Boundary Conditions for the Microscopic FENE Models
Chun Liu, Hailiang Liu · SIAM Journal on Applied Mathematics · 2008
We consider the microscopic equation of finite extensible nonlinear elasticity (FENE) models for polymeric fluids under a steady flow field. It is shown that for the underlying Fokker–Planck type of equations, any preassigned distribution on the boundary will become redundant once the nondimensional number $\text{{\it Li\/}} := \frac{Hb}{k_BT} \geq 2$, where H is the elasticity constant, $\sqrt{b}$ is the maximum dumbbell extension, T is the temperature, and $k_B$ is the usual Boltzmann constant. Moreover, if the probability density function is regular enough for its trace to be defined on the sphere $|m| = \sqrt{b}$, then the trace is necessarily zero when $\text{{\it Li\/}} > 2$. These results are consistent with our numerical simulations as well as some recent well-posedness results by preassuming a zero boundary distribution.