIX]{x1D70C}$ the density of the liquid. When the Péclet number $Pe=a^{2}/D\unicode[STIX]{x1D70F}_{0}$ is larger than unity, the liquid substrate motion reinforces the surface tension gradient, triggering a self-sustained instability insensitive to diffusional regularisation. Several experimental illustrations of the phenomenon are given, both qualitative and quantitative, including a precise study of the first instants of the unstable dynamics made by controlled perturbations of a Savart sheet at large $Pe$.

The diffusive strip method (DSM) is a near-exact numerical method for mixing computations initially developed in two dimensions (Meunier & Villermaux, J. Fluid Mech., vol. 662, 2010, pp. 134–172). The method, which consists of following stretched material lines to compute the resulting scalar field a posteriori, is extended here to three-dimensional flows. We describe the procedure and its three-dimensional peculiarity, which relies on the Lagrangian advection of a triangulated surface from which the stretching rate is extracted to infer the scalar field. The method is first validated at moderate Péclet number against a classical pseudospectral method solving the advection–diffusion equation for a Batchelor vortex, and then applied to a simple Taylor–Couette experimental configuration with non-rotating boundary conditions at the top-end disk, bottom-end disk and outer cylinder. This motion, producing an elaborate although controlled steady three-dimensional flow, relies on Ekman pumping arising from the rotation of the inner cylinder. A recurrent two-cell structure is separated by the horizontal mid-plane and formed by stream tubes shaped as nested tori under laminar flow conditions. A scalar blob in the flow experiences a Lagrangian oscillating dynamics undergoing stretchings and compressions, driving the mixing process. The DSM enables the calculation of the blob elongation and scalar concentration distributions through a single variable computation along the advected blob surface, capturing the rich evolution observed in the experiments. Interestingly, the mixing process in this axisymmetric and steady three-dimensional flow leads to a linear growth of surfaces in time similar to the one obtained in a two-dimensional shear. The potentialities, limits and extension of the method to more general flows are finally discussed.

We experimentally investigate mixing in sheared particulate suspensions by measuring a crucial kinematic quantity of the flow: the stretching laws of material lines in the suspending liquid. High-resolution particle image velocimetry (PIV) measurements in the fluid phase are performed to reconstruct, following the Diffusive Strip Method (Meunier & Villermaux, J. Fluid Mech., vol. 662, 2010, pp. 134–172), the stretching histories of the fluid material lines. In a broad range of volume fractions $20\,\%\leqslant \unicode[STIX]{x1D719}\leqslant 55\,\%$ , the nature of the elongation law changes drastically from linear, in the absence of particles, to exponential in the presence of particles: the mean and the standard deviation of the material line elongations are found to grow exponentially in time and the distribution of elongations converges to a log-normal. A multiplicative stretching model, based on the distribution of local shear rates and on their persistence time, is derived. This model quantitatively captures the experimental stretching laws. The presence of particles is shown to accelerate mixing at large Péclet numbers ( ${\gtrsim}10^{5}$ ). However, the wide distribution of stretching rates results in heterogeneous mixing and, hence, broadly distributed mixing times, in qualitative agreement with experimental observations.

A general theory for predicting the distribution of scalar gradients (or concentration differences) in heterogeneous flows is proposed. The evolution of scalar fields is quantified from the analysis of the evolution of elementary lamellar structures, which naturally form under the stretching action of the flows. Spatial correlations in scalar fields, and concentration gradients, hence develop through diffusive aggregation of stretched lamellae. Concentration levels at neighbouring spatial locations result from a history of lamella aggregation, which is partly common to the two locations. Concentration differences eliminate this common part, and thus depend only on lamellae that have aggregated independently. Using this principle, we propose a theory which envisions concentration increments as the result of a deconstruction of the basic lamella assemblage. This framework provides analytical expressions for concentration increment probability density functions (PDFs) over any spatial increments for a range of flow systems, including turbulent flows and low-Reynolds-number porous media flows, for confined and dispersing mixtures. Through this deconstruction principle, scalar increment distributions reveal the elementary stretching and aggregation mechanisms building scalar fields.

A liquid jet falling by gravity ultimately destabilizes by capillary forces. Accelerating as it falls, the jet thins and stretches, causing a capillary instability to develop on a spatially varying substrate. We discuss quantitatively the interplay between instability growth, jet thinning and longitudinal stretching for two kinds of perturbations, either solely introduced at the jet nozzle exit or affecting the jet all along its length. The analysis is conducted for any values of the liquid properties for a sufficiently large flow rate. In all cases, we determine the net gain of the most dangerous perturbation for all downstream distances, thus predicting the jet length, the wavelength at breakup and the resulting droplet size.

The forced radial expansion of a spherical liquid shell by an exothermic chemical reaction is a prototypical configuration for the explosion of cohesive materials in three dimensions. The shell is formed by the capillary pinch off of a thin liquid annular jet surrounding a jet of reactive gaseous mixture at ambient pressure. The encapsulated gas in the resulting liquid bubble is a mixture of hydrogen and oxygen in controlled relative proportions, which is ignited by a laser plasma aimed at the centre of the bubble. The strongly exothermic combustion of the mixture induces the expansion of the hot burnt gas, pushing the shell radially outwards in a violently accelerated motion. That motion triggers the instability of the shell, developing thickness modulations ultimately piercing it in a number of places. The capillary retraction of the holes concentrates the liquid constituting the shell into a web of ligaments, whose breakup leads to stable drops. We offer a comprehensive description of the overall process, from the kinematics of the shell initial expansion, to the final drop size distribution as a function of the composition of the gas mixture, the initial shell radius and thickness.

We develop a general framework for modelling mixing in porous media flows, in which the scalar mixture is represented as an ensemble of lamellae evolving through stretching, diffusion and coalescence. Detailed numerical simulations in Darcy scale heterogeneous permeability fields are used to analyse the lamella deformation process, which controls the local concentration gradients and thus the evolution of the concentration mixture through stretching enhanced diffusion. The corresponding Lagrangian deformation process is shown to be well modelled by a Langevin equation with multiplicative noise, which can be coupled with diffusion to predict the temporal evolution of the concentration probability density function (PDF). At late times, lamella interaction is enforced by confinement of the mixture within the dispersion area. This process is shown to be well represented by a random aggregation model, which quantifies the frequency of lamella coalescence and allows us to predict the temporal evolution of the concentration PDF in this regime. The proposed theoretical framework provides an accurate prediction of the concentration PDFs at all investigated times, heterogeneity levels and Péclet numbers. In particular, it relates the temporal behaviour of mixing, as quantified by concentration moments, scalar dissipation rate or spatial increments of concentration, to the degree of structural heterogeneity.

The edge of a stationary radially expanding liquid sheet and the receding rim bordering a plane sheet are naturally indented. It presents a collection of cusps at the extremity of which the liquid concentrates and is expelled. An experimental description of these cusps for a stationary flat inviscid Savart sheet is given. We identify the stable node–jet structure responsible for the deflection of the incoming flow at the rim and demonstrate how these cusps are the structures that accommodate for both mass and momentum conservation at the sheet edge. Their shape, their number around the sheet, and the residual momentum carried by the ejected liquid are computed.

We study the viscous version of the planar Savart sheet problem, using an impacting liquid jet up to 300 times more viscous than water. Two surprising observations are made, contrasting with the traditional case introduced by Savart where viscosity plays no role. First, if the radius of a viscous sheet is typically reduced compared to the case of water for a given jet radius and impacting velocity, the smooth–flapping transition is delayed, allowing for smooth sheet radii substantially bigger than those permitted with water at large impacting Weber number. Second, the drop size distribution is bimodal, with a substantial fraction of the drops having a very small, well-defined diameter. We understand these two facts in terms of an additional model experiment, and simple physical arguments.

A planar Savart water sheet uniformly seeded with small air bubbles in a large surface concentration is studied as a model experiment of the so-called ‘effervescent’ atomization process. This two-dimensional setup allows for a quantitative observation of all the steps of the sheet’s disintegration into a collection of disjointed droplets. The bubbles are heterogeneous nucleation sites which puncture the sheet with holes. The dynamics of the opening of holes competes with the simultaneous nucleation rate of new holes in a statistically stationary fashion. The liquid constituting the sheet is then transiently concentrated in a web of ligaments of various lengths and diameters, at the junction between adjacent holes. Their breakup produces the final spray. We provide a complete description of the ligament web statistics when nucleation is synchronous, and we show that the drop size dispersion from the breakup of a single ligament is responsible for the shape of the overall spray drop size distribution.

Stationary axi-symmetrical pressurized bells formed by the impact of a liquid jet on a solid disc (so-called Savart water bells) can exhibit uncommon sharp and pointed shapes. They are characterized by two successive inflections of the two-dimensional bell generator profile, corresponding to partially concave bells. We show that this shape is incompatible with the usual assumption that the detail of the flow across the liquid sheet constitutive of the bell is unimportant. We consider the equilibrium of a curved liquid sheet of finite thickness sustaining a pressure difference between both sides, and show that several curvatures of the interface may be a solution under given flow conditions. The inflection of the bell profile is then explained in terms of a spontaneous transition from a ‘negative’ to a ‘positive’ curvature which conserves mass flow, linear and angular momenta. That inflection is also a transition from a super to a subcritical flow (with respect to capillary waves), having the status of a capillary hydraulic jump on a freely suspended sheet, a novel object in fluid mechanics. The azimuthal wrinkles forming at the jump result from the inertial destabilization of the sheet due to the centripetal acceleration fluid particles experience as they flow along the highly curved bell profile in the vicinity of the fold. This finding also explains the singular shape at the edge of freely flapping sheets.

We depict and analyse the complete evolution of an air bubble formed in a water bulk, from the time it emerges at the liquid surface, up to its fragmentation into dispersed drops. To this end, experiments describing the drainage of the bubble cap film, its puncture and the resulting bursting dynamics determining the aerosol formation are conducted on tapwater bubbles. We discover that the mechanism of marginal pinching at the bubble foot and associated convection motions in the bubble cap, known as marginal regeneration, both drive the bubble cap drainage rate, and are responsible for its puncture. The resulting original film thickness evolution law in time, supplemented with considerations about the nucleation of holes piercing the film together culminate in a determination of the cap film thickness at bursting , where is the bubble cap radius of curvature, and a length which we determine. Subsequent to a hole nucleation event, the cap bursting dynamics conditions the resulting spray. The latter depends both on the bubble shape prescribed by , where is the capillary length based on gravity, and on . The mean drop size , the number of drops generated per bubble and the drop size distribution are derived, comparing well with measurements. Combined with known bubble production rates over the ocean, our findings offer an adjustable parameter-free prediction for the aerosol flux and spray structure caused by bubble bursting in this precise context.

The present study aims at documenting, making use of an original set-up allowing to acquire well-converged data, the coarsening of foams in two dimensions. Experiments show that a foam behaves quite differently depending on the way it has been prepared. We distinguish between an initially quasi-monodisperse foam and a polydisperse foam. The coarsening laws are initially different, although both foams reach a common, time-dependent asymptotic regime.

The ageing process relies on exchanges between adjacent foam cells (von Neumann's law), and on topological rearrangement (1 and 2 processes) whose rates are measured in all regimes. We attempt to make their contribution to the evolution of the area S and facet number n distribution of probability P(S, n, t) quantitative. The corresponding mean field theory predictions represent well the phenomenon qualitatively, and are sometimes in quantitative agreement with the measurements. A simplified version of this theory, taking the form of a Langevin model, explains in a straightforward manner the different scaling laws in the different regimes, for the different foams.