A numerical study is presented of the motion of two-dimensional, doubly periodic, dilute and concentrated emulsions of liquid drops with constant surface tension, subject to a simple shear flow. The numerical method is based on a boundary integral formulation that employs a Green's function for doubly periodic Stokes flow, computed using the Ewald summation method. Under the assumption that the viscosity of the drops is equal to that of the ambient fluid, the motion is examined in a broad range of capillary numbers, volume fractions, and initial geometrical configurations. The latter include square and hexagonal lattices of circular and closely packed hexagonal drops with rounded corners. Based on the nature of the asymptotic motion at large times, a phase diagram is constructed separating regions where periodic motion is established, or the emulsion is destabilized due to continued elongation or coalescence of intercepting drops. Comparisons with previous computations for bounded systems illustrate the significance of the walls on the evolution and rheological properties of an emulsion. It is shown that the shearing flow is able to stabilize a concentrated emulsion against the tendency of the drops to become circular and coalesce, thereby allowing for periodic evolution even when the volume fraction of the suspended phase might be close to that for dry foam. This suggests that the imposed shearing flow plays a role similar to that of the disjoining pressure for stationary foam. At high volume fractions, the geometry of the microstructure and flow at the Plateau borders and within the thin films separating adjacent drops are illustrated and discussed with reference to the predictions of the quasi-steady theory of foam. Although the accuracy of certain fundamental assumptions underlying the quasi-steady theory is not confirmed by the numerical results, we find qualitative agreement regarding the basic geometrical features of the evolving microstructure and effective rheological properties of the emulsion.
Simulations of Coulomb systems with slab geometry using an efficient 3D Ewald summation method