General Ellipsoidal Model for Deformable Drops in Viscous Flows

Abstract

In honor of Professor Bill Schowalter, we present a study which builds on his pioneering work on emulsion flows of deformable drops. Each drop is approximated as an ellipsoid, with its orientation and the lengths of its axes varying because of its time history in an imposed flow. The ordinary differential equations governing the time dependence of the ellipsoidal shape at small Reynolds number are derived based on volume conservation and a least-squares minimization of the difference between the ellipsoidal approximation and the exact boundary-integral description of normal velocities over the entire drop surface. Two-dimensional tables for interpolation of the coefficients in the resulting linear system are introduced to avoid solving the boundary-integral equation at each step, thereby greatly reducing the calculation expense for describing the drop behavior, from O(N2) (where N is the number of triangular elements for the discretized drop surface) to O(1). For single drops in steady shear, time-dependent shear, and extensional flows, excellent agreement in the dynamic drop shapes and stresses predicted by the ellipsoidal approximation and the exact boundary-integral calculations for subcritical deformation is obtained. Viscometric functions, describing the effective viscosity and normal stress differences, are presented for steady shear flow of semidilute emulsions with a wide range of drop-to-medium viscosity ratios and capillary numbers.