Fully-implicit log-conformation formulation of constitutive laws

TL;DR

This paper introduces a new fully-implicit log-conformation formulation for constitutive laws in viscoelastic fluid models, enabling more efficient and stable numerical simulations while maintaining accuracy.

Contribution

It derives a self-contained, non-iterative set of equations for constitutive laws using the log-conformation approach, improving computational efficiency and stability.

Findings

Comparable accuracy to existing methods in benchmark tests

Enables use of Newton-Raphson algorithm for simulations

Reduces computational effort in viscoelastic flow modeling

Abstract

Subject of this paper is the derivation of a new constitutive law in terms of the logarithm of the conformation tensor that can be used as a full substitute for the 2D governing equations of the Oldroyd-B, Giesekus and other models. One of the key features of these new equations is that - in contrast to the original log-conf equations given by Fattal and Kupferman (2004) - these constitutive equations combined with the Navier-Stokes equations constitute a self-contained, non-iterative system of partial differential equations. In addition to its potential as a fruitful source for understanding the mathematical subtleties of the models from a new perspective, this analytical description also allows us to fully utilize the Newton-Raphson algorithm in numerical simulations, which by design should lead to reduced computational effort. By means of the confined cylinder benchmark we will show that a finite element discretization of these new equations delivers results of comparable accuracy to known methods.