  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.
