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.
Highlights
Introduction w vieOn (a) Foam structures in a confined systemAs was demonstrated by [9], for a given driving pressure the velocity at which the liquid-foam flows through a confined plates geometry (Hele-Shaw cell), depends upon how the bubbles are arranged topologically, exhibiting discontinuities in the resulting velocities at the transition between the different topological structures such as bamboo, staircase, and double staircase structures
Applications combining liquid foams with microfluidics occur in processes like enhanced oil recovery (EOR) [4] and soil remediation [5], where the foam is used as a driving fluid to sweep a specific material, colloid pollutant or particles from porous media [6,7,8]
Even though the new steady solution branch itself may be dynamically unstable, locating and tracking it through the domain pb ≤ p∗b can still be worthwhile. By demonstrating that it joins up with the original stable solution, we prove the existence of the saddle-node bifurcation, verifying in turn that for pb > p∗b there is no longer a corresponding steady state solution
Summary
As was demonstrated by [9], for a given driving pressure the velocity at which the liquid-foam flows through a confined plates geometry (Hele-Shaw cell), depends upon how the bubbles are arranged topologically, exhibiting discontinuities in the resulting velocities at the transition between the different topological structures such as bamboo, staircase, and double staircase structures (see Figure 1). As already mentioned, for a high flow rate ( higher driving pressure), T 1 topological transformations took place in the curved bend, making the foam structure unstable, both in experiment and simulation [1] This differs from the situation of an infinite staircase in a straight channel as described earlier. In the limit of high back pressures the geometry would cease to change, with the structure migrating faster and faster as back pressure increased thereafter (see Figure S 9 in the supplementary material section S 3(a) for details of such a structure) This notwithstanding, such behaviour was never observed in the case of the simple lens, which is too drastic a truncation of the infinite staircase [12]. The mode of break http://mc.manuscriptcentral.com/prsa rspa.royalsocietypublishing.org Proc R Soc A 0000000
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