Fundamentals of the Oldroyd-B model revisited: Tensorial vs. vectorial theory

TL;DR

This paper revisits the Oldroyd-B model, proposing a simpler closure based on the first moment, which maintains physical consistency and may reduce numerical instabilities at high flow rates.

Contribution

It introduces an alternative, simpler closure for the Oldroyd-B model based on the first moment, ensuring thermodynamic consistency and potentially alleviating high Weissenberg number problems.

Findings

Both closures are physically sound and thermodynamically consistent.

The new model has regular free energy and dissipation at zero conformation tensor.

Potential to reduce numerical instabilities at large flow rates.

Abstract

The standard derivation of the Oldroyd-B model starts from a coupled system of the momentum equation for the macroscopic flow on the one hand, and Fokker-Planck dynamics for molecular dumbbells on the other. The constitutive equation is then derived via a closure based upon the second moment of the end-to-end vector distribution. We here present an alternative closure that is rather based upon the first moment, and gives rise to an even simpler constitutive equation. We establish that both closures are physically sound, since both can be derived from (different) well-defined non-equilibrium ensembles, and both are consistent with the Second Law of thermodynamics. In contrast to the standard model, the new model has a free energy and a dissipation rate that are both regular at vanishing conformation tensor. We speculate that this might perhaps alleviate the well-known high Weissenberg number problem, i. e. severe numerical instabilities of the standard model at large flow rates. As the new model permits a trivial solution (vanishing conformation tensor, vanishing polymer stress), an extension may be needed, which includes Langevin noise in order to model thermal fluctuations.