A DEFORMATION TENSOR MODEL OF BROWNIAN SUSPENSIONS OF ORIENTABLE PARTICLES - THE NONLINEAR DYNAMICS OF CLOSURE MODELS

A new model is developed for the evolution of the orientation distribu tion within a flowing Brownian suspension of orientable particles, e.g . fibers, disks, rods, etc. Rather than solving the full Fokker-Planck equation for the orientation distribution function for the suspended phase, and in place of the usual approach of developing a moment closu re model, a new approach is taken in which an evolution equation is de veloped for an approximate, simplified deformation of the orientable p articles associated with a material point. The evolution equation for the remaining degrees of freedom in the assumed class of deformations is developed from the Fokker-Planck equation; it is as quick to integr ate as direct moment tensor evolution equations. Because the deformati on is restricted to a special class, one can show a priori that the mo del always gives physically sensible results, even in complicated flow s of practical interest. The nonlinear dynamics of the model in unstea dy, three-dimensional flows is considered. It is shown that the model predicts bounded deformations in all flows. The model has a unique glo bal attractor in any steady, three -dimensional flow, provided the rot ary Brownian diffusivity is non-zero. In contrast, commonly used momen t closure approximations can only be shown to have a unique global att ractor when the rotary Brownian diffusivity is large enough (i.e. when the flow is weak enough). This may be the explanation of why a supuri ous (multiple) attractor was recently observed at high Peclet number i n uniform shear flow of a dilute suspension modeled using the first co mposite closure of Hinch and Leal, by Chaubal, Leal and Fredrickson (J . Rheol., 39(1) (1995) 73-103).