Effective boundary conditions for the creeping flow of a suspension

While the average equations describing a suspension of particles have been continuously improved from year to year, it seems that no such progress has been made concerning the relevant boundary conditions. We here focus on the suspension slip velocity, and show how it can be deduced either from the momentum balance of the fluid phase, or from the momentum conservation of the whole suspension. An important role is played by the modifications of the particle distribution close to the boundaries. We show how to depict that parietal microstructure, using either the number density of particles, or the density of interfaces, or the particles volume fraction. Numerical results have been obtained long ago for dilute suspensions and we collect these pioneering results together with more recent ones obtained for shear flows and sedimentation flows. Experimental results for the slip velocity in non-dilute suspensions are scarce but they agree with one of the theoretical expressions derived here. Key Words: suspensions of rigid particles, boundary conditions, slip velocity, microstructure close to boundaries.