Chaotic streamlines in steady bounded three-dimensional Stokes flows

Accurate control of hyperbolic trajectories in any dimension

“…(10), (12), and (13). This we would need to have available either through an appropriate model (such as Hill's spherical vortex [27,31,32,85] or the Hadamard-Rybczynski solution [35,36,86] if examining hyperbolic trajectories at the poles of droplets moving in an anomalous fluid), as is assumed in many fluid mechanical studies [27,31,31,35,36,85]. An alternative in a purely experimental flow would be to obtain the uncontrolled velocities using PIV measurements, and then use these to impute the higher derivatives required for using our control velocity formulas by a numerical differentiation process.…”

“…A common spherical configuration is the Hadamard-Rybczynski solution for Stokes flows [31,32,35,36,66], whose kinematic structure is similar to the classical Hill's spherical vortex [75][76][77][78][79][80][81][82][83][84][85] for Euler flows with an additive solid rotation. Particle trajectories of this flow satisfẏ…”

“…where r 2 = x 2 + y 2 + z 2 and ω z is a constant [36]. Motion is confined to level sets of the Stokes stream function…”

“…For droplets traveling within an unsteady flow (such as nano-or microdroplets currently under intensive investigation for their promise in processing single cells [25]), such elongation can be achieved by positioning the droplet at a hyperbolic trajectory location, whose control is therefore desirable. While the hyperbolic trajectory described above is exterior to the droplet, many droplet models in the literature themselves possess on the droplet's surface a saddle-like hyperbolic trajectory [26][27][28][29][30][31][32][33][34][35][36] whose motion influences intradroplet chaotic transport because its attached stable and unstable manifolds undergo nontrivial motion. There is currently only one method in the scientific literature which can control manifold paths [37], but it is limited to two-dimensional nearly steady flows with one-dimensional stable and unstable manifolds.…”

“…IV we apply the control method to the fluid dynamical example of a "two-cell" droplet [31,32,35,36,66], which is the Hadamard-Rybczynski solution to a three-dimensional Stokes flow problem. This model has importance in the quest for improving mixing within microdroplets [31,32,35,36,66], which can be achieved by breaking the invariant manifolds and causing them to intersect in complicated ways. We show how we can make the hyperbolic trajectories at the north and south poles to move independently, according to our specifications which are designed to make their stable and unstable manifolds mingle to achieve good intradroplet transport.…”