Boundary Conditions for the Microscopic FENE Models

We consider the microscopic equation of FENE (finite extensible nonlinear elasticity) models for polymeric fluids under a steady flow field. This paper will show for the underlying Fokker-Planck type of equations, any pre-assigned distribution on boundary will become redundant once the non-dimensional number Li := Hb kBT ≥ 2, where H is the elasticity constant, √ b is the maximum dumbbell extension, T is the temperature and kB is the usual Boltzmann constant. Moreover, if the probability density function is regular enough for its trace to be defined on the sphere |m| = √ b, then the trace is necessarily zero when Li > 2. These results are consistent with our numerical simulations as well as some recent well-posedness results by pre-assuming a zero boundary distribution.