A discrete geometric approach for simulating the dynamics of thin viscous threads

TL;DR

This paper introduces a discrete geometric numerical model for simulating the complex dynamics of thin viscous threads, capturing effects like inertia, bending, twisting, and surface tension with high robustness and efficiency.

Contribution

It develops a novel discrete Lagrangian model based on a centerline/spin representation, incorporating viscous stress and ensuring consistency with classical equations.

Findings

Validated against steady coiling solutions

Enables simulation of unsteady viscous jet behavior

Handles large rotations and surface tension effectively

Abstract

We present a numerical model for the dynamics of thin viscous threads based on a discrete, Lagrangian formulation of the smooth equations. The model makes use of a condensed set of coordinates, called the centerline/spin representation: the kinematical constraints linking the centerline's tangent to the orientation of the material frame is used to eliminate two out of three degrees of freedom associated with rotations. Based on a description of twist inspired from discrete differential geometry and from variational principles, we build a full-fledged discrete viscous thread model, which includes in particular a discrete representation of the internal viscous stress. Consistency of the discrete model with the classical, smooth equations is established formally in the limit of a vanishing discretization length. The discrete models lends itself naturally to numerical implementation. Our numerical method is validated against reference solutions for steady coiling. The method makes it possible to simulate the unsteady behavior of thin viscous jets in a robust and efficient way, including the combined effects of inertia, stretching, bending, twisting, large rotations and surface tension.