Abstract

The growth of a foam which is simultaneously draining and bursting is analysed mathematically using the foam drainage equation. The bursting is implemented via a novel upper boundary condition: foam films are assumed to possess a well defined tensile strength. Since capillary suction forces on films depend on the cross-sectional area of surrounding Plateau borders, this is equivalent to assuming a limiting Plateau border area at the top of the foam. The case of a gas-liquid foam being blown up into a vertical column is studied in detail, the results being of relevance to froth stability column measurements. Two regimes of behaviour are possible. The first is where the upward velocity of air relative to the column exceeds the downward gravitational velocity of liquid relative to gas at the top of the foam. Such a foam is predicted to grow indefinitely, although the growth rate at late times is markedly less than that at early times. The second regime is where the upward velocity of the air relative to the column is less than the downward gravitational velocity of liquid at the top: capillary suction always retains a role in determining the net liquid motion in such cases, and the foam grows to a finite steady height. The two respective regimes can also be considered in terms of the froth stability, with the first corresponding to stable foams, and the second to unstable ones. At a fixed air flow velocity, decreasing the foam stability and thereby the film tensile strength, increases the critical Plateau border area for film bursting, and thereby the downward gravitational velocity: this can shift a system from the first regime to the second. We focus on the second regime, and in particular on the case where the foam is nearly stable, in the sense that the final froth height is large (compared to a well defined capillary length scale inherent to the foam itself). The steady state height of the foam and the steady state profile of liquid content versus vertical position are computed. A perturbation eigenvalue analysis is used to determine the exponential rate of approach to steady state. The approach to steady state is predicted to be extremely slow in the case of nearly stable froths, for which the rate of approach eigenvalue becomes vanishingly small. (c) 2005 Elsevier B.V. All rights reserved.