Foam drainage on a sloping weir

Foam drainage is considered in a froth flotation tank with a sloping weir. The drainage is shown to be gravity dominated in most of the foam, except for thin boundary layers at the base of the froth, and along the sloping weir. The mathematical reason for the boundary layers is that capillary suction is a much weaker effect than gravity, but cannot be ignored altogether, because it represents a singular perturbation. The relative weakness of capillary suction with respect to gravity is represented by a key dimensionless parameter, denoted K, which satisfies K<<1. The volumetric flow at any point along the weir boundary layer is the accumulation of all liquid that has rained onto the weir above the point in question: typically, this flow is linear in distance measured downward from the weir lip. All liquid raining onto the weir is ultimately returned to the pulp phase as a high-speed jet. The jet velocity scales with the (2/3) power of distance from the weir lip, and is O(K(-2/3)) times larger than the typical velocity in the gravity-dominated flow in the bulk of the flotation tank. The liquid volume fraction in the jet is likewise O(K(-2/3)) larger than that in the bulk. Across the jet, the foam exhibits a known profile of liquid fraction vs. distance from the weir: this is known as the equilibrium profile. The foam requires a distance equivalent to O(K(4/3)) weir lengths to dry out significantly from the wetness value on the weir, but a larger O(K) distance to fall back to a wetness comparable with that in the bulk of the froth.