Internal circulation and mixing within tight-squeezing deformable droplets.

The internal flow and mixing properties inside deformable droplets, after reaching the steady state within two types of passive droplet traps, are visualized and analyzed as dynamical systems. The first droplet trap (constriction) is formed by three spheres arranged in an equilateral triangle, while the second consists of two parallel spherocylinders (capsules). The systems are assumed to be embedded in a uniform far-field flow at low Reynolds number, and the steady shapes and interfacial velocities on the drops are generated using the boundary-integral method. The internal velocity field is recovered by solving the internal Dirichlet problem, also via a desingularized boundary-integral method. Calculation of 2D streamlines within planes of symmetry reveals the internal equilibria of the flow. The type of each equilibrium is classified in 3D and their interactions probed using passive tracers and their Poincaré maps. For the two-capsule droplet, saddle points located on orthogonal symmetry planes influence the regular flow within the drop. For the three-sphere droplet, large regions of chaos are observed, embedded with simple periodic orbits. Flow is visualized via passive dyes, using material lines and surfaces. In 2D, solely the interface between two passive interior fluids is advected using an adaptive number of linked tracer particles. The reduction in dimension decreases the number of required tracer points, and also resolves arbitrarily thin filaments, in contrast to backward cell-mapping methods. In 3D, the advection of a material surface, bounded by the droplet interface, is enabled using an adaptive mesh scheme. Off-lattice 3D contour advection allows for highly resolved visualizations of the internal flow and quantification of the associated degree of mixing. Analysis of the time-dependent growth of material surfaces and 3D mixing numbers suggests the three-sphere droplet exhibits superior mixing properties compared to the two-capsule droplet.