Flow along two dimensions of liquid pulses in foams: Experiment and theory

– Experiments on foam drainage have so far only been performed in essentially one-dimensional flow geometries aligned with the direction of gravity. Here a foam-filled HeleShaw cell is used to examine pulsed drainage, which is the flow of a finite liquid volume, both along and perpendicular to the direction of gravity. An exact similarity solution to the generalized foam drainage equation exists, and an asymptotic analysis is presented to elucidate the nonlinear dynamics of the model. Good qualitative and quantitative agreement between theory and experiments on aqueous foams made with SDS surfactant is found when the nodedominated foam drainage model is applied. Introduction. – Liquid separates the bubbles of foams and forms a continuous interconnected network. In sufficiently dry foams, almost all of the liquid resides in the channels (or Plateau borders), which are the regions between three touching bubbles, and in nodes (or vertices), which are the junctions of four channels. The liquid volume fraction is a macroscopic quantity, the ratio of liquid volume to total foam volume averaged over several bubbles, and varies spatially and temporally as liquid flows through the channels and nodes. The flow of liquid through foams is resisted by viscous drag and driven by surface tension and (usually) gravity and is often called foam drainage [1] even in the absence of gravity (e.g., [2]). The simplest model makes an analogy to flow through porous media. In [3] we propose that foams have a permeability k that depends on according to k( ) = KχL χ , (1) where L is the length of a channel in the (monodisperse) foam, and χ andKχ are dimensionless numbers that depend on the nature of the foam. The characteristic exponent χ depends on the boundary condition at the liquid/gas interface [4,5]. In the no-slip model [2,6], Poiseuille-type flow through the channels dominates the viscous drag and χ = 1. Alternatively, for the node-dominated foam drainage model [3, 7], the interfaces are mobile, flow through the channels is plug-like, and the dominant viscous dissipation occurs in the nodes because of the merging and bending of the flows from four channels entering the node. Here we primarily investigate the case χ = 1/2, and refer the reader elsewhere [8–10] for the case χ > 1/2.