Chaotic streamlines in a translating drop with a uniform electric field

Abstract

A drop translating in the presence of a uniform electric field is studied both theoretically and experimentally to determine qualitative properties of three-dimensional chaotic particle trajectories and mixing in bounded Stokes flows. The flow is a combination of a Hadamard–Rybczynski and a Taylor circulation due to the translation and electric field respectively. The three-dimensional trajectories are generated by tilting the electric field relative to the drop translational motion by an angle $\alpha$. The numerical analysis includes qualitative analysis of the degree of mixing by Poincaré mapping, quantitative estimates of the largest mixed volume fraction and the rate of mixing characterized by the largest Lyapunov exponent. Experiments are performed using a castor oil/silicone oil system for the continuous and dispersed phases respectively. Single trajectories are studied by visualizing small neutrally buoyant glass particles inside the dispersed phase using a stereoscopic particle tracking technique. Drops are approximately 5 mm in diameter, settling velocities are $O$(0.1 mm s$^{-1}$) and the electric fields are $O$(10 V mm$^{-1}$). We observe crossings of the unperturbed separatrix and particle trajectories that show evidence of a symmetry plane, both important features of the theory.