Volume scavenging of networked droplets

Highlights: • Droplet array experiencing capillarity-driven pressure/volume changes. • All equilibria of the corresponding dynamical system (odes) determined. • Constrained optimization used for stability; non-hyperbolic cases discussed. • Changes in long-term dynamics with fluid rheology addressed numerically. • Ordering of equilibria by surface area shown; bifurcating rest states analyzed. -- Abstract: A system of $N$ spherical-cap fluid droplets protruding from circular openings on a plane is connected through channels. This system is governed by surface tension acting on the droplets and viscous stresses inside the fluid channels. The fluid rheology is given by the Ostwald–de Waele power law, thus permitting shear thinning. The pressure acting on each droplet is caused by capillarity and given in terms of the droplet volume via the Young–Laplace law. Liquid is exchanged along the network of fluid conduits due to an imbalance of the Laplace pressures between the droplets. In this way some droplets gain volume at the expense of others. This mechanism, christened "volume scavenging," leads to interesting dynamics. Numerical experiments show that an initial droplet configuration is driven to a stable equilibrium exhibiting 1 super-hemispherical droplet and $N-1$ sub-hemispherical ones when the initial droplet volumes are large. The selection of this "winning" droplet depends not only on the channel network and the fluid volume, but also notably on the fluid rheology. The rheology is also observed to drastically change the transition to equilibrium. For smaller droplet volumes the long-term behavior is seen to be more complicated since the types of equilibria differ from those arising for larger volumes. These observations motivate our analytical study of equilibria and their stability for the corresponding nonlinear dynamical system. The identification of equilibria is accomplished by locating the zeros of a mass polynomial, defined through the constant volume/mass constraint. The key tool in our stability analysis is a pressure–volume work functional, related to the total surface area, which serves as a Lyapunov function for the dynamical system. This functional is useful since equilibria are typically not hyperbolic and linearization techniques not available. Equilibria will be shown to be hierarchically organized in terms of size of the pressure–volume work functional. For larger droplet volumes this ordering exhibits one hierarchy of equilibria. Two hierarchies exist when the volumes are smaller. The minimizing equilibria in either case are asymptotically stable.