The Matovich-Pearson equations revisited


The Matovich-Pearson equations are a well-known asymptotic regime of the axisymmetric Navier-Stokes equations with moving boundary. They arise as the slender body approximation of the full equations for a highly viscous, Newtonian fluid and are used to describe the dynamics and evolution of thin, viscous fluid filaments, e.g. in the context of fiber spinning. While these equations can be systematically derived, additional boundary conditions have to be imposed to make the resulting initial-boundary value problem well-posed. In this work we shall extend existing results and techniques for the Matovich- Pearson equations to the case of a prescribed pulling force at the downstream boundary. We will prove a local well-posedness result and show that global existence of solutions holds true. Even though this latter result relies on earlier techniques developed for the classical initial-boundary value problem of the Matovich-Pearson equations with prescribed take-up velocity, it requires additional arguments to overcome the absence of a priori bounds on the filament velocity. Global existence is, at its core, based on the fundamental long-term behavior of viscous fluids in extension when other factors are omitted: Viscous fluid filaments do not break.

Publication Title

Dynamical Systems, Number Theory and Applications: A Festschrift in Honor of Armin Leutbecher's 80th Birthday