Linear and non-linear wall friction of wet foams

We study the wall slip of aqueous foams with a high liquid content. We use a set-up where, driven by buoyancy, a foam creeps along an inclined smooth solid wall which is immersed in the foaming solution. This configuration allows the force driving the bubble motion and the bubble confinement in the vicinity of the wall to be tuned independently. First, we consider bubble monolayers with small Bond number Bo < 1 and measure the relation between the friction force F and the bubble velocity V. For bubbles which are so small that they are almost spherical, the friction law F proportional to V is Stokes-like. The analysis shows that the minimal thickness of the lubricating contact between the bubble and the wall is governed by DLVO long-range forces. Our results are the first evidence of this predicted linear friction regime for creeping bubbles. Due to buoyancy, large bubbles flatten against the wall. In this case, dissipation arises because of viscous flow in the dynamic meniscus between the contact film and the spherical part of the bubble. It leads to a non-linear Bretherton-like friction law F proportional to V-2/3, as expected for slipping bubbles with mobile liquid-gas interfaces. The Stokes-like friction dominates for capillary numbers Ca larger than the crossover value Ca* similar to Bo(3/2). The overall friction force can be expressed as the sum of these two contributions. On this basis, we then study 3D foams close to the jamming transition with osmotic pressures P small compared to the capillary pressure Pc. We measure the wall shear stress tau as a function of the capillary number, and we evidence two friction regimes that are consistent with those found for the monolayer. Similarly to this latter case, the total shear stress can be expressed as the sum of the Stokes-like friction term tau proportional to Ca and the Bretherton-like one tau proportional to Ca-2/3. However, for a 3D foam, the crossover at a capillary number Ca** between both regimes is governed by the ratio of the osmotic pressure to the capillary pressure, such that Ca** similar to (Pi/P-c)(3/2).