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.

Figure 1

Side and top views of two trapped-droplet systems embedded in a far-field flow (of uniform velocity ${\mathit{u}}_{\infty }$) and with characteristic length $L$. (a) A droplet ($\tilde{a}=0.6$) trapped between three spheres ($\text{Ca}=0.9$, $\lambda =4$, $\epsilon =0.25$). (b) A droplet ($\tilde{a}=0.5$) trapped between two capsules ($\text{Ca}=0.7$, $\lambda =4$, $\epsilon =0.5$, $\mathcal{L}=6$).

Figure 2

Summary of how an advecting material surface bounded by the drop interface is visualized. At any given time step, we begin with a material surface (yellow) within a droplet volume (transparent grey). The material surface is used to divide the volume into separate parts (one of which shown in opaque grey). Finally, the two parts can be rendered separately to mimic translucent dyes (red and blue).

Figure 3

Cutaways of trapped drops reveal internal dynamics, with velocity magnitude shown in color. (a) A sixth, defined using planes of symmetry, is cut from the three-sphere droplet to reveal two-dimensional streamlines and equilibria (red and blue). (b) A quadrant is cut from the two-capsule droplet, revealing two equilibria (green and orange) on orthogonal symmetry planes.

Figure 4

Internal flow dynamics within a quadrant of the two-capsule droplet revealed by tracers. (a) The interaction between the (1,2) saddle focus (green) and a (2,1) saddle focus (orange) is highlighted by a trajectory (black). In addition, 100 tracers starting near the drop center (placed randomly inside a cube of side length 0.01) are shown in color. See Fig. 3 for location of saddle points with respect to the drop interface. (b) The nonexistence of a heteroclinic orbit between the two saddle foci is shown, with a forward-time trajectory from the green point and reverse-time trajectory from the orange point.

Figure 5

Visualizations of the flow within a quadrant of the two-capsule drop. Axes in (b) apply to entire figure. (a) The Poincaré section for 100 tracers starting near the droplet center, intersecting with the $xy$ plane that contains the drop center (shown in black). Points within the blue section result from the trajectories shown in (b). A KAM torus is shown in orange. (b) Another view of the trajectories shown in Fig. 4. (c) An example of a complex periodic orbit. (d) The orbit of a single tracer initialized near the center of the manifold in (a). Shaded to emphasize position along the $z$ axis. The Poincaré section of this trajectory is shown in orange in (a).

Figure 6

Dynamic behavior within the three-sphere droplet. (a) Intersections with the $xy$ plane containing the droplet center, for 100 tracers initialized near the droplet center, over a duration of $5\times {10}^5$ (shown in black). The Poincaré map of a single tracer initialized outside of the large chaotic sea is also shown in orange, over a trajectory of the same duration. Dotted green lines indicate planes of symmetry. (b) The orbit from frame (b) is shown, colored to indicate position along the $z$ axis. Also shown in light blue is part of a chaotic orbit (over a duration of $2\times {10}^4$). Two orthogonal views shown.

Figure 7

Reverse-time trajectories from the two (2,1) saddle foci located on one symmetry plane of the three-sphere droplet. In both cases, relatively simple orbits terminate at the stagnation point at the top of the drop, indicating a heteroclinic connection.

Figure 8

Advection of a 2D passive dye within the axial symmetry plane of the two-capsule droplet, using the dye interface as a material line. The material line is initialized at the droplet center.

Figure 9

Advection of a passive material surface within the two-capsule droplet (solid particles hidden). The initial surface lies on the $xy$ plane passing through the drop center. Images are created using the procedure described in Fig. 2, which corresponds to $t=25$. At $t=650$, the material surface consists of $\approx 48\text{K}$ nodes ($\approx 95\text{K}$ triangles).

Figure 10

Advection of a passive material surface separating red and blue dyes within the three-sphere droplet (solid particles hidden), where each face of the material surface is colored according to the dye with which it is in contact. The initial interface lies on the $xy$ plane, passing through the drop center. For $t>225$, half the droplet is hidden to reveal the internal dynamics (the full droplet is shown as the smaller image at $t=325$). Visually, this threefold-symmetric system appears considerably more mixing than that of the two-capsule droplet, including the formation of thin layers of each dye by $t=500$ that could be mixed by diffusion at finite Péclet number. At $t=500$, the material surface consists of $\approx 150\text{K}$ nodes ($\approx 300\text{K}$ triangles).

Figure 11

Mixing metrics for the systems shown in Figs. 9 and 10. (a) Growth of the interfacial area as a function of time. The rate of increase is a measure of system complexity. (b) The mixing number, a measure of how quickly a system can be mixed by diffusion, decays faster for the three-sphere droplet, indicating superior mixing properties.