A thermodynamically consistent Johnson-Segalman-Giesekus model: numerical simulation of the rod climbing effect

TL;DR

This paper introduces a thermodynamically consistent Johnson-Segalman-Giesekus model for viscoelastic fluids, demonstrating its accuracy in simulating the rod climbing effect and providing an open-source finite-element implementation.

Contribution

A novel thermodynamically consistent variant of the Johnson--Segalman model is developed, validated against experimental data, and made available as open-source software.

Findings

The new model captures experimental rod climbing data more accurately.

The standard Johnson--Segalman model is incompatible with the second law of thermodynamics.

The implementation is robust, efficient, and available on GitHub.

Abstract

Viscoelastic rate-type fluids represent a popular class of non-Newtonian fluid models due to their ability to describe phenomena such as stress relaxation, non-linear creep, and normal stress differences. The presence of normal stress differences in a simple shear flow gives rise to forces acting in directions orthogonal to the primary flow direction. The rod climbing effect, i.e. the rise of a fluid along a rod rotating about its axis, is associated with this phenomenon. Within the class of viscoelastic rate-type fluids that includes the Oldroyd-B and Giesekus models with Gordon--Schowalter convected derivatives, we show -- by means of thermodynamical analysis and numerical simulations -- that a thermodynamically consistent variant of the Johnson--Segalman model captures experimental data exceedingly well and is therefore superior to other models in this class, including the standard Johnson--Segalman model, which is widely used in engineering applications but is shown here to be incompatible with the second law of thermodynamics. We release a robust and computationally efficient higher-order finite-element implementation as open-source software on GitHub. The implementation is based on an arbitrary Lagrangian--Eulerian (ALE) formulation of the governing equations and is developed using the Firedrake library.