One-dimensional reduction of viscous jets. I. Theory

TL;DR

This paper develops a comprehensive tensor-based formalism to model thin viscous jets as one-dimensional objects, including first-order corrections for curvature and elliptic sections, extending classical models with new effects and corrections.

Contribution

It introduces a tensor algebra approach to derive a detailed one-dimensional model of viscous jets, including first-order curvature corrections and elliptic section effects, which were not previously fully characterized.

Findings

First-order corrections lead to elliptic sections in curved jets.

The model recovers standard axisymmetric results and extends them to include curvature effects.

A missing term in viscous torque expressions is identified and corrected.

Abstract

We build a general formalism to describe thin viscous jets as one-dimensional objects with an internal structure. We present in full generality the steps needed to describe the viscous jets around their central line, and we argue that the Taylor expansion of all fields around that line is conveniently expressed in terms of symmetric trace-free tensors living in the two dimensions of the fiber sections. We recover the standard results of axisymmetric jets and we report the first and second corrections to the lowest order description, also allowing for a rotational component around the axis of symmetry. When applied to generally curved fibers, the lowest order description corresponds to a viscous string model with circular sections. However, when including the first corrections we find that curved jets generically develop elliptic sections. Several subtle effects imply that the first corrections cannot be described by a rod model, since it amounts to selectively discard some corrections. For completeness, we also recover the constitutive relations for forces and torques in rod models and exhibit a missing term in the lowest order expression of viscous torque. Given that our method is based on tensors, the complexity of all computations has been beaten down by using an appropriate tensor algebra package such as {\it xAct}, allowing us to obtain a one-dimensional description of curved viscous jets with all the first order corrections consistently included. Finally, we find a description for straight fibers with elliptic sections as a special case of these results, and recover that ellipticity is dynamically damped by surface tension. An application to toroidal viscous fibers is presented in the companion paper [Pitrou, Phys. Rev. E 97, 043116 (2018)].