Viscous froth model applied to the motion and topological transformations of two-dimensional bubbles in a channel : three-bubble case

Torres-Ulloa, C. and Grassia, P. (2022) Viscous froth model applied to the motion and topological transformations of two-dimensional bubbles in a channel : three-bubble case. Proceedings of the Royal Society A : Mathematical, Physical and Engineering Sciences, 478 (2258). 20210642. ISSN 1471-2946 (https://doi.org/10.1098/rspa.2021.0642)

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Abstract

The viscous froth model is used to predict rheological behaviour of a two-dimensional (2D) liquid-foam system. The model incorporates three physical phenomena: the viscous drag force, the pressure difference across foam films and the surface tension acting along them with curvature. In the so-called infinite staircase structure, the system does not undergo topological bubble neighbour-exchange transformations for any imposed driving back pressure. Bubbles then flow out of the channel of transport in the same order in which they entered it. By contrast, in a simple single bubble staircase or so-called lens system, topological transformations do occur for high enough imposed back pressures. The three-bubble case interpolates between the infinite staircase and simple staircase/lens. To determine at which driving pressures and at which velocities topological transformations might occur, and how the bubble areas influence their occurrence, steady-state propagating three-bubble solutions are obtained for a range of bubble sizes and imposed back pressures. As an imposed back pressure increases quasi-statically from equilibrium, complex dynamics are exhibited as the systems undergo either topological transformations, reach saddle-node bifurcation points, or asymptote to a geometrically invariant structure which ceases to change as the back pressure is further increased.

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