Finite volume approach for the instationary Cosserat rod model describing the spinning of viscous jets

TL;DR

This paper introduces a finite volume numerical method for simulating the dynamic behavior of viscous jets using a Cosserat rod model, enabling analysis across a broad parameter range with free or fixed jet ends.

Contribution

It presents the first instationary simulations of a Cosserat rod in rotational spinning, addressing numerical challenges posed by the slenderness ratio and extending applicability.

Findings

Successful simulation of viscous jet spinning in various parameter regimes.

Implementation of a finite volume scheme with mixed difference methods.

Use of Radau methods for stable time integration.

Abstract

The spinning of slender viscous jets can be described asymptotically by one-dimensional models that consist of systems of partial and ordinary differential equations. Whereas the well-established string models possess only solutions for certain choices of parameters and set-ups, the more sophisticated rod model that can be considered as $\epsilon$-regularized string is generally applicable. But containing the slenderness ratio $\epsilon$ explicitely in the equations complicates the numerical treatment. In this paper we present the first instationary simulations of a rod in a rotational spinning process for arbitrary parameter ranges with free and fixed jet end, for which the hitherto investigations longed. So we close an existing gap in literature. The numerics is based on a finite volume approach with mixed central, up- and down-winded differences, the time integration is performed by stiff accurate Radau methods.