Leonid Beryland, Department of Mathematics, Penn State
"Effective Properties of Fluid/Solid Composites"
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