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.

在生物医学和工业应用中经常出现的具有非线性粘度的不可压缩流动问题，在规则性要求，边界条件，基质预处理等方面提出了一些数值挑战。特别是，标准的分步或投影方案将速度和压力去耦对于广义牛顿流体并不那么有效，因为由于非零粘度梯度而产生的附加项再次将所有速度分量耦合在一起。此外，经典的压力校正方法与非牛顿设置不一致，这可能会导致数值伪像，例如伪压力边界层。尽管最近已经开发出一致的公式，但是粘性应力张量所需的其他投影步骤会产生可观的计算开销。在这项工作中，我们提出了一个新的时间分割框架来处理这些重要问题，从而提供了一种有效而准确的数值工具。实现这一目标的两个关键因素是对粘性和对流非线性的适当显式-隐式处理，以及具有完全一致的边界条件和适合有限元素的正则性要求的压力泊松问题的推导。我们介绍了为此目的量身定制的一阶和高阶步进方案，以及各种数字示例，它们展示了所提出框架的稳定性，准确性和效率。实现这一目标的两个关键因素是对粘性和对流非线性的适当显式-隐式处理，以及具有完全一致的边界条件和适合有限元素的正则性要求的压力泊松问题的推导。我们介绍了为此目的量身定制的一阶和高阶步进方案，以及各种数字示例，它们展示了所提出框架的稳定性，准确性和效率。实现这一目标的两个关键因素是对粘性和对流非线性的适当显式-隐式处理，以及具有完全一致的边界条件和适合有限元素的正则性要求的压力泊松问题的推导。我们介绍了为此目的量身定制的一阶和高阶步进方案，以及各种数字示例，它们展示了所提出框架的稳定性，准确性和效率。