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

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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.

Original languageEnglish
Article number20210642
Number of pages27
JournalProceedings of the Royal Society A : Mathematical, Physical and Engineering Sciences
Issue number2258
Early online date9 Feb 2022
Publication statusPublished - 9 Feb 2022


  • physics of bubbles
  • foam rheology
  • multiphase fluids
  • non-Newtonian fluids
  • viscous froth model

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