Viscous force exerted on a foam at a solid boundary : influence of the liquid fraction and of the bubble size

– We study experimentally the pressure drop needed to push a bubble train in a millimetric channel, as a function of the velocity. For dry liquid foams, the influence of the amount of liquid and of the bubble size is pointed out and we predict theoretically that this influence is closely related to the power law obtained for the force/velocity relation. This model is in fair agreement with our experimental data and provides a new interpretation of previous results. Introduction. – The study of the dissipation induced by foam flows in confined geometries was initially mainly motivated by the petroleum industry and the need to push multiphasic liquids through porous rocks [1, 2]. The domains of application recently enlarged to very different fields as microfluidics [3], 2D foams in Hele Shaw cells, seen as model systems for complex fluids [4–6], or even lung surfactant foam flows in pulmonary airways [7], thus giving a revived interest to this old problem. When a soap film train is pushed in a channel by a constant gas flux, the pressure drop may be several orders of magnitude larger than the one needed to push only water, at the same flux. Moreover, if the amount of gas in the bubble train increases, reaching more than 95% in volume, the pressure drop even increases. This puzzling multiphase flow behavior is governed by low Reynolds number hydrodynamical processes, with subtle non linearities related to the deformable air/liquid interfaces. Despite a large number of theoretical and experimental studies, for isolated bubbles [8–10] or bubble trains [11–14], a coherent understanding of the phenomenon over the whole parameter range explored is still missing. Analytical approaches, mainly based on the Landau Levich lubrication theory [15] predict a pressure drop scaling as the bubble velocity at the power 2/3 [8]. Nevertheless, as already mentioned by Schwartz et al [10], a phenomenological 1/2 power law very often better agrees with experimental observations. In a recent paper [14], these two behaviors were experimentally observed. The change between the 1/2 and the 2/3 power laws was obtained by changing the surfactant monolayer rigidity, a parameter neglected in the previous theoretical treatments. The aim of this paper is to determine experimentally, for dry foams, the relation between the foam velocity, its texture (liquid fraction and bubble size) and the viscous force exerted on the channel wall. Once rescaled by an adimensional number involving a non trivial combination of these parameters, all the experimental data are superimposed