Mathematical modelling of fibre coating

TL;DR

This paper develops a mathematical model for the evolution of a thin liquid film coating a wire in extrusion, deriving an approximate analytical solution and validating it against numerical results.

Contribution

It introduces a simplified third-order differential equation model for the coating process and provides an analytical solution validated by numerical comparison.

Findings

Approximate analytical solution matches numerical results within 5% error.

Key control parameters include initial film height, surface tension, viscosity, wire velocity, and exit angle.

Model simplifies complex fluid dynamics to facilitate process understanding.

Abstract

In this report we formulate and analyse a mathematical model describing the evolution of a thin liquid film coating a wire via an extrusion process. We consider the Navier-Stokes equations for a 2D incompressible Newtonian fluid coupled to the standard equation relating the fluid surface tension with the curvature. Taking the lubrication theory approximation and assuming steady state, the problem is reduced to a single third-order differential equation for the thin film height. An approximate analytical solution for the final film height is derived and compared with a numerical solution obtained by means of a shooting scheme. Good agreement between the two solutions is obtained, resulting in a relative error of around 5\%. The approximate solution reveals that the key control parameters for the process are the initial film height, the fluid surface tension and viscosity, the wire velocity and the angle of exit at the extruder.