Tailored mixing inside individual droplets could be useful to ensure that reactions within microscopic discrete fluid volumes, which are used as microreactors in "digital microfluidic" applications, take place in a controlled fashion. In this paper we consider a translating spherical liquid drop to which we impose a time periodic rigid-body rotation. Such a rotation not only induces mixing via chaotic advection, which operates through the stretching and folding of material lines, but also offers the possibility of tuning the mixing by controlling the location and size of the mixing region. Tuned mixing is achieved by judiciously adjusting the amplitude and frequency of the rotation, which are determined by using a resonance condition and following the evolution of adiabatic invariants. As the size of the mixing region is increased, complete mixing within the drop is obtained.

The ability to generate complete, or almost complete, chaotic mixing is of great interest in numerous applications, particularly for microfluidics. For this purpose, we propose a strategy that allows us to quickly target the parameter values at which complete mixing occurs. The technique is applied to a time periodic, two-dimensional electro-osmotic flow with spatially and temporally varying Helmoltz-Smoluchowski slip boundary conditions. The strategy consists of following the linear stability of some key periodic pathlines in parameter space (i.e., amplitude and frequency of the forcing), particularly through the bifurcation points at which such pathlines become unstable.

Enhancing and controlling chaotic advection or chaotic mixing within liquid droplets is crucial for a variety of applications including digital microfluidic devices which use microscopic "discrete" fluid volumes (droplets) as microreactors. In this work, we consider the Stokes flow of a translating spherical liquid droplet which we perturb by imposing a time-periodic rigid-body rotation. Using the tools of dynamical systems, we have shown in previous work that the rotation not only leads to one or more three-dimensional chaotic mixing regions, in which mixing occurs through the stretching and folding of material lines, but also offers the possibility of controlling both the size and the location of chaotic mixing within the drop. Such a control was achieved through appropriate tuning of the amplitude and frequency of the rotation in order to use resonances between the natural frequencies of the system and those of the external forcing. In this paper, we study the influence of the orientation of the rotation axis on the chaotic mixing zones as a third parameter, as well as propose an experimental set up to implement the techniques discussed.

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Periodic forcing of nonlinear oscillators generates a rich and complex variety of behaviors, ranging from regular to chaotic behavior. In this work we seek to control, i.e., either suppress or generate, the chaotic behavior of a classical reference example in books or introductory articles, the Duffing oscillator. For this purpose, we propose an elegant strategy consisting of simply adjusting the shape of the time-dependent forcing. The efficiency of the proposed strategy is shown analytically, numerically and experimentally. In addition due to its simplicity and low cost such a work could easily be turned into an excellent teaching tool.