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

In this work, we design and analyse 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, which are associated with possibly different Sobolev exponents r is an element of (1, infinity) and s is an element of (1, infinity). 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.