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

Viscous froth model applied to the dynamic simulation of bubbles flowing in a channel: three-bubble case

A two-dimensional foam system comprised of three bubbles is studied via simulations with the viscous froth model. Bubbles are arranged in a so-called staircase configuration and move along a channel due to imposed driving back pressure. This flowing three-bubble system has been studied previously on the basis that it interpolates between a simpler staircase structure (a simple lens, which breaks up via so-called topological transformations if driven at high pressure) and an infinite staircase (which sustains arbitrarily large driving pressure without breaking). Depending on bubble size relative to channel size, different solution branches for the three-bubble system were found: certain branches terminate (as for the simple lens) in topological transformations and others reach (as for an infinite staircase) a geometrically invariant migrating state. The methodology used previously was, however, a purely steady state one, and hence did not interrogate stability of the various branches, nor the role of imposing different driving pressures upon topological transformation type. To address this, unsteady state three-bubble simulations are realized here. Stable solution branches without topological transformation exist for comparatively low driving pressures. For sufficiently high imposed back pressures, however, topological transformations occur, albeit with imposed pressure now influencing the transformation type.

Analysis of a model for surfactant transport around a foam meniscus

A model developed by Bussonnière & Cantat [1] is considered for film-to-film surfactant transport around a meniscus within a foam, with the transport rate dependent upon film-to-film tension difference. The model is applied to the case of a five-film device, in which motors are used to compress two peripheral films on one side of a central film and to stretch another two peripheral films on the central film's other side. Moreover, it is considered that large amounts of compression or stretch are imposed on peripheral films, and also that compression or stretch might be imposed at high velocities (relative to a characteristic velocity associated with physico-chemical properties of the foam films themselves). The actual strain that results on elements within each film might differ from the imposed strain, with the instantaneous film length coupled to the actual strain determining the amount of surfactant currently on each film (and hence also the amount of surfactant that has transferred either from or onto films). Quite distinct surfactant transport behaviour is predicted for the stretched film compared with the compressed one. In particular, when a film is stretched sufficiently at high enough velocity, surfactant flux onto it is predicted to become extremely 'plastic', increasing significantly.

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

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.