#### Abstract

The talk will consist of two parts.

First we will present the results of a joint work with E. Khruslov on
mathematical model of a Newtonian viscous incompressible fluid with
particles which interact due to surface forces (e.g., Van der Waals or
London forces). We show that in the limit when the number of the particles
becomes very large and their diameters become very small, the solution of
such suspension can be described by a single medium, which is an
anisotropic, *non-Newtonian viscoelastic fluid* with memory described
by the integral (relaxation) term. We derive formulas for calculation of
the effective viscosity tensor and the relaxation integral kernel. In the
case of a periodic array of particles we computed the distribution of
relaxation times explicitly. The latter distribution is of particular
interest for the rheological community. We also showed how the shape of
the particles affects this distribution.

In the second part of the talk (if time permits) we will present the
results of joint work with L. Borcea and A. Panchenko, which is in
progress, on developing a vectorial analog of the discrete network
approximation recently suggested in our joint paper with A. Kolpakov
(Arch , Rat. Mech, 2001) for a scalar problem. Our main objective is to
obtain and justify efficient computational formulas for the dependence of
the effective viscosity on the geometrical parameters such as the
interparticle distances and the radii of particles. We will also discuss
the comparison of the contributions to the effective viscosity from the
translational and rotational displacements of particles, which is the key
question specifically pertinent to the vectorial problems.

*TUESDAY, December 4, 2001 *

**Time:** 1:30 P.M.

**Location:** Physical Plant Building 300