Viscous froth lens

Abstract

Microscale models of foam structure traditionally incorporate a balance between bubble pressures and surface tension forces associated with curvature of bubble films. In particular, models for flowing foam microrheology have assumed this balance is maintained under the action of some externally imposed motion. Recently, however, a dynamic model for foam structure has been proposed, the viscous froth model, which balances the net effect of bubble pressures and surface tension to viscous dissipation forces: this permits the description of fast-flowing foam. This contribution examines the behavior of the viscous froth model when applied to a paradigm problem with a particularly simple geometry: namely, a two-dimensional bubble "lens." The lens consists of a channel partly filled by a bubble (known as the "lens bubble") which contacts one channel wall. An additional film (known as the "spanning film") connects to this bubble spanning the distance from the opposite channel wall. This simple structure can be set in motion and deformed out of equilibrium by applying a pressure across the spanning film: a rich dynamical behavior results. Solutions for the lens structure steadily propagating along the channel can be computed by the viscous froth model. Perturbation solutions are obtained in the limit of a lens structure with weak applied pressures, while numerical solutions are available for higher pressures. These steadily propagating solutions suggest that small lenses move faster than large ones, while both small and large lens bubbles are quite resistant to deformation, at least for weak applied back pressures. As the applied back pressure grows, the structure with the small lens bubble remains relatively stiff, while that with the large lens bubble becomes much more compliant. However, with even further increases in the applied back pressure, a critical pressure appears to exist for which the steady-state structure loses stability and unsteady-state numerical simulations show it breaks up by route of a topological transformation.

Diffusion of curvature on a sheared semi-infinite film

“…In our case (similar also to what is observed in [8]) surface tension acts to smooth out the corner and the film immediately evolves into a curve. This system (of a film that turns through a right angle) becomes the same as the sheared 2 Values of λ can be quite difficult to determine and are sensitive to details of a particular experiment [18,19], but reference [9] has quoted a value 290 kg m Figure 3. Sketch of the long-time asymptotic solution of the sheared semi-infinite system, which corresponds to a similarity solution of an infinite film with a right-angled corner at first, from which curvature diffuses away over time.…”

“…Rather it is injected onto an already curved section of film and its θ and S values evolve accordingly. 9 …”

“…In other words, any overall increase in film length here, is due to new film elements being extracted from a fluid reservoir (the reservoir corresponds physically to a so-called Plateau border located at the film endpoint that is being dragged along), but after an element is thereby extracted, it begins to shrink. Taking dX to represent an element of film with length ds, and differentiating the relationship (ds) 2 = dX · dX with respect to time and substituting t = dX/ds and dn/ds = κt, yields the shrinkage rate over a material domain [9]:…”

“…Among such models, the viscous froth model has proved to be a powerful tool to simulate the rheology of dry foams, predicting realistic bubble shapes and agreeing with experimental results for 'two-dimensional' foam flows in channels, i.e. foam monolayers confined between upper and lower plates [8][9][10]. In this geometry, a foam film can be represented mathematically by a curve in the 'two-dimensional' plane.…”

“…In this geometry, a foam film can be represented mathematically by a curve in the 'two-dimensional' plane. The viscous froth model [3,[8][9][10][11][12][13][14] applied to such 'two-dimensional' systems is based on a force balance between the pressure difference across the two sides of the film ( P), the film tensions (γ ) and the viscous drag on each film element (the drag being associated with moving the films over the confining upper and lower plates): λv = Pn − cγ n, (1.1) where λ is the viscous drag coefficient per unit length of film, v is the velocity of a film element, n is the unit vector normal to the film element and c is the 'two-dimensional' film curvature. One particularly attractive feature of the viscous froth model over others is its capability to account for film curvatures which are not simply arcs of circles.…”