Schematic representations of (a) the experimental channels, with a 1 and a 2 the channel widths and b the separation distance between channels, and (b) the complete experimental setup.
Image of the foam flow from left to right through two enclosed channels of length 15 cm in a test section of length 22 cm.
Images of foam flow through asymmetric enclosed channels (flow from bottom to top), with widths satisfying a 1 + a 2 = 4 cm. As the smaller channel becomes narrower, the foam structure explores a range of ordered and disordered arrangements. Ordered arrangements are shown for narrow channel widths of (a) a 2 = 1.33 cm, (b) a 2 = 0.72 cm, (c) a 2 = 0.59 cm, and (d) a 2 = 0.22 cm, resulting in double staircase, single staircase, mixed staircase/wide bamboo, and narrow bamboo structures, respectively. The wider channel contains a random two-dimensional foam in each case.
Dependence of (a) the velocity ratio, V 2/V 1, and (b) the flux ratio, Q 2/Q 1, on the channel width ratio, a 2/a 1, for the enclosed channels. The bubble size is R = 0.25 cm and a 1 + a 2 = 4 cm. The dotted vertical lines indicate the limits of the transition zone in which a mixture of single staircase and bamboo foam was observed in the narrow channel. The regions of the graph in which only bamboo or where only staircase or random foams were observed are labeled.
Velocity, non-dimensionalised by the reference velocity V 0, versus dimensionless time for both the enclosed and open-ended channels. (a) For a channel of width 0.38 cm and (b) for a channel of width 0.12 cm. Large fluctuations are observed for open-ended channels. The mean value V 2 for each case is indicated with a dashed line. The velocity in the enclosed-channel data is offset by +2 to improve clarity.
Velocity ratio, V 2/V 1, plotted as a function of the channel width ratio, for open-ended channels.
Schematic diagram of the experiment showing the points at which the pressures P up, , , , , and P down are defined.
A meniscus of length L (i) and orientation θ(i), travelling at velocity v. The normal velocity, v ⊥ = (v · n)n, and the viscous force acting on the film, f, are indicated.
Sketch of a staircase structure.
Surface Evolver 36 simulation of a bamboo film emerging from a channel with a rectangular cross-section a × h with h = 0.8 a. Main graph: Ratio of the capillary pressure across the film to 2γ/a, as a function of the bubble volume external to the channel Ωext divided by a 3. For comparison, we show the result obtained for a cylindrical tube of radius a/2 (dashed line). Inset: Image of the emergent film. The film is pinned all around the exit from the channel.
(a) Comparison of the experimental data of Fig. 4 with theoretical predictions. Analytical prediction: horizontal solid line—random foam structure (R) in the narrow channel; dashed line—staircase structure (S.S.), Eq. (17) ; and dotted–dashed line—bamboo structure (B. S.), Eq. (16) (A = 0.19 cm2, h = 0.1 cm and a 2 = 4x/(1 + x) in cm, with x = a 2/a 1). The functions are plotted in the domains of existence of the corresponding structure, as observed experimentally. The vertical dashed lines that separate these domains are not strict frontiers, and some fluctuations are observed. In the “random domain,” adjacent to the staircase domain, one data point corresponds to a double staircase structure (D.S.), for which a special treatment would lead to a better theoretical prediction. The prediction of Eq. (14) , determined from the experimental values of the foam structure and the velocity in the wide channel, is given as red dots. The curved solid line is the velocity ratio expected for a Newtonian fluid. 37 The analytical expression is given in the Appendix. (b) Same data for the fluxes (without the result for a Newtonian fluid).
Images of the foam entering a narrow channel (a 2 = 0.49 cm) to form a wide bamboo structure. The average bubble area is 0.25 ± 0.05 cm2, with an equivalent diameter of 0.51 ± 0.06 cm. Images (a)–(c) show three consecutive frames (Δt = 0.05 s) of an experiment in which a transient staircase structure is formed at the entrance to the channel before making the transition to a wide bamboo structure. Image (d) shows a single film being stretched down the centre of the channel—this phenomena only occurs when the foam forms a wide bamboo structure.
Velocity of a bamboo film in a narrow channel as a function of time. (•) Experimental data. Solid line, for t > t 1: prediction of Eq. (25) v(t) = v 0[1 + (t − t 1)/τ]−3, with the adjustable parameter τ = 2s. v 0 is the maximal velocity, reached by the lamella between t = 0, the time at which a pinned bubble burst, and t 1 = 1.24 s, the time at which a new film was pinned at the channel outlet.
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