Summary: We study the suspension of force free particles in a viscous incompressible fluid. Under the hypothesis of a locally periodic structure of the suspension at the initial time, the homogenization method furnishes a rigorous deductive procedure for obtaining the macroscopic equations and the bulk constitutive law.
Equations for the bulk medium have homogenized coefficients in the inertial terms and the stress tensor depending on the microstructure and the applied couples. The antisymmetrical part of the macroscopic stress tensor is expressed in terms of the couples acting on particles. The non- Newtonian behavior of the suspension arises from the microstructure evolution, the evolution equations of the microstructure include the bulk velocity and the couples. Limiting case of dilute suspension is studied, when particles are identical spheres the Einstein viscosity and the vortex viscosity formula are found.