Three-dimensional computational rheometry of viscoplastic fluids in vane-in-cup fixtures
Abstract
We develop a computational framework to investigate the flow kinematics of viscoplastic materials, described by the Herschel-Bulkley model, in vane-in-cup rheometers. Numerical simulations are performed using augmented Lagrangian finite element and regularised finite volume methods for two- and three-dimensional configurations, respectively, to characterise the hydrodynamics and yielding behaviour over a broad range of effective parameters. Three types of vanes are investigated: conventional 4- and 6-arm bespoke vanes and a 24-arm fractal vane (Owens et al., J. Rheol., vol. 64(3), 2020, 643-662). Our previous two-dimensional study (Chaparian et al., J. Non-Newtonian Fluid Mech., vol. 307, 2022, 104857) demonstrated the superiority of the fractal vane over conventional N-arm designs. The bespoke vanes produce highly non-uniform yielding, leading to significant deviations from the circular Couette analogy, whereas the 24-arm fractal vane generates nearly axisymmetric yield surfaces that effectively "cloak" the vane geometry. Consequently, the resulting flow closely resembles the azimuthal shear in a conventional "bob & cup" rheometer, providing substantially more accurate rheological measurements. Building on these findings, the present study revisits the torque-to-stress and rotation-to-strain rate conversions used in vane-in-cup rheometry. In particular, we re-examine the circular Couette analogy that underpins both the physical interpretation of the flow and the extraction of flow curves from raw rheometric data. Based on our extensive two- and three-dimensional simulations, we identify limitations of the existing conversion relations and propose improved closure expressions that enable more accurate determination of material's rheological properties. Furthermore, the three-dimensional flow structures and yielding patterns within the vane-in-cup fixtures are characterised.
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BibTeX
@misc{malcolm2026threedimensional,
title = {{Three-dimensional computational rheometry of viscoplastic fluids in vane-in-cup fixtures}},
author = {James Malcolm and Clare McColgan and Emma Mullarkey and Emad Chaparian},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02115},
eprint = {2611.02115},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02115},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Three-dimensional computational rheometry of viscoplastic fluids in vane-in-cup fixtures
AU - James Malcolm
AU - Clare McColgan
AU - Emma Mullarkey
AU - Emad Chaparian
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02115
UR - https://arxiv.org/abs/2611.02115
AB - We develop a computational framework to investigate the flow kinematics of viscoplastic materials, described by the Herschel-Bulkley model, in vane-in-cup rheometers. Numerical simulations are performed using augmented Lagrangian finite element and regularised finite volume methods for two- and three-dimensional configurations, respectively, to characterise the hydrodynamics and yielding behaviour over a broad range of effective parameters. Three types of vanes are investigated: conventional 4- and 6-arm bespoke vanes and a 24-arm fractal vane (Owens et al., J. Rheol., vol. 64(3), 2020, 643-662). Our previous two-dimensional study (Chaparian et al., J. Non-Newtonian Fluid Mech., vol. 307, 2022, 104857) demonstrated the superiority of the fractal vane over conventional N-arm designs. The bespoke vanes produce highly non-uniform yielding, leading to significant deviations from the circular Couette analogy, whereas the 24-arm fractal vane generates nearly axisymmetric yield surfaces that effectively "cloak" the vane geometry. Consequently, the resulting flow closely resembles the azimuthal shear in a conventional "bob & cup" rheometer, providing substantially more accurate rheological measurements. Building on these findings, the present study revisits the torque-to-stress and rotation-to-strain rate conversions used in vane-in-cup rheometry. In particular, we re-examine the circular Couette analogy that underpins both the physical interpretation of the flow and the extraction of flow curves from raw rheometric data. Based on our extensive two- and three-dimensional simulations, we identify limitations of the existing conversion relations and propose improved closure expressions that enable more accurate determination of material's rheological properties. Furthermore, the three-dimensional flow structures and yielding patterns within the vane-in-cup fixtures are characterised.
ER -