A Constitutive Equation for the Droplet Distribution in the Unidirectional Flow of a Dilute Emulsion of Deformable Drops Suspended in a Second Order Fluid

This work elucidates, for the first time, the first effect of the non-Newtonian behaviors of the fluids in an emulsion on the distribution of droplets in a unidirectional flow, by employing (in part) the classic single-droplet-drift work of Gary from the late 70s. The limit considered in the work is (Ca,λ)<<φ<<1, where Ca is the capillary number, λ is the Weissenberg number that compares the intrinsic relaxation time of the fluid to the convective time scale of fluid motion, and φ is the volume fraction of the drops. In this limit, the droplet drift normal to the unidirectional flow arises due to deformation of the drop, and the second-order, non-Newtonian behavior of the drop fluids. The particle flux can then be written as the sum of the drift fluxes due to the above two mechanisms, and the flux due to shear-induced migration. The drift of the droplet is modeled using the theoretical results of Chan and Leal [J. Fluid Mech., 92, 131-170, 1979]. The flux due to shear-induced migration is modeled using the suspension balance approach of Nott and Brady [J. Fluid Mech., 275, 157-199, 1994] , as explained in a prior publication [Ramachandran, Loewenberg and Leighton, Phys. Fluids, 2010]. As an example, the model is applied to the tube Poiseuille flow of a dilute emulsion. It is demonstrated that the unsteady concentration distribution of droplets resulting from arbitrary time-dependent average velocity follows a self-similar solution in certain limits.