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Projection-based solver for viscoelastic Stokes flow using Fast Fourier Transforms

Georg Rempfer1, Mae Nesenberend2, Chengkai Zhu3, Bart Stam3, Debabrata Panja3, and Joost de Graaf2,*

  • 1Institute for Computational Physics, Universität Stuttgart, Allmandring 3, 70569 Stuttgart, Germany
  • 2Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands
  • 3Department of Information and Computing Sciences, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands

  • *Contact author: j.degraaf@uu.nl
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Phys. Rev. Fluids 11, 084901 – Published 24 August, 2026

DOI: https://doi.org/10.1103/gqfw-8413

Abstract

Understanding the flow of complex media is relevant for a wide range of research fields and industrial applications. Several numerical approaches exist by which approximate solutions can be determined for the Stokes equations that describe microhydrodynamic flows at the continuum level. However, achieving efficiency and accuracy for an incompressible fluid remains challenging. Here, we present an algorithm for solving the Stokes equations for an Oldroyd-B fluid using Fourier transforms. We gain efficiency by leveraging the “Fastest Fourier Transform in the West” (FFTW). We validate our approach for the well-characterized four-roll mill, which exhibits nearly singular points of stress at the extensional points of the flow. We capture this divergence and showcase the potential of our method without making the usual diffusive renormalization. We also focus on characterizing the power-law behavior and numerically assess the divergence criterion. Future work will concentrate on active systems, the introduction of moving boundaries, and application to microfluidic devices.

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Phys. Rev. Fluids 11, 084901– Published 24 August, 2026
  • Received 1 September 2025
  • Accepted 5 August 2026
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