An efficient split-step framework for non-Newtonian incompressible flow problems with consistent pressure boundary conditions

Douglas R.Q. Pacheco*, Richard Schussnig, Thomas Peter Fries

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Incompressible flow problems with nonlinear viscosity, as they often appear in biomedical and industrial applications, impose several numerical challenges related to regularity requirements, boundary conditions, matrix preconditioning, among other aspects. In particular, standard split-step or projection schemes decoupling velocity and pressure are not as efficient for generalised Newtonian fluids, since the additional terms due to the non-zero viscosity gradient couple all velocity components again. Moreover, classical pressure correction methods are not consistent with the non-Newtonian setting, which can cause numerical artifacts such as spurious pressure boundary layers. Although consistent reformulations have been recently developed, the additional projection steps needed for the viscous stress tensor incur considerable computational overhead. In this work, we present a new time-splitting framework that handles such important issues, leading to an efficient and accurate numerical tool. Two key factors for achieving this are an appropriate explicit–implicit treatment of the viscous and convective nonlinearities, as well as the derivation of a pressure Poisson problem with fully consistent boundary conditions and finite-element-suitable regularity requirements. We present first- and higher-order stepping schemes tailored for this purpose, as well as various numerical examples showcasing the stability, accuracy and efficiency of the proposed framework.

Original languageEnglish
Article number113888
JournalComputer Methods in Applied Mechanics and Engineering
Publication statusPublished - 15 Aug 2021


  • Finite element methods
  • Incompressible flow
  • non-Newtonian fluids
  • Pressure boundary conditions
  • Pressure Poisson equation
  • Split-step schemes

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Computer Science Applications

Fields of Expertise

  • Information, Communication & Computing


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