Field-assisted self-assembly, motion, and manipulation of droplets have gained much attention in the past decades. We exhibit an electric field manipulation of the motion of a liquid metal (mercury) droplet submerged in a conductive liquid medium (a solution of sulfuric acid). A mercury droplet moves toward the cathode and its path selection is always given by the steepest descent of the local electric field potential. Utilizing this unique behavior, we present several examples of droplet motions, including maze solving, electro-levitation, and motion on a diverted path between parallel electrodes by controlling the conductivity of the medium. We also present an experimental demonstration of Fermat's principle in a non-optical system, namely a mercury droplet moving along a refracted path between electrodes in a domain having two different conductivities.

We develop a numerical method for realizing mean curvature motion of interfaces separating multiple phases, whose areas are preserved throughout time. The foundation of the method is a thresholding algorithm of the Bence-Merriman-Osher type. The original algorithm is reformulated in a vector setting, which allows for a natural inclusion of constraints, even in the multiphase case. Moreover, a new method for overcoming the inaccuracy of thresholding methods on non-adaptive grids is designed, since this inaccuracy becomes especially prominent in area-preserving motions. Formal analysis of the method and numerical tests are presented.

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We investigated self-propelled motions of thin filaments atop water, where we focused on understanding pendulum-type oscillations and synchronization. The filaments were produced from a commercial adhesive (consisting mainly of nitrocellulose and acetone), and exhibited deformable motions. One end of each filament was held on the edge of a quadrangular water chamber while the other was left free. Acetone and other organic molecules from the nitrocellulose filament develop on the water surface and decrease the surface tension. The difference in the surface tension around the filament becomes the driving force of the self-propelled motions. When a single filament was placed in the water chamber, a pendulum-type oscillation in the deformation of the filament was observed. When two filaments were placed in parallel in the chamber, in-phase, out-of-phase, and no-synchronization motions were observed. It was found that the class of motions depends on the distance between the two fixed points of the filaments. Mathematical modeling and numerical simulations were also used in order to further understand the dynamics of the surface active molecules and the filament motions. We propose a mathematical model equation and reproduce various behaviors exhibited by soft self-propelled matters through numerical simulation.

Turing instability is a general and straightforward mechanism of pattern formation in reaction–diffusion systems, and its relevance has been demonstrated in different biological phenomena. Still, there are many open questions, especially on the robustness of the Turing mechanism. Robust patterns must survive some variation in the environmental conditions. Experiments on pattern formation using chemical systems have shown many reaction–diffusion patterns and serve as relatively simple test tools to study general aspects of these phenomena. Here, we present a study of sinusoidal variation of the input feed concentrations on chemical Turing patterns. Our experimental, numerical and theoretical analysis demonstrates that patterns may appear even at significant amplitude variation of the input feed concentrations. Furthermore, using time-dependent feeding opens a way to control pattern formation. The patterns settled at constant feed may disappear, or new patterns may appear from a homogeneous steady state due to the periodic forcing.