Numerical study of the natural formulation of the tensor with the inclusion of the generalized Lie derivative

• Grant number: 25/03324-9
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: May 01, 2025
• End date: April 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: José Alberto Cuminato
• Grantee: Fernando Henrique Tadashi Himeno
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 13/07375-0 - CeMEAI - Center for Mathematical Sciences Applied to Industry, AP.CEPID

Abstract

This research project aims to address challenges in viscoelastic flow simulations, such as numerical instabilities and spurious solutions, particularly in geometries with singularities. These problems are associated with high stresses near the singularities and the High Weissenberg Number Problem (HWNP), which occurs when the Weissenberg number exceeds a critical value, generating instabilities and even interruptions in the simulations.To overcome these obstacles, we propose the reformulation of the constitutive equations of viscoelastic fluids by combining the Natural Stress Formulation (NSF) with the Generalized Lie Derivative (GLD). The NSF expands the non-Newtonian tensor into components based on the dyadic product involving the velocity field u and a unit vector v, orthogonal to u, and the GLD reformulates the convected upper derivative according to a Lagrangian approach. These procedures aim to improve the accuracy, stability, and temporal performance of the simulations. In addition, the kernel-conformation stabilizing method will be used, which applies matrix decompositions to the conformation tensor, aiming to complement the stabilization of the constitutive equation, especially in HWNP simulations. The methodology will be evaluated through simulations of reference flows, such as contractions/expansions and stick-slip/die-swell, which present geometric singularities and abrupt changes in boundary conditions. The ultimate goal is to provide an innovative approach that improves the stability, accuracy, and efficiency of viscoelastic fluid simulations in complex geometries.