## Abstract

Liquid foams are an extreme case of multiphase flow systems: capable of flow despite a very high dispersed phase volume fraction, yet exhibiting many characteristics of not only viscoelastic materials but also elastic solids. The non-trivial, well-defined geometry of foam bubbles is at the heart of a plethora of dynamical processes on widely varying length and time scales. We highlight recent developments in foam drainage (liquid dynamics) and foam rheology (flow of the entire gas–liquid system), emphasizing that many poorly understood features of other materials have precisely defined and quantifiable analogues in aqueous foams, where the only ingredients are well-known material parameters of Newtonian fluids and bubble geometry, together with subtle but important information on the surface mobility of the foam. Not only does this make foams an ideal model system for the theorist, but also an exciting object for experimental studies, in which dynamical processes span length scales from nanometres (thin films) to metres (foam continuum flows) and time scales from microseconds (film rupture) to minutes (foam rheology).

## 1. Introduction

When the gas volume fraction of a gas/liquid multiphase flow (i.e. a bubbly liquid) is increased, interactions between bubbles become a more and more prominent feature in describing the dynamics and physical properties of the system. Hydrodynamic forces between bubbles can become important even at relatively low gas fraction *ϕ* (Wijngaarden 1972; Mazzitelli *et al.* 2003), and the speed of sound in a bubbly liquid is famously lowered dramatically even for *ϕ*≪10^{−3} (Caflisch *et al.* 1985; Commander & Prosperetti 1989). When *ϕ* reaches a critical value, bubbles start to interact directly by contact, i.e. they start to touch. For monodisperse bubbles, this happens around the density of random close packing of spheres, *ϕ*_{rcp}≈0.64 (Torquato *et al.* 2000). For , we can start to think of the bubbly liquid as a *foam*, where bubbles touch continually and, crucially, are therefore deformed from their individual spherical equilibrium shapes. The foam systems discussed in this work will thus be *liquid foams*, where the continuous phase between bubbles is a liquid, as opposed to solid or solidified foams such as polyurethane or styrofoam. Industrial and everyday applications for foams abound: the foam head on beer or soda and soapy dishwater come to mind immediately, but there is also a huge market for liquid foams as oil recovery agents or fractionation media for refining metallic ores in a process called foam flotation (Gibson & Ashby 1997; Evans *et al.* 1998). Studies of multiphase flow in foams can focus on either the liquid moving by bubble interfaces (*foam drainage*) or the bubbles moving through the liquid in some externally defined geometry (*foam rheology*). In this contribution, we highlight recent key results in both fields, pointing out how foam multiphase dynamics, despite being typically confined to creeping flow, gives rise to non-trivial behaviour that can be understood in terms of very few, well-known parameters (figure 1).

### (a) Geometry parameters

The deformations from spherical bubble shape usually lead to the formation of bubble–bubble contact areas that appear as flat, or slightly curved, facets shared by the contacting bubbles that, strictly speaking, do not touch but are still separated by a thin film of liquid. If the gas–liquid interfaces are clean, this is a very unstable situation. As figure 2*b* shows, the curvature of the bubbles in the non-flat areas necessitates a Young–Laplace pressure difference with respect to the fluid, while the fluid pressure in the thin film is equal to that of the bubble. Consequently, a pressure gradient in the fluid is set up that draws liquid from the thin film into the interstitial space between bubbles. The film rapidly thins and eventually breaks as thickness perturbations bridge the film width. This process of *film drainage* takes only a few milliseconds for millimetre-sized air bubbles in water (Narsimhan & Ruckenstein 1996).

This is why foam needs to be stabilized by means of a surfactant. The surfactant molecules cover all liquid–gas interfaces; in particular, they reside on both sides of the thin film. The two surfactant layers repel each other by one of a number of possible processes (Bergeron 1999), most prominently Coulomb repulsion or steric interaction. The repulsive force can be interpreted as an effective pressure inside the film, the *disjoining pressure*, counteracting the suction of the Young–Laplace pressure. Typically, equilibrium is reached for aqueous films of thickness 20–100 nm (Mysels *et al.* 1959; Bergeron 1999). With surfactant-stabilized films, bubble coalescence is very effectively inhibited and the gas volume fraction can be pushed beyond 99%. In these cases of *dry foams*, we prefer to give the liquid volume fraction as *ϵ*≡1−*ϕ*.

Figure 2*a* shows the typical geometry of a dry foam bubble: it is a polyhedron whose faces are made up of stabilized thin films and whose edges are liquid channels known as *Plateau borders* (PBs). A Plateau border has a characteristic scalloped-triangle cross section (figures 2*a* and 3) with a radius of curvature *a* related to a typical edge length *L* via the liquid fraction by , where *δ*_{ϵ}≈0.171, as shown by Koehler *et al.* (2000). An important special case is a quasi-two-dimensional foam, e.g. a single layer of bubbles confined between rigid plates (figure 3*a*). Seen through the plates, such bubbles have a polygonal cross section. The vertical PBs spanning the gap between plates (figure 3*b*) are called *internal* PBs; the horizontal PBs (those spread out on one of the plates; figure 3*c*) are called *external* PBs and relate to the liquid fraction via a different *δ*_{ϵ}≈0.43 (see Koehler *et al.* 2004). These foam geometries provide the framework for dynamical studies in the foam system: note that the knowledge of bubble size (and thus *L*) and liquid volume fraction *ϵ* is enough to quantify in detail the non-trivial geometrical features displayed in figures 2 and 3.

### (b) Material parameters

The presence of most surfactants does not measurably compromise either the character of water as a Newtonian fluid or even the value of its viscosity, so that liquid flow is always Newtonian fluid dynamics. As bubbles in experiments tend to be millimetre-sized and the cross-sectional dimensions of PBs are consequently 10–200 μm for typical *ϵ*∼10^{−3}−10^{−2}, the resulting Reynolds numbers are small and liquid flow in foams is almost always viscous (Stokes), entirely characterized by the viscosity of water, *μ*≈10^{−3} Pa s. In foam rheology, bubbles move as a whole and will generally deform. These deformations result in restoring forces due to the Young–Laplace pressure, so that the surface tension *γ* is the parameter that governs *elastic* properties of the foam dynamical system. We can infer here that foams generally behave as viscoelastic materials, yet unlike other viscoelastic systems (such as clays, polymer solutions, polymer melts, slurries and many others; see Larson (1999)) there are no hard-to-determine parameters depending on intricate properties of macromolecules or other complicated constituents of the system—*μ* and *γ*, together with bubble size, will usually describe the system entirely.

One caveat needs to be made with respect to the above statement: the *boundary conditions* of the fluid flow crucially depend on the surfactant properties, more specifically the surfactant *mobility*. This will be discussed in more detail in §2.

## 2. Liquid dynamics: foam drainage

As the film equilibrium thickness is very small, liquid drainage in dry foams occurs predominantly through the PB between bubbles, which form a network of channels as in a porous medium; however, PB cross sections are variable. In foam drainage, we seek a description of the temporal and spatial variation of the liquid fraction *ϵ*.

### (a) Nonlinear flow behaviour in Stokes flow

The driving forces for drainage flow are gravity and capillarity. Note that the capillary forces (for a given bubble size *L*) are dependent on the Young–Laplace pressure and thus, via the radius of curvature *a*, on *ϵ*. These driving forces can formally be combined into an effective pressure gradient(2.1)As is appropriate for Stokes flow, this driving is opposed by viscous friction in the fluid of viscosity *μ* (flowing at velocity *v*), which in porous media is typically expressed as *μv*/*k*, with a permeability *k*, which is a function of the cross section of liquid-carrying channels. In foams, however, this permeability is not a constant, but does itself depend on *ϵ*, as the cross section of the PBs is liquid fraction dependent. Making use of the continuity equation in the form , we can therefore write(2.2)Clearly, the resulting PDE for *ϵ* will be nonlinear in *both* the advection *and* the diffusion terms, posing a non-trivial problem. It was shown that this foam drainage equation sustains solitary wave solutions (Koehler *et al.* 1999) as well as subdiffusive spreading of pulses (Koehler *et al.* 2000, 2001).

The exact form of the permeability *ϵ*(*k*) depends on the nature of flow resistance in the PB. Interestingly, the most important factor here is the mobility of the interface.

### (b) Interfacial rheology

A clean gas–liquid interface will readily move in-plane if the liquid is driven to flow. In the case of a Plateau border, the bulk liquid inside the PB can eliminate its resistance by flowing with uniform velocity (plug flow, figure 4*b*). This is, of course, in contradiction to the assumption of Stokes flow: without viscous dissipation, the flow would have to accelerate until inertial terms become important. Such an acceleration is never observed and two different mechanisms can prevent it.

Surfactants at a gas–liquid interface will, in general, affect its *mobility*. In particular, large molecule surfactants such as proteins tend to interlock at the interface forming a ‘skin’ that is not easily moved in-plane. In such a case, exemplified by the use of bovine serum albumin (BSA) in Koehler *et al.* (2002), the PB walls behave as if they were rigid, providing no-slip boundary conditions. The resulting flow is a Poiseuille flow, the resistance of which can be accurately calculated from the shape of the PB cross section (figure 4*a*). The permeability becomes *k*(*ϵ*)∝*ϵ* and leads to nonlinear exponents 2 and 3/2 for the advection and diffusion terms of the foam drainage equation, respectively,(2.3)Here, *K*_{1}≈0.0063 is a permeability constant following from the geometry outlined above. Because the viscous resistance in this case of immobile interfaces is dominated by the PB (channels), this has been called the channel-dominated foam drainage equation. It was first derived by Gol'dfarb *et al.* (1988) and was further analysed by Verbist *et al.* (1996).

Other classes of surfactants (in particular small, soap-like molecules) do not provide great resistance to in-plane motion of the interface. Indeed, in some cases, the rheology of a clean interface may be almost recovered. Here, the flow is stopped from accelerating because all PBs have a finite length (of order *L*) and meet in junctions (called *nodes*). Plateau's rules (Plateau 1873) demand that each node is the meeting point of precisely four PBs and, furthermore, that the angles under which they meet are maximally symmetric (i.e. tetrahedral angles in three-dimensional foams and 120° angles in a quasi-two-dimensional foam, with the fourth PB spanning the gap between the plates; figure 3). This ensures that the liquid flowing from one PB to one or several connected PBs must change direction and has to split into several flows while merging with others. As a result, even if the flow in the PBs themselves is essentially a plug flow, viscous dissipation is present in the nodes. The difference is that the dissipation occurs over a much smaller volume than with the channel-dominated case above (namely, the volume of the nodes only). Consequently, the effective permeability for a foam with *mobile* interfaces changes qualitatively to *k*(*ϵ*)∝*ϵ*^{1/2}, which results in the *node-dominated* foam drainage equation,(2.4)where *K*_{1/2}≈0.0023 is another permeability constant (see Koehler *et al.* 2000). Note that the nonlinearity is now restricted to the advection (gravity) term—the liquid in a perfectly mobile foam shows pure diffusion dynamics perpendicular to the direction of gravity.

To quantify the mobility of the interfaces, it is convenient to introduce a dimensionless parameter that is the ratio of viscous bulk resistance in the PB to interfacial resistance, the latter described by an effective surface viscosity *μ*_{s}. The interfacial mobility parameter *M*=*μa*/*μ*_{s} is ≪1 for cases where the channel-dominated equations hold, while mobile-interface surfactants such as sodium dodecyl sulphate (SDS) show *M*>1. These mobilities have been tested experimentally by Koehler *et al.* (2002), directly measuring the liquid flow profiles and verifying that BSA leads to Poiseuille flow, while SDS approximates a plug flow. Experimentally, mobilities do not rise above *M*≈2, so that the velocity profile still retains a sizable drop towards the edges of the PB (figure 5*a*). This is also reflected in the fact that the permeability *k*(*ϵ*)∝*ϵ*^{ξ} for SDS is best fit with an exponent *ξ*≈0.6, larger than that for an ideally mobile interface. The same is true for the mobile surfactant in the commercial detergent Dawn, which is used in many experiments, including those discussed below. The work of Durand *et al.* (1999) showed that, by adding a co-surfactant to SDS, the effective permeability exponent could be varied between 0.6 and nearly 1, thus changing *M* and the drainage behaviour continually from almost node dominated to entirely channel dominated.

### (c) T1 dynamics

The same importance of interfacial rheology has been demonstrated in the dynamics of relative bubble motion (Weaire & Hutzler 2000): when two neighbouring bubbles in a foam are subject to sufficiently different forces to move one bubble with respect to the other, these bubbles will at some point cease to be neighbours and become neighbours to other bubbles in the foam. Such a neighbour-switching (T1) process is outlined in figure 6. Its dynamics is that of an activated process: usually, the nearly symmetric configuration between the initial and end states (where a number of PBs greater than four would have to meet in a node) is energetically much less favourable than either the initial or end configurations. Once it is surpassed, though, the T1 process proceeds spontaneously towards the end configuration. This spontaneous dynamics was investigated by Durand & Stone (2006) and found to be strongly dependent on interfacial rheology.

## 3. Foam dynamics: a first-principle study in rheology

The insights gained in experiments on foam drainage (§2) have been discussed extensively in the literature pertaining to drainage processes in foam flotation or physical chemistry (Prud'homme & Khan 1996) and have been extended to include shear-thinning and viscoelastic fluids (Safouane *et al.* 2006). However, the principal outcomes (nonlinear rheology and dependence on interfacial mobility) are just as valid when the bubbles are moving with respect to the fluid.

In this section, we will concentrate on pressure-driven quasi-two-dimensional foams, i.e. exactly the case discussed by Bretherton (1961) in his famous study of viscous resistance of a bubble driven in a tube or between parallel plates. For Bretherton's case of ideally mobile interfaces, it is the flow around the front and back regions of the bubble (analogous to the nodes in node-dominated drainage) that provides the viscous resistance, not the flow in the uniform film between the bubble and the tube wall (figure 5*a*). One important result is that the viscous resistance of the moving bubble scales with *Ca*^{2/3}, where *Ca*=*μv*/*γ* is the capillary number. For realistic surfactants, one expects the mobility of the interface to be at least somewhat impaired, so that dissipation in the thin film regions between bubbles and plate (figure 5*b*) becomes important. Indeed, Denkov *et al.* (2005) showed theoretically and experimentally that, in the opposite limit of rigid interfaces, viscous resistance scales as *Ca*^{1/2}.

Thus, in both the mobile and immobile limits, the quasi-two-dimensional foam is a rheological system in which the viscous resistance (balancing the driving pressure) depends on the power of the velocity (or capillary number). As this power is less than 1, it is a shear-thinning power-law fluid. Other fluids, such as certain polymer solutions (Larson 1999), have been characterized as power-law fluids, but often with only empirical evidence to back up this assumption of a constitutive relation. With quasi-two-dimensional foams, we have a system that can be shown, from first principles, to be a power-law fluid.

### (a) Hele-Shaw flows

Of the experiments used to study the rheology of foam, most have used shear-driven flows, e.g. in rheometers. Denkov *et al.* (2005) have measured the shear stress of wet foam (*ϵ*≈10%) with small bubble size (*L*∼30 μm) in a parallel-plate rheometer. By contrast, we use a pressure-driven flow set-up, namely a Hele-Shaw cell (Hele-Shaw 1898), and reproduce the quasi-two-dimensional set-up theoretically introduced by Bretherton (1961), whose experimental results were for tubes, not parallel plates.

In our set-up (figure 7), two large parallel rectangular glass plates enclose a rectangular channel of approximately *L*_{c}=1.0 m length and *w*=10 cm width. The gap between the plates is of *b*=1 mm thickness and can be filled through an inclined feeder channel with a foam of uniform, controlled bubble size (size monodispersity better than 5%) and controlled wetness (liquid fraction *ϵ*=0.01). Details of the set-up will be reported elsewhere. The aqueous foam is made with Dawn dishwashing detergent, the same surfactant that was found to result in nearly completely mobile bubbles in foam drainage experiments (Koehler *et al.* 1999, 2000). The experiment combines a number of features of important previous work: the pressure driving of Bretherton's original studies of bubble flow resistance, the ability to measure viscous resistance (but via an applied pressure rather than applied shear as in Denkov *et al.* (2005)) and the general aspect of a Saffman–Taylor fingering experiment, which has been tried for aqueous foams in circular-plate set-ups (Park & Durian 1994; Lindner *et al.* 2000), but not for a controlled rectangular channel. Once filled into the gap, the foam is driven from one end of the channel by injection of air at a fixed pressure (applied by a hydrostatic pressure head). For very low pressures, this leads to uniform displacement of the entire foam. For pressures larger than a critical value, fingering of air in the foam is observed (see §3*b*).

We take care to fill the gap between the plates with only one layer of bubbles, i.e. we generate a quasi-two-dimensional foam with bubble diameters *D*≈3 mm>*b*. The bubble geometry in side view is sketched in figure 7. Can we verify the power-law rheology for the pressure-driven foam? Can we determine the coefficients of viscous friction, and are they compatible with the existing results on shear flows and foam drainage?

The material parameters of the foam are as follows: the viscosity is very close to that of pure water (*μ*≈0.001 Pa s) and the surface tension was determined, by means of the pendant-drop method1, to be *γ*≈0.025 N m^{−1}. The bubble size *D* translates into Plateau border lengths of *L*≈2 mm. Using the geometry of bubbles and both exterior and interior PBs (see figure 3 and Koehler *et al.* 2004), we derive that the radius of curvature of the PBs is *a*≈140 μm. The foam is driven by pressure heads of between *P*=10 and 100 cmH_{2}O, i.e. *P*=1000–10 000 Pa, resulting in bubble velocities relative to the plate walls of between *v*=0.1 and 5 cm s^{−1} for these measurements of viscous resistance. The capillary number *Ca* therefore ranges between 4×10^{−5} and 2×10^{−3}, well within the applicable range of small *Ca* theories such as those in Bretherton (1961) and Denkov *et al.* (2005).

### (b) Measuring rheology

A total of 16 runs with different driving pressure *P* were analysed, each at eight different positions along the Hele-Shaw cell. It was found that data closer than approximately 15 cm to the inlet or 20 cm from the outlet of the Hele-Shaw cell were influenced by end effects, but the centre portion of the foam yielded consistent results. The functional dependence of *P* on *v* was determined via a least-squares best fit. The resulting exponent *β* in *P*=*R*(*L*_{foam})*v*^{β} is *β*≈0.63 (see figure 8). This is just slightly below 2/3, indicating—in agreement with the foam drainage results—that the Dawn foam has almost entirely mobile interfaces. The resistance prefactor *R* itself depends on the length *L*_{foam} of the foam pushed by the pressure. It fitted well with the linear relation *R*=*αL*_{foam}+*C*, where the constant offset *C* is due to the fact that the air not only pushes bubbles in the main, horizontal Hele-Shaw cell (where *L*_{foam} is variable), but also down the attached feeder channel (figure 7), which presents a significant additional resistance that does not change. The prefactor is determined as *α*^{*}≈0.20±0.05, with the driving pressure given in cmH_{2}O, *L*_{foam} in cm and *v* in cm s^{−1}. This should now be compared with the Bretherton (1961) theory.

### (c) Evaluating rheology

Bretherton (1961) derived an expression for the viscous resistance of a bubble in a circular tube as well as for a bubble between parallel plates. The latter results in a dynamic pressure drop per bubble of(3.1)where *p*_{c}=*γκ* is the capillary pressure with the mean curvature *κ* and *Ca*=*μv*/*γ* is a capillary number (cf. Denkov *et al.* 2005). In Bretherton's work, *κ*=1/*R*_{b} for a cylindrical bubble between plates, where *R*_{b} is the bubble radius. In our case, where the foam is dry, the radius of curvature determining the capillary pressure is much smaller, namely *κ*=1/*a*. Taking this into account, we conclude that each of our bubbles carries a pressure drop of *δP*≈840*Ca*^{2/3} Pa. Given a length *L*_{foam} of foam and a bubble diameter *D*≈3 mm, the total pressure drop is *P*=(*L*_{foam}/*D*)*δP* and the proportionality factor in *R* becomes *α*≈280 kPa m^{−1}. Expressed in our experimental units, we obtain *α*^{*}≈0.15, very close to the experimental value. Note that the estimate presented here contains uncertainties about the role of overall bubble deformation (as compared with the Bretherton case) as well as the role of finite bubble size in the direction perpendicular to the motion. It is noteworthy, though, that our pressure-driven quasi-two-dimensional experiments do not seem to need a significant correction factor, in contrast to the shear stress experiments by Denkov *et al.* (2005), where an additional factor of approximately 4 was needed to reconcile experiments with Bretherton's results. This is perhaps to be expected as the current experiment (other than being devised for tightly packed foam bubbles) is much more closely related to Bretherton's. The remaining discrepancy in *α*^{*} can be at least partially explained by the finite mobility of the interfaces, as a smaller *β*<2/3 leads to a compensating larger prefactor.

## 4. Foam dynamics: discrete processes and solid behaviour

Beyond a characterization as a continuum, viscoelastic medium, foam offers other important insights: viscoelasticity is usually a consequence of a *mesostructure* being present in the medium, i.e. a non-trivial organizational level of discrete objects between the atomic or molecular and the macroscopic length scales (cf. Witten 1999). For polymer solutions or melts, the mesoscale is that of macromolecules; for clays, it is microscopic platelets; and for granular media, grains. For foam, this is obviously the scale of individual bubbles, which is conveniently accessible in experiments (on length scales of mm).

### (a) Foam dynamics by T1 processes

This allows for unprecedented access to the *solid* characteristics of the viscoelastic medium. The geometrical arrangement of the mesostructure is important in setting the shear modulus, yield stress and other quantities associated with solids. The order of magnitude of the yield stress *σ*_{Y} in the foam is simply given by the generic pressure scale *γ*/*L*, while it was shown in Kraynik & Hansen (1986) that the *O*(1) prefactor *c* in *σ*_{Y}=*cγ*/*L* is determined by the relative orientation between the principal axes of the foam and the direction of shear. This yielding process is, microscopically, associated with neighbour flips of individual bubbles, i.e. T1 processes (see §2*c*). In fact, foam rafts have long been used to quantitatively study the formation and motion of dislocations in solids (always associated with neighbour changes of atoms), starting with Bragg & Nye (1947) and continuing until recently (Gouldstone *et al.* 2001). We have also observed dislocation formation and motion in the fingering experiments mentioned above. Figure 9 illustrates how T1 processes advance the air/foam boundary by one bubble length at a time.

Together with the power-law shear-thinning behaviour characterizing any bubble motion involving liquid viscous dissipation, the yield stress property characterizes a quasi-two-dimensional foam as a Herschel–Bulkley fluid in in-plane flow (as opposed to the pure power-law behaviour for mere motion with respect to the confining plates), leading to the constitutive relation(4.1)determined from first principles through surface tension, liquid viscosity and bubble geometry only.

### (b) Foam dynamics by film rupture

While the fingering process described above proceeds without breakage of films, a foam can also yield by breaking the thin films between bubbles. One way of transitioning between the T1 yielding and the film breakage is by applying forces at different rates (Arif *et al.* 2008). Another is by applying driving forces strong enough to overcome the cohesive elastic forces due to surface tension as follows.

We again make use of Bretherton's formula for viscous resistance, taking into account now that the radius of curvature (and implicitly the thickness of the liquid film along the plate) varies, within one bubble, owing to the shape of the (horizontal) PB lying along the plates. Figure 10*a* illustrates how these PBs experience differential viscous resistance force, largest at the thinnest part of the PB and smallest at the ends of the PBs, where they are connected to others. For the same driving force, therefore, the ends tend to move faster than the centres of the PBs. In figure 10*b*, we show an experimental confirmation of that fact, showing a ‘counter-intuitive’ curving of the PBs opposite to the direction of driving. Such a curvature, however, results in surface tension providing a restoring force trying to straighten the PB. As long as surface tension is strong enough to overcome the viscous force differential, all parts of the bubble still move at the same speed, and bubble integrity is preserved. But if the viscous force differential is greater than the maximum surface tension force (e.g. that occurring for a semicircular PB), the PB and the attached film will be elongated due to the velocity differential between the centre and end parts, resulting in the rupture of the film.

The mean curvature varies between approximately 1/a at the centre of the PB to approximately 1/(2*a*) at the end (cf. Koehler *et al.* 2000). With (3.1), we expect the viscous stress difference to be 4.70(*γ*/2*a*)*Ca*^{2/3} for a single bubble. However, the maximum Laplace–Young pressure able to balance this force is reached when the horizontal PB has formed a semicircle (of radius *L*/2), for which *p*_{LY}∼2*γ*/*L*. Equating both stresses yields a purely geometric criterion for a maximum capillary number without rupture(4.2)With the experimental values (*L*≈2 mm for the hexagonal bubbles of *D*=3 mm), we obtain a critical velocity of *v*_{c}≈0.36 m s^{−1}. Film breakage (propagation without T1 transitions) was observed for bubbles with local speed above *v*≈0.4 m s^{−1}, in good agreement with the estimate (4.2). Figure 10*b* shows that the curved PBs are indeed attached to films for which breakage is imminent.

## 5. Conclusions

We have emphasized here the many unique qualities of a foam both as a model system and as a soft material in its own right: it is a non-Newtonian, viscoelastic material, the rheology of which is described by a power-law fluid model (when driven uniformly between parallel plates) and a Herschel–Bulkley model (when deformed in the plane between the plates). The models can be derived from first principles, where yield stress and the power of shear thinning are directly related to simple material parameters (surface tension and viscosity) and information about the mobility of the gas–liquid interfaces. This mobility is a crucial feature of a liquid foam: although set on a very small scale by the physical chemistry of surfactant molecules, it determines macroscopic behaviour such as liquid drainage (fundamentally altering the power laws of the drainage dynamics) as well as microscopic behaviour on the single-bubble scale (such as the dynamics of T1 processes). We have shown that Bretherton's formulae give very good estimates for pressure-driven quasi-two-dimensional foam rheology. The accessibility of the single-bubble scale, including the positions and shapes of all bubbles, allow for observation and interpretation of discrete processes, from T1s to film rupture, as an analogue to what happens to atoms in a solid. In this fashion, foams not only model complex fluids, but also allow an unprecedented, detailed look into the flowing and yielding behaviour of solids, bridging the gap between multiphase flow phenomena and the solid mechanics of yielding.

## Footnotes

One contribution of 11 to a Theme Issue ‘New perspectives on dispersed multiphase flows’.

↵We thank Prof. Ken Shull for the opportunity to obtain these accurate surface tension measurements.

- © 2008 The Royal Society