Ellipsoidal model for deformable drops and application to non-Newtonian emulsion flow

In honor of the extraordinary career of Professor Andreas Acrivos, we are pleased to present a study which combines his pioneering work on dilute emulsions of drops having small, ellipsoidal deformations with boundary-integral methods for single drops having large deformations, to model dilute emulsion flows of drops with large deformations. Each drop is approximated as an ellipsoid, with its orientation and the lengths of its axes varying due to its time history in a locally linear external flow field. The ordinary differential equations governing the time dependence of the coefficients which describe the ellipsoidal shape 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 the normal velocities at a small number of points on the drop interface. For single drops in linear shear and extensional flows, very good agreement in the dynamic drop shapes predicted by the ellipsoidal approximation and the full (exact) boundary-integral calculations is obtained. The utility of the ellipsoidal model is demonstrated by the example application of flow of a dilute emulsion of many deformable drops past a rigid sphere. Although the fluids internal and external to the drops are both Newtonian, the resultant behavior for deformable drops is weakly non-Newtonian. The drag force at small Reynolds number decreases slightly with increasing deformation, and the streamlines are perturbed from the Stoke’s solution and lose fore-and-aft symmetry due to drop deformation.