A numerical study of the Kernel-conformation transformation for transient viscoelastic fluid flows

This work presents a numerical application of a generic conformation tensor transformation for simulating highly elastic flows of non-Newtonian fluids typically observed in computational rheology. In the Kernel-conformation framework [14], the conformation tensor constitutive law for a viscoelastic fluid is transformed introducing a generic tensor transformation function. The numerical stability of the application of the Kernel-conformation for highly elastic flows is ultimately related with the specific kernel function used in the matrix transformation, but also to the existence of singularities introduced either by flow geometry or by the characteristics of the constitutive equation. In this work, we implement this methodology in a free-surface Marker-And-Cell discretization methodology implemented in a finite differences method. The main contributions of this work are two fold: on one hand, we demonstrate the accuracy of this Kernel-conformation formulation using a finite differences method and free surfaces; on the other hand, we assess the numerical efficiency of specific kernel functions at high-Weissenberg number flows. The numerical study considers different viscoelastic fluid flow problems, including the Poiseuille flow in a channel, the lid-driven cavity flow and the die-swell free surface flow. The numerical results demonstrate the adequacy of this methodology for high Weissenberg number flows using the Oldroyd-B model