A deformation tensor model of Brownian suspensions of orientable particles—the nonlinear dynamics of closure models

Abstract A new model is developed for the evolution of the orientation distribution 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 closure model, a new approach is taken in which an evolution equation is developed for an approximate, simplified deformation of the orientable particles 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 integrate as direct moment tensor evolution equations. Because the deformation is restricted to a special class, one can show a priori that the model always gives physically sensible results, even in complicated flows of practical interest. The nonlinear dynamics of the model in unsteady, three-dimensional flows is considered. It is shown that the model predicts bounded deformations in all flows. The model has a unique global attractor in any steady, three-dimensional flow, provided the rotary Brownian diffusivity is non-zero. In contrast, commonly used moment closure approximations can only be shown to have a unique global attractor when the rotary Brownian diffusivity is large enough (i.e. when the flow is weak enough). This may be the explanation of why a supurious (multiple) attractor was recently observed at high Peclet number in uniform shear flow of a dilute suspension modeled using the first composite closure of Hinch and Leal, by Chaubal, Leal and Fredrickson (J. Rheol., 39(1) (1995) 73–103).