The Cauchy-Dirichlet problem for the FENE dumbbell model of polymeric fluids

TL;DR

This paper establishes local well-posedness for the FENE dumbbell model of polymeric fluids with boundary conditions, analyzing the behavior of the polymer density near the boundary for different parameter regimes.

Contribution

It introduces sharp boundary conditions for the FENE model and characterizes the boundary behavior of the polymer density for various parameter values.

Findings

Density approaches zero faster than d for b>2

Density approaches zero faster than d|ln d| for b=2

Density approaches zero as d^{b/2} for 0<b<2

Abstract

The FENE dumbbell model consists of the incompressible Navier-Stokes equation and the Fokker-Planck equation for the polymer distribution. In such a model, the polymer elongation cannot exceed a limit $\sqrt{b}$, yielding all interesting features near the boundary. In this paper we establish the local well-posedness for the FENE dumbbell model under a class of Dirichlet-type boundary conditions dictated by the parameter $b$. As a result, 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 boundary, necessarily faster than the distance function $d$ for $b>2$, faster than $d|ln d|$ for $b=2$, and as fast as $d^{b/2}$ for $0<b<2$. Moreover, the sharp boundary requirement for $b\geq 2$ is also sufficient for the distribution to remain a probability density.