A theory of the effective yield stress of foam in porous media: the motion of a soap film traversing a three-dimensional pore

The effective yield stress of foam in porous media depends on the capillary resistance of the soap films between bubbles, or lamellae, to forward movement. This resistance depends in turn on the shapes lamellae take as they move across pores. Even in idealized, radially symmetric pores, lamellae spontaneously jump to asymmetric shapes in their drive to minimize their surface area. These shapes affect the overall capillary resistance to foam movement. Earlier theoretical study of quasi-static lamella movement in two dimensions (2D) is extended here to three dimensions (3D) using the Surface Evolver computer program. Whereas in 2D, the lamella can take flat, asymmetric shapes in the pore body; in 3D, it can take a sequence of saddle-shapes of increasingly negative mean curvature as the trailing edge of the lamella approaches the middle of the pore. The results based on 2D lamellae are altered in detail but not in essence: the asymmetric jump increases the capillary resistance to foam movement, and for small bubbles in small pores, the minimum pressure gradient required to drive gas flow in foam is substantial.