Continuum model for linked fibers with alignment interactions

TL;DR

This paper develops a continuum model for linked fibers with alignment interactions, deriving a kinetic model and analyzing its diffusion limit to obtain nonlinear diffusion equations describing fiber density and orientation.

Contribution

It introduces a novel individual-based fiber model with linking/unlinking and alignment, deriving a macroscopic nonlinear diffusion system from the kinetic framework.

Findings

The macroscopic model is elliptic in the case of homogeneous fiber density.

The kinetic model effectively captures fiber linking, unlinking, and alignment behaviors.

The diffusion limit provides a simplified yet accurate description of fiber dynamics.

Abstract

We introduce an individual-based model for fiber elements having the ability to cross-link or unlink each other and to align with each other at the cross links. We first formally derive a kinetic model for the fiber and cross-links distribution functions. We then consider the fast linking/unlinking regime in which the model can be reduced to the fiber distribution function only and investigate its diffusion limit. The resulting macroscopic model consists of a system of nonlinear diffusion equations for the fiber density and mean orientation. In the case of a homogeneous fiber density, we show that the model is elliptic.