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How foams flow…





Aqueous foams are complex fluids which exhibit unique structural and mechanical properties. These are intimately linked to gravity or shear induced flows at the scale of the bubbles, films or liquid-gas interfaces. Two scientists of the team “Multiscale Mechanics of Soft Solids” and a colleague from the laboratory IFFSTAR have recently reviewed the state of the art in experimental and theoretical studies of the multiscale couplings that drive foam drainage and mechanics..

Foams are concentrated dispersions of gas bubbles in a soapy solution (fig. 1). Their structure is organized over several characteristics length scales, ranging from the bubble size, contact soap film thickness, down to the nanometric scale of the surfactant molecules which form monolayers adsorbed at the liquid-has interfaces. These surfactants not only stabilize the foam but also confer specific mechanical properties to the interfaces. Physical foam properties are governed by couplings between processes at these different length scales. The review presents foam drainage, an ageing process that arises from internal flows, as well as mechanical properties that also involve local complex flows, such as foam flow, plasticity or viscoelastic relaxations.

GIF Figure 1
Foams with different liquid volume fractions as indicated. a) and c) In the vicinity of the jamming transition, where contacts between neighboring bubbles vanish. b) and d) Foams with small liquid fraction. The structures can be disordered : a) and b), or ordered : c) f.c.c. structure, d) c.c. structure.






Due to their high specific surface, foams are metastable and undergo ageing processes. Under gravity, the liquid contained in a foam is drained such that a foam column dries up at the top while liquid accumulates at the bottom (fig. 2a). This non-linear process is described as the flow of liquid through a deformable porous medium. The permeability of this particular porous material is strongly coupled to the resistance of the interfaces against deformation, which sets the hydrodynamic boundary conditions of the flow in the constitutive geometric elements : the films, the edges at the junctions between adjacent films, or the vertices at the junction between edges (fig. 2b). One of the major current challenges is to model the permeability of these elements as a function of the interfacial resistance against shear or dilation/compression, in order to predict the law that governs foam drainage at the macroscopic scale.








Figure 2
Drainage of a vertical column of foam under gravity. Initially, the liquid content φl is homogenous, and foam becomes dry as time goes on. Bubbles become polyhedral at the top, and spherical at the bottom where liquid accumulates. b) Velocity profile u of the flow along the edge at the junction between three films. The liquid-gas interface is entrained (as indicated by the arrows) by the underlying bulk flow. The velocity is maximum at the center, and zero at the junction between the edge and the films. The edge permeability is modeled using the Boussinesq number that combines the liquid viscosity, interfacial viscosity and edge radius of curvature RPb..

Although foams are only constituted of fluids, they can exhibit either solid-like or liquid-like mechanical behavior depending on the applied stress and on the liquid volume fraction. Unlike usual solids, their elasticity arises solely from their specific surface and their surface tension. In the linear regime, foams behave as rigid solids on short time scales, due to a coupling between interfacial elasticity and the macroscopic response. In contrast, at low frequency, foam slowly flows at the rate of irreversible structural rearrangements induced by the coarsening (an ageing process due to inter-bubble gas exchanges). Bubble rearrangements can also be induced by applied stress if it exceeds the yield stress where plastic flow sets in. Their dynamics involve local flows at the film and edge scales. They determine the macroscopic non-linear friction law that governs stationary foam flow. In the case of interfaces with low rigidity, this friction is dominated by viscous flows in the contact films between neighbouring bubbles (fig. 3a et 3b). The macroscopic friction grows with increasing interfacial rigidity (fig. 3c). However, this link is not yet entirely understood. As liquid is added to foam, contacts between bubbles progressively disappear up to the point where the packing looses its elasticity and the slightest stress triggers flow. Close to this jamming transition, the mechanical behavior of foam with interfaces of low rigidity is similar to that of a concentrated suspension of solid grains (fig. 3d).








Figure 3
a) a) Schematic model of stationary flow in an ordered foam. The applied shear (indicated by the arrows) translates adjacent bubble layers relative to each another and thus induces structural rearrangements at the bubble scale. The liquid flow in the contact films between neighbouring bubbles governs the local friction law. The macroscopic friction law given by the normalized viscous shear stress (where the yield stress has been subtracted) as a function of the normalized strain rate is represented for : b) Foam with low interfacial rigidity and a concentrated emulsion ; c) Foam with interfaces that are very rigid with respect to dilation/compression ; d) Foam near the jamming transition (cf. fig. 1a) and a concentrated suspension of solid grains under the same conditions.

Beyond its fundamental interest, foam multiscale mechanics is the basis of many practical applications or industrial processes on a very large scale, such as in ore flotation or enhanced oil recovery.

« Flow in foams and flowing foams » Sylvie Cohen-Addad, Reinhard Höhler, Olivier Pitois Annual Review of Fluid Mechanics, vol. 45 (2013), p. 241-267