A multi-scale framework to model dry foam dynamics
Seminar Room 1, Newton Institute
AbstractCo-author: Robert I. Saye (Department of Mathematics, University of California, Berkeley)
We describe a mathematical and computational framework for modeling soap bubble dynamics. We use a scale-separation approach to split the problem into three distinct phases that cycle over and over. During the rearrangement phase, a cluster of bubbles readjusts itself as surface tension in the membranes pushes on air in the pockets according to multi-phase incompressible flow, until a new macroscopic equilibrium is reached. Once this large-scale equilibrium is reached, we then invoke a drainage stage in which liquid in the lamellae drains into the Plateau borders according to thin film equations. Once one of the membranes becomes too thin, the model ruptures the equilibrium by removing that membrane, moving the cluster far from equilibrium, and leading back to the rearrangement stage.
This approach relies on several different computational methodologies, including (1) a Voronoi Implicit Interface Method (VIIM) to track the moving interface as a single PDE defined on a fixed mesh; (2) a second order projection method to solve the incompressible Navier-Stokes equations during the macroscopic rearrangement phase, describing the transport of fluid in the membranes; (3) a finite element formulation of a set of thin film equations for the fluid in the interfaces themselves, defined on the lamellae and Plateau borders, and linked together through coupled boundary conditions to describe drainage and (4) a rupture mechanism which includes topological rearrangement. We present results from a series of computations, including bubble cascades and thin film interference from a cluster of collapsing bubbles.