Multiscale simulations for suspensions of rod-like molecules

We study the Doi model for suspensions of rod–like molecules. The Doi model couples a microscopic Fokker–Planck type equation (Smoluchowski equation) to the macroscopic Stokes equation. The Smoluchowski equation describes the evolution of the distribution of the rod orientations; it comes as a drift–diffusion equation on the sphere at every point in physical space. For sufficiently high macroscopic shear rates (high Deborah numbers), the solution of the coupled system develops internal layers in the macroscopic strain rate (the spurt phenomenon). In the high Deborah numbers regime, the drift term in the Smoluchowski equation is dominant. We thus introduce a finite–volume type discretization of the microscopic Smoluchowski equation which is motivated by transport dominated PDEs. We carry out direct numerical simulations of the spurt phenomenon both in the dilute and concentrated regimes. Below the isotropic–nematic transition, the solution structure is identical to the one described by purely macroscopic models (JSO model). For higher concentrations, we observe the formation of microstructure coming from a position–dependent tumbling rate. We also investigate the 2–d stability of the spurted solution.