The Cauchy-Dirichlet Problem for the Finitely Extensible Nonlinear Elastic Dumbbell Model of Polymeric Fluids

The finitely extensible nonlinear elastic (FENE) dumbbell model consists of the incompressible Navier–Stokes equation for the solvent and the Fokker–Planck equation for the polymer distribution. In such a model, the polymer elongation cannot exceed a limit m0 which yields all interesting features of solutions near this limit. This work is concerned with the sharpness of boundary conditions in terms of the dimensionless parameter b = Hm0 kBT . Through a careful analysis of the Fokker–Planck operator coupled with the Navier–Stokes equation, we establish a local well-posedness for the full coupled FENE dumbbell model under a class of Dirichlet-type boundary conditions dictated by the parameter b. For each b > 0 we identify a sharp boundary requirement for the underlying density distribution, while the sharpness follows from the existence result for each specification of the boundary behavior. It is shown that the probability density governed by the Fokker–Planck equation approaches zero near the boundary, necessarily faster than the distance function d for b > 2, faster than d|lnd| for b = 2, and as fast as db/2 for 0 < b < 2. Moreover, the sharp boundary requirement for b ≥ 2 is also sufficient for the distribution to be a probability density.