In this article we present a numerical method for the Stokes flow of an Oldroyd-B fluid. The viscoelastic stress evolves according to a constitutive law formulated in terms of the upper convected time derivative. A finite difference method is used to discretise along fluid trajectories to approximate the advection and deformation terms of the upper convected derivative in a simple, cheap and cohesive manner, as well as ensuring that the discrete conformation tensor is positive definite. A full implementation with coupling to the fluid flow is presented, along with detailed discussion of the issues that arise with such schemes. We demonstrate the performance of this method with detailed numerical experiments in a lid-driven cavity setup. Numerical results are benchmarked against published data, and the method is shown to perform well in this challenging case.

翻译：本文提出了一种针对Oldroyd-B流体Stokes流动的数值方法。黏弹性应力根据以上对流时间导数形式的本构方程演化。我们采用有限差分法沿流体轨迹进行离散，以简单、高效且连贯的方式近似上对流导数中的对流项与变形项，同时确保离散构象张量保持正定性。本文给出了耦合流体流动的完整实现方案，并详细讨论了此类格式中出现的典型问题。通过方腔顶盖驱动流中的精细数值实验验证了该方法的性能，数值结果与已发表数据进行了基准对比，证明该方法在该挑战性算例中表现优异。