Simple shear flow of suspensions of liquid drops

The shearing motion of monodisperse suspensions of two-dimensional deformable liquid drops with uniform interfacial tension is studied by means of numerical simulations. In the theoretical model, the drops are distributed randomly within a square that is repeated periodically in two directions yielding a doubly periodic flow. Under the assumption that inertial effects are negligible and the viscosity of the drops is equal to that of the suspending fluid, the motion is investigated as a function of the area fraction of the suspended drops and of the capillary number. The evolution of the suspension from an initial configuration with randomly distributed circular drops is computed using an improved implementation of the method of interfacial dynamics which is based on the standard boundary integral formulation for Stokes flow. The numerical procedure incorporates the method of multipole expansions to account for far-drop interactions, and interpolation through tables for computing the doubly periodic Green's function; the latter allows considerable savings in the cost of the computations. Dynamic simulations are carried out for suspensions with up to 49 drops within each periodic cell, for an extended period of time up to kt = 60, where k is the shear rate. Comparisons with previous numerical results for solid particles reveal that particle deformability and interfacial mobility play an important role in the character of the motion. The effects of particle area fraction and capillary number on the effective rheological properties of the suspension are discussed, and the statistics of the drop motion is analysed with reference to the drop-centre pair distribution function and probability density functions of drop aspect ratio and inclination. It is found that the effective rheo-logical properties may be predicted with remarkable accuracy from a knowledge of the instantaneous mean drop deformation and orientation alone, even at high area fractions. Cluster formation is not as important as in suspension of solid particles. The apparent random motion of the individual drops, when viewed at a sequence of time intervals that are large compared to the inverse shear rate, is described in terms of an effective non-isotropic long-time diffusivity tensor, and the transverse component of this tensor is computed from the results of the simulations with some uncertainty.