{"title":"Discretisation of an Oldroyd-B viscoelastic fluid flow using a Lie derivative formulation","authors":"Ben S. Ashby, Tristan Pryer","doi":"10.1007/s10444-024-10211-x","DOIUrl":null,"url":null,"abstract":"<div><p>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.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 1","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10211-x.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Computational Mathematics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10444-024-10211-x","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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Abstract
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.
期刊介绍：
Advances in Computational Mathematics publishes high quality, accessible and original articles at the forefront of computational and applied mathematics, with a clear potential for impact across the sciences. The journal emphasizes three core areas: approximation theory and computational geometry; numerical analysis, modelling and simulation; imaging, signal processing and data analysis.
This journal welcomes papers that are accessible to a broad audience in the mathematical sciences and that show either an advance in computational methodology or a novel scientific application area, or both. Methods papers should rely on rigorous analysis and/or convincing numerical studies.
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