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

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.