Virtual Element methods for non-Newtonian shear-thickening fluid flow problems

TL;DR

This paper develops a Virtual Element method for simulating incompressible non-Newtonian shear-thickening fluids, ensuring divergence-free velocity fields and mesh flexibility, with theoretical analysis and numerical validation.

Contribution

It extends previous shear-thinning analysis to shear-thickening flows, introducing new stability tools and stabilization techniques for nonlinear regimes.

Findings

Method achieves divergence-free velocity fields.

Theoretical stability analysis for shear-thickening regime.

Numerical results confirm practical effectiveness.

Abstract

In this work, we present a comprehensive theoretical analysis for Virtual Element discretizations of incompressible non-Newtonian flows governed by the Carreau-Yasuda constitutive law, in the shear-thickening regime (r > 2) including both degenerate (delta = 0) and non-degenerate (delta > 0) cases. The proposed Virtual Element method features two distinguishing advantages: the construction of an exactly divergence-free discrete velocity field and compatibility with general polygonal meshes. The analysis presented in this work extends a previous work, where only shear-thinning behavior (1 < r < 2) was considered. Indeed, the theoretical analysis of the shear-thickening setting requires several novel analytical tools, including: an inf-sup stability analysis of the discrete velocity-pressure coupling in non-Hilbertian norms, a stabilization term specifically designed to address the nonlinear structure as the exponent r > 2; and the introduction of a suitable discrete norm tailored to the underlying nonlinear constitutive relation. Numerical results demonstrate the practical performance of the proposed formulation.