Dynamics and stability of volume-scavenging drop arrays: Coarsening by capillarity

Abstract

N drops, pinned by circular contact lines, are arranged in an array and coupled by a network of conduits. Inertialess exchange of volume among drops is driven by capillarity through the minimization of total surface energy. Drops scavenge volume from one another based on pressure differences, proportional to the surface tension, and arising from curvature differences. The system coarsens in the sense that, with time, volume is increasingly localized and ends up in a single ‘winner’ drop. Numerical simulations show that the identity of the winner can depend discontinuously on the initial condition and connectivity network. This motivates a study of the corresponding N -dimensional nonlinear dynamical system. All fixed points and their linear stabilities, obtained analytically, are found to be independent of connectivity. To determine which of the stable fixed points will be the winner, manifolds separating the attracting regions are found using a method which combines local information (eigenvectors at fixed points) with global information (invariant manifolds due to symmetry). This method is demonstrated for three N = 3 systems with various connectivity networks, and is used to explain the numerical observations.