Discretisation of an Oldroyd-B viscoelastic fluid flow using a Lie derivative formulation

• Published in: Advances in computational mathematics Vol. 51; no. 1
• Main Authors: Ashby, Ben S., Pryer, Tristan
• Format: Journal Article
• Language: English
• Published: New York Springer Nature B.V 01.02.2025
• Subjects: Finite difference method; Fluid flow; Mathematical analysis; Numerical methods; Stokes flow; Stokes law (fluid mechanics); Tensors; Viscoelastic fluids
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-024-10211-x

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 a 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.