Reply to Comment by S. J. Cox and D. Weaire on Free Drainage of Aqueous Foams: Container Shape Effects on Capillarity and Vertical Gradients

Cox and Weaire [1] rightly emphasize that our solution of the drainage equation for the “Eiffel Tower” geometry does not treat the boundary conditions. There should be a no- flow condition at the top, and, after leakage begins, the liquid fraction should be pegged to εc ≈ 0.36 at the bottom. They then show how approximating the no-flow conditions at the top can improve agreement with numerical solution. But as argued in [2], we maintain that the neglect of capillarity coming from boundary conditions at the bottom dominates, and that this cannot explain our measurements. At short times, capillarity can delay the onset of leakage, and at long times it can counter gravity and retain liquid in the foam indefinitely; in either case, leakage is slower than our approximate solution, contrary to experiment. Therefore, we speculated that the discrepancy arose from neglect of coarsening, whereby the average bubble size increases via gas diffusion from smaller to larger bubbles. This is an important puzzle because, while the drainage equation successfully predicts forced-drainage experiments, it fails dramatically for free-drainage experiment

Free Drainage of Aqueous Foams: Container Shape Effects on Capillarity and Vertical Gradients

The standard drainage equation applies only to foam columns of constant cross-sectional area. Here, we generalize to include the effects of arbitrary container shape and develop an exact solution for an exponential, Eiffel Tower , sample. This geometry largely eliminates vertical wetness gradients, and hence capillary effects, and should permit a clean test of dissipation mechanisms. Agreement with experiment is not achieved at late times, however, highlighting the importance of both boundary conditions and coarsening

Instabilities in a Liquid-Fluidized Bed of Gas Bubbles

Gas bubbles in an aqueous foam can be unjammed, or fluidized, by introducing a forced flow of the continuous liquid phase at a sufficiently high rate. We observe that the resulting bubble dynamics are spatially inhomogeneous, exhibiting a sequence of instabilities vs increasing flow rate. First irregular swirls appear, then a single convective roll, and finally a series of stratified convection rolls each with a different average bubble size

Mechanical probing of liquid foam aging

We present experimental results on the Stokes experiment performed in a 3D dry liquid foam. The system is used as a rheometric tool : from the force exerted on a 1cm glass bead, plunged at controlled velocity in the foam in a quasi static regime, local foam properties are probed around the sphere. With this original and simple technique, we show the possibility of measuring the foam shear modulus, the gravity drainage rate and the evolution of the bubble size during coarsening

Electrical conductivity of dispersions: from dry foams to dilute suspensions

We present new data for the electrical conductivity of foams in which the liquid fraction ranges from two to eighty percent. We compare with a comprehensive collection of prior data, and we model all results with simple empirical formul\ae. We achieve a unified description that applies equally to dry foams and emulsions, where the droplets are highly compressed, as well as to dilute suspensions of spherical particles, where the particle separation is large. In the former limit, Lemlich's result is recovered; in the latter limit, Maxwell's result is recovered

Decorated vertices with 3-edged cells in 2D foams: exact solutions and properties

The energy, area and excess energy of a decorated vertex in a 2D foam are calculated. The general shape of the vertex and its decoration are described analytically by a reference pattern mapped by a parametric Moebius transformation. A single parameter of control allows to describe, in a common framework, different types of decorations, by liquid triangles or 3-sided bubbles, and other non-conventional cells. A solution is proposed to explain the stability threshold in the flower problem.Comment: 13 pages, 17 figure

The spreading of hydrosoluble surfactants on water

International audienceHeterogeneities in the distribution of surfactants at an interface between two fluids create a gradient of interfacial tension, which triggers the Marangoni effect, i.e., a bulk flow in the two phases surrounding the interface. The Marangoni effect is used to enhance the spreading of liquids on substrates, and some living organisms rely on this to move at the surface of water. It can also impair processes such as surface coating