Droplet size scaling of water-in-oil emulsions under turbulent flow

The size of droplets in emulsions is important in many industrial, biological, and environmental systems, as it determines the stability, rheology, and area available in the emulsion for physical or chemical processes that occur at the interface. While the balance of fluid inertia and surface tension in determining droplet size under turbulent mixing in the inertial subrange has been well established, the classical scaling prediction by Shinnar half a century ago of the dependence of droplet size on the viscosity of the continuous phase in the viscous subrange has not been clearly validated in experiment. By employing extremely stable suspensions of highly viscous oils as the continuous phase and using a particle video microscope (PVM) probe and a focused beam reflectance method (FBRM) probe, we report measurements spanning 2 orders of magnitude in the continuous phase viscosity for the size of droplets in water-in-oil emulsions. The wide range in measurements allowed identification of a scaling regime of droplet size proportional to the inverse square root of the viscosity, consistent with the viscous subrange theory of Shinnar. A single curve for droplet size based on the Reynolds and Weber numbers is shown to accurately predict droplet size for a range of shear rates, mixing geometries, interfacial tensions, and viscosities. Viscous subrange control of droplet size is shown to be important for high viscous shear stresses, i.e., very high shear rates, as is desirable or found in many industrial or natural processes, or very high viscosities, as is the case in the present study.