Faxén-like relations for a non-uniform suspension

The first part of the paper shows how ensemble averages that correspond to a prescribed statistically nonuniform spatial distribution of particles can be evaluated starting from a statistically uniform ensemble. The method consists of attributing to each realization of the uniform ensemble a suitable weight which is explicitly constructed. As an application of this general procedure, in the second part of the paper, the behavior of particles subjected to force or torque in a statistically nonuniform suspension and the behavior of a suspension subjected to a uniform shear are studied. In particular, it is shown how the average translational and angular velocities of the particles with respect to the mixture satisfy Faxén-like relations. Furthermore, it is pointed out that several quantities which vanish in an identical way in the case of a uniform suspension are nonzero in the presence of spatial nonuniformities.