Quasi-one-dimensional and two-dimensional drainage of foam

Foam drainage is considered in a froth flotation tank containing a flowing froth. A simplified theory (the quasi-one-dimensional theory) exists in which horizontal variations across the tank of the foam's typical Plateau border area can be neglected. Moreover, drainage is shown to be gravity dominated over most of the froth. For weakly decelerated foam flows, the gravity dominated drainage equation admits constant Plateau border area solutions, both horizontally and vertically. For strongly decelerated flows there are two solution branches: one with constant Plateau border area, and another branch with kinked solutions. The kinked branch is the relevant one in the case of a foam with non-uniform bubble sizes. The gravity dominated solutions are required to match onto boundary layer solutions at the base of the foam. Capillary suction becomes important in this layer, and the solutions have a well-known structure: that of a soliton on an already wetted foam. The simple quasi-one-dimensional solutions are shown to have remarkable agreement with full two-dimensional drainage simulations.