The transition from two-dimensional to three-dimensional foam structures

Curvature driven motion of a bubble in a toroidal Hele-Shaw cell

We investigate the equilibrium properties of a single area-minimizing bubble trapped between two narrowly separated parallel curved plates. We begin with the case of a bubble trapped between concentric spherical plates. We develop a model which shows that the surface energy of the bubble is lower when confined between spherical plates than between flat plates. We confirm our findings by comparing against Surface Evolver simulations. We then derive a simple model for a bubble between arbitrarily curved parallel plates. The energy is found to be higher when the local Gaussian curvature of the plates is negative and lower when the curvature is positive. To check the validity of the model, we consider a bubble trapped between concentric tori. In the toroidal case, we find that the sensitivity of the bubble's energy to the local curvature acts as a geometric potential capable of driving bubbles from regions with negative to positive curvature.

Strain-induced yielding in bubble clusters

We study how shearing clusters of two or four bubbles induces bubble separation or topological rearrangement. The critical deformation at which this yielding occurs is measured as a function of shear rate, liquid composition, and liquid content in the cluster. We establish a geometrical yield criterion in the quasistatic case on the basis of these experimental data as well as simulations. In the dynamic regime, the deformation where the cluster yields increases with the strain rate, and we derive a scaling law describing this phenomenon based on the dynamical inertial rupture of the liquid meniscus linking the two bubbles. Our experiments show that the same scaling law applies to two- and four-bubble clusters.

The transition from three-dimensional to two-dimensional foam structures

A two-dimensional foam consists of a monolayer of bubbles. It can be created by squeezing the more familiar three-dimensional foam between two parallel glass plates. We describe and explain the minimum plate separation H which must be reached to fully effect the transition from three- to two- dimensional foam. We find that H/V(1/3) is close to one, where V is the average bubble volume, and increases slightly when the side-walls of the container are taken into account.

Rapid Plateau border size variations expected in three simple experiments on 2D liquid foams

Up to a global scaling, the geometry of foams squeezed between two solid plates (2D GG foams) essentially depends on two independent parameters: the liquid volume fraction and the degree of squeezing (bubble thickness to diameter ratio). We describe it in two main asymptotic regimes: fully dry floor tiles, where the Plateau border radius is smaller than the distance between the solid plates, and dry pancakes, where it is larger. We predict a rapid variation of the Plateau border radius in one part of the pancake regime, namely when the Plateau border radius is larger than the inter-plate distance but smaller than the geometric mean of that distance and the bubble perimeter. This rapid variation is not related to any topological change in the foam: in all the regimes we consider, the bubbles remain in mutual lateral contact through films located at mid-height between both plates. We provide asymptotic predictions in different types of experiments on such 2D GG foams: when foam is being progressively dried or wetted, when it is being squeezed further or stretched, when it coarsens through film breakage or through inter-bubble gas diffusion. Our analysis is restricted to configurations close to equilibrium, as we do not include stresses resulting from bulk viscous flow or from non-homogeneous surfactant concentrations. We also assume that the inter-plate distance is sufficiently small for gravity to be negligible. The present work does not provide a method for measuring small Plateau border radii experimentally, but it indicates that large (and easily observable) Plateau borders should appear or disappear rather suddenly in some types of experiments with small inter-plate gaps. It also gives expected orders of magnitude that should be helpful for designing experiments on 2D GG foams.

Yield drag in a two-dimensional foam flow around a circular obstacle: effect of liquid fraction

We study the two-dimensional flow of foams around a circular obstacle within a long channel. In experiments, we confine the foam between liquid and glass surfaces. In simulations, we use a deterministic software, the Surface Evolver, for bubble details and a stochastic one, the extended Potts model, for statistics. We adopt a coherent definition of liquid fraction for all studied systems. We vary it in both experiments and simulations, and determine the yield drag of the foam, that is, the force exerted on the obstacle by the foam flowing at very low velocity. We find that the yield drag is linear over a large range of the ratio of obstacle to bubble size, and is independent of the channel width over a large range. Decreasing the liquid fraction, however, strongly increases the yield drag; we discuss and interpret this dependence.

Planar soap bubble clusters with a cavity

We construct local energy-minimizing bubble clusters in the plane that are not simply-connected. Numerical evidence suggests that these minima are not isolated.

Instabilities in liquid foams

Instabilities play a central role in the physics of foams. Some that change the topology of a dry foam are indicated by the laws promulgated by Plateau in his 1873 book. Their occurrence is less clearly predictable in wet foams. Various other instabilities are related to gravitational loading and gas compressibility. We gather up many examples as a guide to future research and identify problems that remain, including what we call instabilities, which occur before they are expected on the basis of Plateau's laws.

Dissipative flows of 2D foams

We analyze the flow of a liquid foam between two plates separated by a gap of the order of the bubble size (2D foam). We concentrate on the salient features of the flow that are induced by the presence, in an otherwise monodisperse foam, of a single large bubble whose size is one order of magnitude larger than the average size. We describe a model suited for numerical simulations of flows of 2D foams made up of a large number of bubbles. The numerical results are successfully compared to analytical predictions based on scaling arguments and on continuum medium approximations. When the foam is pushed inside the cell at a controlled rate, two basically different regimes occur: a plug flow is observed at low flux whereas, above a threshold, the large bubble migrates faster than the mean flow. The detailed characterization of the relative velocity of the large bubble is the essential aim of the present paper. The relative velocity values, predicted both from numerical and from analytical calculations that are discussed here in great detail, are found to be in fair agreement with experimental results from the preprint Experimental evidence of flow destabilization in a 2D bidisperse foam by the present authors (2005).

Foams in contact with solid boundaries: equilibrium conditions and conformal invariance

A liquid foam in contact with a solid surface forms a two-dimensional foam on the surface. We derive the equilibrium equations for this 2D foam when the solid surface is curved and smooth, generalising the standard case of flat Hele-Shaw cells. The equilibrium conditions at the vertices in 2D, at the edges in 3D, are invariant by conformal transformations. Regarding the films, conformal invariance only holds with restrictions, which we explicit for 3D and flat 2D foams. Considering foams confined in thin interstices between two non-parallel plates, normal incidence and Laplace's law lead to an approximate equation relating the plate profile to the conformal map. Solutions are given for the logarithm and power laws in the case of constant pressure. The paper concludes on a comparison with available experimental data.

Two-dimensional flow of foam around an obstacle: force measurements

A Stokes experiment for foams is proposed. It consists of a two-dimensional flow of a foam, confined between a water subphase and a top plate, around a fixed circular obstacle. We present systematic measurements of the drag exerted by the flowing foam on the obstacle versus various separately controlled parameters: flow rate, bubble volume, bulk viscosity, obstacle size, shape, and boundary conditions. We separate the drag into two contributions: an elastic one (yield drag) at vanishing flow rate and a fluid one (viscous coefficient) increasing with flow rate. We quantify the influence of each control parameter on the drag. The results exhibit in particular a power-law dependence of the drag as a function of the bulk viscosity and the flow rate with two different exponents. Moreover, we show that the drag decreases with bubble size and increases proportionally to the obstacle size. We quantify the effect of shape through a dimensional drag coefficient, and we show that the effect of boundary conditions is small.

The interfacial energy of two-dimensional bidisperse cellular fluids

Abstract. The mixing energy of two biphasic fluid systems (dispersions of two liquids or a liquid and a gas, A and B in a liquid C), confined between two parallel plates, is calculated. Our attention is limited to concentrated and monodisperse systems, i.e. emulsions and foam/emulsions consisting of equal-size (if of the same composition) cells separated by a thin liquid film. It is demonstrated that the multiphase mixtures ordered into regular patterns can be stable in a wide range of interfacial tensions acting along A-C and B-C interfaces and also in a wide range of volume fractions of fluid A. Anisotropic properties of such well-ordered structures are also demonstrated.