Warner or "FENE" spring:(2.2)
This spring has a maximum length of . It can describe
many nonlinear rheological properties, but is difficult to
handle analytically. (FENE = finitely extensible nonlinear
Here the ratio in the denominator is averaged at the local
conditions. "P" stands for Anton Peterlin who used a similar
When the spring constant is allowed to go to infinity, the
Fraenkel spring becomes a rigid rod of length L.
Whatever model is chosen, the beads are presumed to be acted on by a Stokes law type of drag force, with a drag coefficient .
2a. The Liouville equation for general bead-spring models
(i.e., models with any connectivity and complexity)