A foam film propagating in a confined geometry: analysis via the viscous froth model

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.

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.

Pressure-driven growth in strongly heterogeneous systems

The pressure-driven growth model for advance of a foam front through an oil reservoir during foam improved oil recovery is considered: specifically the limit of strong heterogeneity in the reservoir permeability is treated, such that permeability variation with depth more than outweighs the tendency of the net pressure driving the front to decay with depth. This means that the fastest moving part of the front is not at the top of the solution domain, but rather somewhere in the interior. Moreover the location of the foam front on the top boundary of the system can no longer be specified as a boundary condition, but instead must be determined as part of the solution of the problem. Numerical solutions obtained from the pressure-driven growth model under these circumstances are compared with approximate analytic solutions. An early-time approximate solution is found to break down remarkably quickly (far more quickly than breakdown would occur in the analogous homogeneous system). Numerical solutions agree much better with local quasi-static solutions centred about local maxima in the front shape, each local maximum corresponding to a depth within the reservoir at which a high permeability stratum is found. These individual local solutions meet together at sharp concave corners to cover the entire depth of the foam front. As time continues to progress however, the system evolves towards a long-time, global quasi-static solution, corresponding to the fastest moving of the aforementioned local maxima. Additional key features of the predicted front shapes are elucidated. The foam front is found to meet the top boundary obliquely despite an established convention in pressure-driven growth that the front and top boundary should meet at right angles. In addition, at each sharp concave corner, discontinuous jumps are predicted in the path length that material points travel to reach either side of the corner. Moreover the long-time, global quasi-static solution is found to admit smooth concavities, as opposed to the aforementioned sharp concave corners, which only tend to be prominent earlier on.

Foam front propagation in anisotropic oil reservoirs

The pressure-driven growth model is considered, describing the motion of a foam front through an oil reservoir during foam improved oil recovery, foam being formed as gas advances into an initially liquid-filled reservoir. In the model, the foam front is represented by a set of so-called "material points" that track the advance of gas into the liquid-filled region. According to the model, the shape of the foam front is prone to develop concave sharply curved concavities, where the orientation of the front changes rapidly over a small spatial distance: these are referred to as "concave corners". These concave corners need to be propagated differently from the material points on the foam front itself. Typically the corner must move faster than those material points, otherwise spurious numerical artifacts develop in the computed shape of the front. A propagation rule or "speed up" rule is derived for the concave corners, which is shown to be sensitive to the level of anisotropy in the permeability of the reservoir and also sensitive to the orientation of the corners themselves. In particular if a corner in an anisotropic reservoir were to be propagated according to an isotropic speed up rule, this might not be sufficient to suppress spurious numerical artifacts, at least for certain orientations of the corner. On the other hand, systems that are both heterogeneous and anisotropic tend to be well behaved numerically, regardless of whether one uses the isotropic or anisotropic speed up rule for corners. This comes about because, in the heterogeneous and anisotropic case, the orientation of the corner is such that the "correct" anisotropic speed is just very slightly less than the "incorrect" isotropic one. The anisotropic rule does however manage to keep the corner very slightly sharper than the isotropic rule does.

Diffusion of curvature on a sheared semi-infinite film

The viscous froth model is used to study the evolution of a long and initially straight soap film which is sheared by moving its endpoint at a constant velocity in a direction perpendicular to the initial film orientation. Film elements are thereby set into motion as a result of the shear, and the film curves. The simple scenario described here enables an analysis of the transport of curvature along the film, which is important in foam rheology, in particular for energy-relaxing ‘topological transformations’. Curvature is shown to be transported diffusively along films, with an effective diffusivity scaling as the ratio of film tension to the viscous froth drag coefficient. Computed (finite-length) film shapes at different times are found to approximate well to the semi-infinite film and are observed to collapse with distances rescaled by the square root of time. The tangent to the film at the endpoint reorients so as to make a very small angle with the line along which the film endpoint is dragged, and this angle decays roughly exponentially in time. The computed results are described in terms of a simple asymptotic solution corresponding to an infinite film that initially contains a right-angled corner.

The viscous froth model: steady states and the high-velocity limit

The steady-state solutions of the viscous froth model for foam dynamics are analysed and shown to be of finite extent or to asymptote to straight lines. In the high-velocity limit, the solutions consist of straight lines with isolated points of infinite curvature. This analysis is helpful in the interpretation of observations of anomalous features of mobile two-dimensional foams in channels. Further physical effects need to be adduced in order to fully account for these.