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

TL;DR

This work tackles the numerical approximation of steady incompressible non-Newtonian flows described by the Carreau–Yasuda constitutive law in the shear-thickening regime ($r>2$) using divergence-free Virtual Elements on general polygonal meshes. The authors formulate a robust kernel-based weak problem, prove discrete inf-sup stability and monotone, Lipschitz-controlled behavior of the nonlinear viscous term, and develop a stabilization tailored to the nonlinear structure. They establish a priori error estimates for velocity and pressure that depend on mesh size, regularity, and the degenerate parameter $\delta$, with convergence rates showing a transition between $\mathcal{O}(h^{2k/r})$ and $\mathcal{O}(h^{k/(r-1)})$-type behavior; convex meshes recover standard $W^{1,r}$-norm convergence. Numerical experiments on quadrilateral, Voronoi, and Cartesian meshes corroborate the theory, including degenerate and non-degenerate cases and singular solutions, highlighting the method’s flexibility and reliability for complex polytopal geometries.

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.

Figures (4)

• Figure 1: Example of the adopted polygonal meshes.
• Figure 2: Test 1. Computed errors defined as in \ref{['eq:err_quant']} as a function of the mesh size (loglog scale), for the mesh family QUADRILATERAL. Left panel: $\delta=1$, right panel: $\delta=0$.
• Figure 3: Test 1. Computed errors defined as in \ref{['eq:err_quant']} as a function of the mesh size (log-log scale), for the mesh family RANDOM. Left panel: $\delta=1$, right panel: $\delta=0$.
• Figure 4: Test 2. Computed errors defined as in \ref{['eq:err_quant']} as a function of the mesh size (loglog scale), for the mesh family QUADRILATERAL. Left panel: $\delta=1$, right panel: $\delta=0$.

Theorems & Definitions (43)

• Lemma 1: Continuity and monotonicity of $a$
• proof
• Lemma 2: Inf-sup condition
• Proposition 3: Well-posedness
• proof
• Lemma 4
• Remark 5
• Lemma 6
• proof
• Lemma 7