Efficient and Higher-Order Accurate Split-Step Methods for Generalised Newtonian Fluid Flow

In various practically relevant incompressible flow problems, such as polymer flow or biomedicalengineering applications, the dependence of fluid viscosity on the local shear rate plays an impor-tant role. Standard techniques using inf-sup stable finite elements lead to saddle-point systemsposing a challenge even for state-of-the-art solvers and preconditioners.For efficiency, projection schemes or time-splitting methods decouple the governing equations forvelocity and pressure, resulting in more, but easier to solve linear systems. Doing so, boundaryconditions and correction terms at intermediate steps have to be carefully considered in order toprohibit spoiling accuracy. In the case of Newtonian incompressible fluids, pressure and velocitycorrection schemes of high-order accuracy have been devised (see, e.g. [1, 2]). However, the exten-sion to generalised Newtonian fluids is a non-trivial task and considered an open question. Deteixet al. [3] successfully adapted the popular rotational correction scheme to consider for shear-ratedependent viscosity, but this resulted in substantial numerical overhead caused by necessarily pro-jecting viscous stress components.In this contribution we address this shortcoming and present a split-step scheme, extending pre-vious work by Liu [4]. The new method is based on an explicit-implicit treatment of pressure,convection and viscous terms combined with a Pressure-Poisson equation equipped with fully con-sistent Neumann and Dirichlet boundary conditions. Through proper reformulation, the use ofstandard continuous finite element spaces is enabled due to low regularity requirements. Addition-ally, equal-order velocity-pressure pairs are applicable as in the original scheme.The stability, accuracy and efficiency of the higher-order splitting scheme is showcased in challeng-ing numerical examples of practical interest.[1] Karniadakis, G. E., Israeli, M. and Orszag, S. A. High-order splitting methods for the incom-pressible Navier-Stokes equations. J. Comput. Phys., (1991).[2] Timmermans, L.J.P., Minev, P.D. and Van de Vosse, F. N. An approximate projection schemefor incompressible flow using spectral elements. Internat. J. Numer. Methods Fluids, Vol.22(7), pp. 673–688, (1996).[3] Deteix, J. and Yakoubi, D. Shear rate projection schemes for non-Newtonian fluids, Comput.Methods Appl. Mech. Engrg., Vol. 354, pp. 620–636, (2019).[4] Liu, J. Open and traction boundary conditions for the incompressible NavierStokesequations.J. Comput. Phys., Vol. 228(19), pp. 7250..7267, (2009).