Global-in-time solutions for the isothermal Matovich-Pearson equations

TL;DR

This paper proves the first global existence and uniqueness of solutions for the isothermal Matovich-Pearson equations, modeling glass fiber drawing, by establishing local solutions and ensuring the fluid radius remains positive.

Contribution

It provides the first global existence and uniqueness results for the nonlinear coupled elliptic-hyperbolic system modeling glass fiber drawing.

Findings

Existence of local solutions for the system.

Fluid radius remains strictly positive.

Global existence and uniqueness proved.

Abstract

In this paper we study the Matovich-Pearson equations describing the process of glass fiber drawing. These equations may be viewed as a 1D-reduction of the incompressible Navier-Stokes equations including free boundary, valid for the drawing of a long and thin glass fiber. We concentrate on the isothermal case without surface tension. Then the Matovich-Pearson equations represent a nonlinearly coupled system of an elliptic equation for the axial velocity and a hyperbolic transport equation for the fluid cross-sectional area. We first prove existence of a local solution, and, after constructing appropriate barrier functions, we deduce that the fluid radius is always strictly positive and that the local solution remains in the same regularity class. To the best of our knowledge, this is the first global existence and uniqueness result for this important system of equations.