Simulation of detailed dynamics of suspended particles in the presence of confinements is extremely complicated. The difficulties are twofold: for successful analysis we have to determine 1) the effect of the conduit on the particle-dynamics, and 2) mutual interactions between the particles.
In this talk, we propose a novel formulation which describes the influence of the bounding surface on the dynamics of the colloidal particles. Though at present, our mathematical theory is applied to cylindrical conduits and spherical particles, it can be generalized to other geometries.
We consider spherical particles in a pressure-driven parabolic flow through a cylindrical duct. The particles are assumed to be driven by a known force-field like gravity. In our analysis, we accurately determine the hydrodynamic interactions between the cylindrical confinement and the spherical particles. As a result, we can describe the precise dynamics of a sphere in the presence of the parabolic flow and the force field.
Next, we consider a suspension of equally sized spheres. We calculate the induced stresses due to these spheres and assuming a uniform population density, obtain a statistically averaged stress-field. As a result, we can relate the pressure-drop in the conduit for a given flow-rate of the suspension and predict its rheological properties in terms of effective viscosity. We present the effective viscosity as a function of the ratio between radii of the spherical particles and the cylindrical conduit. In the limit of very small particles our result matches with Einstein's effective viscosity relation.