Tunable turbulence in driven microscale emulsions

We present a tunable, non-equilibrium oil-in-oil emulsion that serves as a model system for investigating the transition from controlled droplet deformation to multiscale flows reminiscent of turbulence. By utilizing a miscible mixture of silicone and motor oils as the continuous phase and the immiscible castor oil as the droplet phase, we isolate electrical conductivity as a single experimental control parameter, varying it by over two orders of magnitude while keeping viscosity and permittivity nearly constant. This high degree of control allows us to systematically traverse the electrohydrodynamic (EHD) phase diagram with dielectric constant and conductivity as control parameters. We validate small-deformation theory at low fields before driving the system into a regime of multiscale, unsteady flows at high fields. We employ three complementary approaches on the same system (particle image velocimetry (PIV), used to map velocity fields, and rheometry and differential dynamic microscopy (DDM), two techniques used to probe viscosity and diffusion) to quantify the emergence of scale invariance in the energy spectra with increasing field strength. Above a threshold field, we find that the spatio-temporal energy spectra obtained by PIV analysis of droplet dynamics display power-law scaling, $E(k) \sim k^{-\alpha_k}$, where $\alpha_k$ approaches the inertial turbulence exponent of $5/3$ at high fields. Energy spectra from rheometry also yield a power law, $S(\nu) \sim \nu^{-\alpha_\nu}$, with $\alpha_\nu = 5/3$ at high fields. Mean square displacement (MSD) analyses on the same datasets reveal super-diffusive behavior, $\mathrm{MSD} \sim t^{\gamma}$, with $\gamma = 3/2$. These observations provide strong evidence of a conductivity-tunable transition to EHD-driven turbulence in a microscale emulsion.