A new law of thinning in foam dynamics

Abstract

A new law for the thinning of surfactant-free lamellae (applicable to metallic and ceramic foams with mobile interfaces) in a cross-section of an arid gas–liquid foam is derived using matched asymptotic analysis. Two limiting cases are identified at small capillary number: the well-known semi-arid foam having unit-order liquid fraction and the arid foam in which it is small. The lamellar thinning rates in both cases exhibit $t^{-2}$ power-law behaviour at long times even though the foam liquid area fractions have different orders of magnitude in capillary number. At early times, arid foam thinning is slowed because the curvature of the capillary quasi-static interfacial region must decrease in order to accommodate the flow from the films. Therefore, the thinning of lamellae feeding into a given Plateau border is coupled and the dynamics is distinct from that of the semi-arid foam.

Approximations of rupture times in arid and semi-arid foams are found by calculating the times for lamellae to thin to a pre-specified thickness. For given initial lamellar thicknesses, and for arid and semi-arid foams that have identical initial lamellar liquid areas, the arid foam ruptures more quickly than the semi-arid foam. On the other hand the rupture of lamellae is significantly delayed in arid foam compared to semi-arid foam if the initial lamellar thickness and capillary number are the same.