Applications and generalizations of the foam drainage equation

Abstract

Some new analytic results concerning the foam drainage equation are described. The temporal approach to the equilibrium pro le of the foam is shown to be exponential, in agreement with numerical calculations. The e¬ect of including the equilibrium distribution of the liquid fraction as an initial condition is discussed, with respect to the progress of a solitary wave of liquid through a foam column. Existing analysis is further extended to allow for a variation of bubble sizes within the column and numerical calculations are compared with experimental results. We then consider the extension of the foam drainage equation into two and three dimensions and in particular the e¬ect of con nement in a cylindrical tube.

A model system for foam fractionation

“…Fairy Liquid has been associated with fairly rigid interfaces (i.e. n ≈ 2) in previous foam drainage experiments [10,13], so the theoretical expectation is approximately realized. It would be interesting to repeat these experiments in combination with other drainage experiments that more directly determine n. Figure 8 shows the dependence of the liquid velocity v on the gas velocity V. Again, the general theoretical prediction (v = k(n)V) is verified, with a linear fit describing the data well.…”

“…Only uniform profiles of steady drainage were considered. The later contributions of Gol'dfarb et al [7], Verbist et al [8], Koehler et al [9], Cox et al [10] and Saint-Jalmes & Langevin [11] developed the field in its full generality. This formulation has been applied with qualitative and semi-quantitative success to a range of drainage experiments, including 'free drainage' and 'forced drainage' [12,13], and foamability tests [14] in a single column.…”

Perspectives on foam drainage and the influence of interfacial rheology

“…This equation has been compared with experiments for one-dimensional (Verbist et al 1996, Cox et al 2000, Koehler et al 1998, 2000 and limited two-dimensional configurations ; much insight can be obtained from other studies of nonlinear convective-diffusion equations. There are generalizations to account for changes in container shape (Saint-Jalmes et al 2000), which requires keeping track of the local number density of channels, for macroscopic foam flow during drainage (Grassia et al 2001), for simultaneous coarsening where the average bubble size changes with position and time Weaire 2000, Hilgenfeldt et al 2001), and for incorporating higherorder terms in the (r, L) relation (Neethling et al 2002).…”

Numerical study of macroscopical drainage process in fabricating foamed aluminum using microscopical method

“…In the studies of the drainage, most researchers addressed the evolution of the liquid fraction in the foams by solving a classical foam drainage equation, and then gained the distribution of the liquid volume fraction. It can be seen as a macroscopical method [4][5] . Cox et al [5] extended the drainage equation to include a variation of the bubble size in time and space.…”

“…It can be seen as a macroscopical method [4][5] . Cox et al [5] extended the drainage equation to include a variation of the bubble size in time and space. Therefore, a great diversity of foams can be modeled.…”