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arXiv · 2601.04866

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

Paola F. Antonietti·Lourenço Beirão da Veiga·Michele Botti·André Harnist·Giuseppe Vacca·Marco Verani
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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.

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Paola F. Antonietti, Lourenço Beirão da Veiga, Michele Botti, André Harnist, Giuseppe Vacca, Marco Verani. 2026-01-08. Virtual Element methods for non-Newtonian shear-thickening fluid flow problems. https://arxiv.org/abs/2601.04866

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