Robust stabilised finite element solvers for generalised Newtonian fluid flows

Abstract Various materials and solid-fluid mixtures of engineering and biomedical interest can be modelled as generalised Newtonian fluids, as their viscosity depends locally on the flow field. Despite the peculiarities of such models, it is common practice to combine them with numerical techniques conceived for Newtonian fluids, which can bring several issues such as spurious pressure boundary layers, unsuitable natural boundary conditions and additional nonlinear terms spoiling the effectiveness of both nonlinear solution procedures and preconditioners. In this context, we present a novel framework dealing with such issues while maintaining low computational cost and simple implementation. The building blocks of the presented algorithm are (i) a novel stabilised formulation for incompressible flow problems preserving consistency for low-order pairs, (ii) robust extrapolation of velocities in the time-dependent case to decouple the rheological law from the overall system, (iii) adaptive time step selection and (iv) a fast physics-based preconditioned Krylov subspace solver, to tackle the relevant range of discretisation parameters including highly varying viscosity. Selected numerical experiments are provided demonstrating the potential of the presented approach in terms of robustness, accuracy and efficiency for problems of practical interest.