A constitutive equation for droplet distribution in unidirectional flows of dilute emulsions for low capillary numbers

Abstract

The concentration distribution of droplets in the unidirectional flow of an emulsion for small capillary numbers (Ca) can be written as a balance between the drift flux arising from droplet deformation and the flux due to shear induced migration. The droplet drift flux is modeled using the O(Ca) theoretical results of Chan and Leal [J. Fluid Mech. 92, 131 (1979)], while the flux due to shear-induced migration is modeled using the suspension balance approach of Nott and Brady [J. Fluid Mech. 275, 157 (1994)], whereby particle migration is ascribed to normal stress gradients in the flowing dilute emulsion. In the limit of vanishingly small capillary numbers, the leading order contribution of the normal stresses in dilute emulsions arises from droplet-droplet interaction and thus scales as ϕ2τ, where ϕ is the droplet volume fraction and τ is the local shear stress. In our model, the normal stress calculations of Zinchenko [Prikl. Mat. Mekh. 47, 56 (1984)] are connected to our gradient diffusivity data computed from droplet trajectories [M. Loewenberg and E. J. Hinch, J. Fluid Mech. 338, 299 (1997)] via a reduced droplet mobility to derive the droplet flux due to shear-induced migration. As an example, the model is applied to the tube Poiseuille flow of a dilute emulsion at small Ca. It is demonstrated that the unsteady concentration distribution of droplets resulting from arbitrary time-dependent average velocity obeys a self-similar solution, provided the thickness of the droplet-depleted region near the walls is always nonzero.