A Hybrid High-Order method for incompressible flows of non-Newtonian fluids with power-like convective behaviour

TL;DR

This paper introduces a Hybrid High-Order (HHO) discretization method tailored for incompressible non-Newtonian fluid flows with power-like convection, providing stability, convergence analysis, and validation for various fluid behaviors.

Contribution

The paper develops a novel HHO scheme for non-Newtonian flows with general viscosity and convection laws, including comprehensive theoretical analysis and validation.

Findings

The HHO scheme is stable and convergent under broad conditions.

Error estimates are provided for shear-thinning fluids and small data.

The method is validated on multiple model problems.

Abstract

In this work, we design and analyze a Hybrid High-Order (HHO) discretization method for incompressible flows of non-Newtonian fluids with power-like convective behaviour. We work under general assumptions on the viscosity and convection laws, that are associated with possibly different Sobolev exponents r > 1 and s > 1. After providing a novel weak formulation of the continuous problem, we study its well-posedness highlighting how a subtle interplay between the exponents r and s determines the existence and uniqueness of a solution. We next design an HHO scheme based on this weak formulation and perform a comprehensive stability and convergence analysis, including convergence for general data and error estimates for shear-thinning fluids and small data. The HHO scheme is validated on a complete panel of model problems.