Aqueous foams coarsen with time due to gas diffusion through the liquid between the bubbles. The mean bubble size grows, and small bubbles vanish. However, coarsening is little understood for foams with an intermediate liquid content, particularly in the presence of surfactant-induced attractive forces between the bubbles, measured by the interface contact angle where thin films meet the bulk liquid. Rigorous bubble growth laws have yet to be developed, and the evolution of bulk foam properties is unclear. We present a quasistatic numerical model for coarsening in two-dimensional wet foams, focusing on growth laws and related bubble properties. The deformation of bubble interfaces is modelled using a finite-element approach, and the gas flow through both films and Plateau borders is approximated. We give results for disordered two-dimensional wet foams with $256$ to $1024$ bubbles, at liquid fractions from $2\,\%$ to $25\,\%$, beyond the zero-contact-angle unjamming transition, and with contact angles up to $10^\circ$. Simple analytical models for the bubble pressures, film lengths and coarsening growth rates are developed to aid interpretation. If the contact angle is non-zero, we find that a prediction of the coarsening rate approaches a non-zero value as the liquid fraction is increased. We also find that an individual bubble's effective number of neighbours determines whether it grows or shrinks to a good approximation.

Foams and bubbles formed in liquid mixtures have lifetimes longer by several orders of magnitude than in pure liquids of similar viscosities. We have shown recently that this effect results from slight differences in molecular concentrations between bulk and surfaces, which give rise to a thickness-dependent surface tension of liquid films. We present a quantitative description of the enhanced stability of liquid films in binary mixtures, based on experimental data and theoretical analysis. Experiments were performed with mixtures of different natures and compositions: foams of stationary heights were obtained by continuous injection of gas, on one hand, and single bubbles were swollen under the surface of a liquid bath, on the other hand. Remarkably, the lifetimes measured in both experiments exhibit the same variations with mixture composition, and follow a power law with the microscopic length that characterises the amplitude of the thickness-dependent surface tension. In addition, the lifetimes vary with the squared film thicknesses at the onset of bursting. We suggest that two stages occur between the birth of liquid films and their rupture. We show how the thickness-dependent surface tension allows an equilibrium shape to be reached at the end of a first stretching stage. We give an analytical description of this shape, which is fully consistent with experimental findings. We suggest a possible mechanism for final rupture of the film, and discuss it in light of existing theoretical predictions. Finally, we compare the properties of liquid mixtures and surfactant solutions, and in particular their surface rheology.

This study validated a two-phase compressible flow model considering elasto-plastic porous foams. The numerical data were compared with the previous experimental results in terms of the interaction between the planar shock wave and the porous foams and the mitigation effect of a porous foam filling a straight tube on a blast wave. The porous foams in the shock tube interacted with a planar shock wave. The drag between the shocked air and the foams reduced the shock wave strength. Moreover, the flexible foam was significantly deformed by the shock wave. The validation results confirmed good agreement and consistency between the numerical and experimental data. The mitigation effect on the blast wave caused by a high explosive, where the main parameter for comparison was the location of a rigid porous foam layer inside the straight tube, was investigated. In the first case, the porous foam plate was placed on the floor, whereas in the second case, the porous foam plates were placed on the floor, sidewalls and ceiling. The total energy transferred between the porous foam and the shocked air was computed to quantitatively understand the mitigation mechanism of the porous foam on the blast wave. The heat transfer was a dominant factor for the energy transfer from the shocked air to the porous foams. The second case further mitigated the blast wave outside, and the increment of the interface area of the air/porous foam greatly affected the blast wave mitigation.

Surfactant transport from foam film to foam film is an essential (yet poorly understood) aspect of the viscoplastic yielding behaviour of flowing foam. Recent experimental and modelling work by Bussonnière & Cantat (J. Fluid Mech., vol. 922, 2021, A25) has, however, helped to advance understanding of the relevant surfactant transport processes: the significance of that work is described herein.

Porous materials have many applications for laser–matter interaction experiments related to inertial confinement fusion. Obtaining new knowledge about the properties of the laser-produced plasma of porous media is a challenging task. In this work, we report, for the first time to the best of our knowledge, the time-dependent measurement of the reflected light of a terawatt laser pulse from the laser-produced plasma of low-Z foam material of overcritical density. The experiments have been performed with the ABC laser, with targets constituted by foam of overcritical density and by solid media of the same chemical composition. We implemented in the MULTI-FM code a model for the light reflection to reproduce and interpret the experimental results. Using the simulations together with the experimental results, we indicate a criterion for estimating the homogenization time of the laser-produced plasma, whose measurement is challenging with direct diagnostic techniques and still not achieved.

Liquid foam exhibits surprisingly high viscosity, higher than each of its phases. This dissipation enhancement has been rationalized by invoking either a geometrical confinement of the shear in the liquid phase, or the influence of the interface viscosity. However, a precise localization of the dissipation, and its mechanism, at the bubble scale is still lacking. With this aim, we simultaneously monitored the evolution of the local flow velocity, film thickness and surface tension of a five-film assembly, induced by different controlled deformations. These measurements allow us to build local constitutive relations for this foam elementary building block. We first show that, for our millimetric foam films, the main part of the film has a purely elastic, reversible behaviour, thus ruling out the interface viscosity in explaining the observed dissipation. We then highlight a generic frustration at the menisci, controlling the interface transfer between neighbour films and resulting in the localization of a bulk shear flow close to the menisci. A model accounting for surfactant transport in these small sheared regions is developed. It is in good agreement with the experiment, and demonstrates that most of the dissipation is localized in these domains. The length of these sheared regions, determined by the physico-chemical properties of the solution, sets a transition between a large bubble regime, in which the films are mainly stretched and compressed, and a small bubble regime, in which they are sheared. Finally, we discuss the parameter range where a model of foam viscosity could be built on the basis of these local results.

This work provides evidence that anisotropic drainage in sheared foam is at the origin of convective instability in very long foam channels. Convective instability occurs in foam under forced drainage when a critical liquid fraction is exceeded. Liquid spontaneously accumulates at one side of the channel. The weight imbalance induces convection rolls in the foam. Experiments in a very long vertical foam channel demonstrate that the critical liquid fraction is smaller than in previous findings by a factor of five. The critical liquid fraction depends on both the channel length and the inhomogeneity of the liquid feed. Well below the critical liquid fraction, a static, elastic shear deformation of the foam structure occurs. At the critical liquid fraction, initial steady convection rolls are located at the lower region of the channel and expand as the liquid fraction further increases. Combining the drainage equation with both the elastic response of the foam and a model for anisotropic drainage, a critical liquid fraction for the growth of an initial liquid imbalance is derived analytically, which corresponds very well to experimental findings. Numerical simulations of the drainage equation and the elastic response of the foam reproduce these experimental and analytical findings.

Liquid foams are, amongst other applications, used to mitigate shock waves. This aspect has received considerable attention at the macroscopic scale. However, the interaction between foam films and shock waves is still poorly understood and may be an important missing local information to build mitigation models. In this paper, we experimentally identify a new process leading to the foam film rupture, which dominates when the film thickness is sufficiently heterogeneous. Using a two-thickness film with a sharp and localised thickness gradient, we record the deformation of the interface between the thick and the thin parts. We observe the growth of an excess liquid area in the thin part and establish an analytical model and scaling laws which account for this phenomenon. Our results in this ideal configuration are consistent with actual rupture processes at stake in heterogeneous foam films.

Chaplygin’s hodograph method of classical fluid mechanics is applied to explicitly solve the Plateau problem of finding minimal surfaces. The minimal surfaces are formed between two mirror-symmetric polygonal frames having a common axis of symmetry. Two classes of minimal surfaces are found: the class of regular surfaces continuously connecting the supporting frames forming a tube with complex shape; and the class of singular surfaces which have a partitioning film closing the tube in between. As an illustration of the general solution, minimal surfaces supported by triangular frames are fully described. The theory is experimentally validated using soap films. The general solution is compared with the known particular solutions obtained by the Weierstrass inverse method.

Cordierite foams were prepared by thermo-foaming of alumina–microsilica–talc powder dispersions in molten D-glucose anhydrous followed by reaction sintering at 1400 °C, which exhibited an interconnected cellular morphology and three-dimensional porous cell walls. The cordierite foam had a porosity of up to 96%, and its corresponding thermal conductivity was as low as 0.057 W/(m·K). The foam structures showed a great promise for gas filtration and gas catalytic support. The formation of interconnected cellular morphology, the variations of cell wall thickness, and cell size were explained from the perspective of viscosity and weak points in this paper. The linear shrinkage of cordierite foams having a density of 0.102–0.226 g/cm3 was in the range of 13.0–6.9%. And the compressive strength (0.05–0.28 MPa) was determined by the large cell size (1.1–1.3 mm), ultra-high porosity (91–96%), and characteristic of cordierite.

A liquid foam is a dispersion of gas bubbles in a liquid matrix containing surface-active agents. Its flow involves the relative motion of bubbles, which switch neighbours during a so-called topological rearrangement of type 1 (T1). The dynamics of T1 events, as well as foam rheology, have been extensively studied, and experimental results point to the key role played by surfactants in these processes. However, the complex and multiscale nature of the system has so far impeded a complete understanding of the mechanisms involved. In this work, we investigate numerically the effect of surfactants on the rheological response of a 2D sheared bubble cluster. To do so, a level-set method previously employed for simulation of two-phase flow has been extended to include the effects of surfactants. The dynamical processes of the surfactants – diffusion in the liquid and along the interface, adsorption/desorption at the interface – and their coupling with the flow – surfactant advection and Laplace and Marangoni stresses at the interface – are all taken into account explicitly. Through a systematic study of the Biot, capillary and Péclet numbers that characterise the surfactant properties in the simulation, we find that the presence of surfactants can affect the liquid/gas hydrodynamic boundary condition (from a rigid-like situation to a mobile one), which modifies the nature of the flow in the volume from a purely extensional situation to a shear. Furthermore, the work done by surface tension (the 2D analogue of the work by pressure forces), resulting from surfactant and interface dynamics, can be interpreted as an effective dissipation, which reaches a maximum for a Péclet number of order unity. Our results, obtained at high liquid fraction, should provide a reference point, with which experiments and models of T1 dynamics and foam rheology can be compared.

The pressure-driven growth model is used to determine the shape of a foam front propagating into an oil reservoir. It is shown that the front, idealised as a curve separating surfactant solution downstream from gas upstream, can be subdivided into two regions: a lower region (approximately parabolic in shape and consisting primarily of material points which have been on the foam front continuously since time zero) and an upper region (consisting of material points which have been newly injected onto the foam front from the top boundary). Various conjectures are presented for the shape of the upper region. A formulation which assumes that the bottom of the upper region is oriented in the same direction as the top of the lower region is shown to fail, as (despite the orientations being aligned) there is a mismatch in location: the upper and lower regions fail to intersect. Alternative formulations are developed which allow the upper region to curve sufficiently so as to intersect the lower region. These formulations imply that the lower and upper regions (whilst individually being of a convex shape as seen from downstream) actually meet in a concave corner, contradicting the conventional hypothesis in the literature that the front is wholly convex. The shape of the upper region as predicted here and the presence of the concave corner are independently verified via numerical simulation data.

We examine the fracture of a quasi-two-dimensional surfactant-laden aqueous foam under an applied driving pressure, using a network modelling approach developed for metallic foams by Stewart & Davis (J. Rheol., vol. 56, 2012, p. 543). In agreement with experiments, we observe two distinct mechanisms of failure analogous to those observed in a crystalline solid: a slow ductile mode when the driving pressure is applied slowly, where the void propagates as bubbles interchange neighbours through the T1 process; and a rapid brittle mode for faster application of pressures, where the void advances by successive rupture of liquid films driven by Rayleigh–Taylor instability. The simulations allow detailed insight into the mechanics of the fracturing medium and the role of its microstructure. In particular, we examine the stress distribution around the crack tip and investigate how brittle fracture localizes into a single line of breakages. We also confirm that pre-existing microstructural defects can alter the course of fracture.

The rupture of soap films is traditionally described by a law accounting for a balance between momentum and surface tension forces, derived independently by Taylor and Culick in the 1960s. This law is highly relevant to the dynamics of thin liquid films of jets when viscous effects are negligible. However, the minute amounts of surfactant molecules present in soap films play a major role in interfacial rheology, and may result in complex behaviour. Petit et al. (J. Fluid Mech., vol. 774, 2015, R3) challenge standard thin film dynamics via intriguing experiments conducted with highly elastic surfactants. Unexpected structures reminiscent of faults are observed.

In order to understand the flow profiles of complex fluids, a crucial issue concerns the emergence of spatial correlations among plastic rearrangements exhibiting cooperativity flow behaviour at the macroscopic level. In this paper, the rate of plastic events in a Poiseuille flow is experimentally measured on a confined foam in a Hele-Shaw geometry. The correlation with independently measured velocity profiles is quantified by looking at the relationship between the localisation length of the velocity profiles and the localisation length of the spatial distribution of plastic events. To complement the cooperativity mechanisms studied in foam with those of other soft glassy systems, we compare the experiments with simulations of dense emulsions based on the lattice Boltzmann method, which are performed both with and without wall friction. Finally, unprecedented results on the distribution of the orientation of plastic events show that there is a non-trivial correlation with the underlying local shear strain. These features, not previously reported for a confined foam, lend further support to the idea that cooperativity mechanisms, originally invoked for concentrated emulsions (Goyon et al., Nature, vol. 454, 2008, pp. 84–87), have parallels in the behaviour of other soft glassy materials.

T1 topological rearrangement, i.e. switching of neighbouring bubbles in a liquid foam, is the elementary process of foam dynamics, and it involves film disappearance and generation. It has been studied extensively as it is crucial in foam rheology or foam collapse. T1 dynamics depends mainly on the surfactants used to generate the foam, and several models taking into account surface viscosity and/or elasticity have been proposed. By performing experiments in a cubic assembly of films, we go a step forward in this global analysis and investigate experimentally the mechanism of formation of the new film. In particular, the flow velocity field is probed by particle tracking and the film thickness is measured by light absorption and interferometric measurements. Two limit behaviours for the film are reported: it may (i) undergo an homogeneous extension, or (ii) resist elongation and remain at rest, new film being created from liquid exchange with connecting meniscus. Both T1 dynamics and film thickness are shown to depend on the competition between these two behaviours. Interestingly, their balance is set by the surfactant solution used, but it is also shown to vary during a single T1 relaxation process.

During improved oil recovery (IOR), gas may be introduced into a porous reservoir filled with surfactant solution in order to form foam. A model for the evolution of the resulting foam front known as ‘pressure-driven growth’ is analysed. An asymptotic solution of this model for long times is derived that shows that foam can propagate indefinitely into the reservoir without gravity override. Moreover, ‘pressure-driven growth’ is shown to correspond to a special case of the more general ‘viscous froth’ model. In particular, it is a singular limit of the viscous froth, corresponding to the elimination of a surface tension term, permitting sharp corners and kinks in the predicted shape of the front. Sharp corners tend to develop from concave regions of the front. The principal solution of interest has a convex front, however, so that although this solution itself has no sharp corners (except for some kinks that develop spuriously owing to errors in a numerical scheme), it is found nevertheless to exhibit milder singularities in front curvature, as the long-time asymptotic analytical solution makes clear. Numerical schemes for the evolving front shape which perform robustly (avoiding the development of spurious kinks) are also developed. Generalisations of this solution to geologically heterogeneous reservoirs should exhibit concavities and/or sharp corner singularities as an inherent part of their evolution: propagation of fronts containing such ‘inherent’ singularities can be readily incorporated into these numerical schemes.

In this work, we study theoretically the thickness of a liquid film (typically made of a surfactant solution) pulled out of a bath at constant speed in the absence of gravity, when it features a viscous or an elastic interfacial rheology. We show that a purely viscous rheology does not lead to the extraction of a steady state film of constant thickness. In contrast, the thickness of the film is well defined in the elastic case, which allows us to compute it. This thickness depends on the capillary number of the experiment, and on the elasticity of the interface. It is always lower than or equal to that obtained for an incompressible interface predicted by Frankel (Mysels, Shinoda and Frankel, Soap Films: Studies of their Thinning and a Bibliography, 1959), which is recovered in the limit of an arbitrary large elasticity.

The stability of a two-dimensional surfactant-free (gas–liquid) foam in a gravitational field is considered. The foam is assumed to have low liquid fraction, so the gas phase can be divided into approximately polygonal bubbles separated by thin liquid films. These free films drain toward accumulations of liquid at the bubble vertices, the Plateau borders, and eventually rupture due to van der Waals intermolecular attractions; this drives foam coarsening through the coalescence of neighbouring bubbles. In particular, we demonstrate how gravitational effects strongly modify the shape of the Plateau border interfaces and enhance the drainage flow in the liquid films, driving non-uniform thinning with exponential decay of the minimum film thickness, significantly faster than the power-law thinning predicted when gravitational effects are negligible.

We consider the stability of a planar gas–liquid foam with low liquid fraction, in the absence of surfactants and stabilizing particles. We adopt a network modelling approach introduced by Stewart & Davis (J. Rheol., vol. 56, 2012, p. 543), treating the gas bubbles as polygons, the accumulation of liquid at the bubble vertices (Plateau borders) as dynamic nodes and the liquid bridges separating the bubbles as uniformly thinning free films; these films can rupture due to van der Waals intermolecular attractions. The system is initialized as a periodic array of equally pressurized bubbles, with the initial film thicknesses sampled from a normal distribution. After an initial transient, the first film rupture initiates a phase of dynamic rearrangement where the bubbles rapidly coalesce, evolving toward a new quasi-equilibrium. We present Monte Carlo simulations of this coalescence process, examining the time intervals over which large-scale rearrangement occurs as a function of the model parameters. In addition, we show that when this time interval is rescaled appropriately, the evolution of the normalized number of bubbles is approximately self-similar.