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A planar two-dimensional computational analysis is presented to qualifytraditional and fractal vane-in-cup geometries for accurate rheometry of simpleviscoplastic fluids with and without slip. Numerical simulations based on anadaptive augmented Lagrangian scheme are used to study the two-dimensional flowfield of yield-stress fluids within and around vane tools with N=3 to 24 armsfor a wide range of Bingham numbers, B (i.e. the ratio of the yield stress overthe characteristic viscous stress). This allows for accurate calculations ofthe velocity and stress fields around vanes with various geometries, as well asdirect comparison to experimental observations of the output torque measured bya rheometer, revealing sources of variation and error. We describe the impactof the vane structure on the fluid velocity field, from few-arm cruciform vanes(N < 6) that significantly perturb the flow away from ideal azimuthalkinematics, to many-arm fractal vanes (N > 12) in which the internal structuralfeatures are successfully ``cloaked" by a yield surface. This results in theshearing of an almost-circular ring of viscoplastic fluid that isindistinguishable from the annular ring of fluid deformed around a slip-freerotating cylindrical bob and leads to more accurate rheometric measurements ofthe material flow curve. Moreover, in direct comparison with data from previousliterature, we show that slip conditions on the vane surface do not impact thevelocity field or measured overall torque T, whereas slip conditions on thesmooth outer wall have significant impact on data, even when using a vanegeometry. Finally, we describe the impact of vane topography and Binghamnumber, B, on the measured torque and rheometric accuracy of vane-in-cupgeometries for ``simple" (inelastic) yield-stress fluids described by eitherthe Bingham plastic or Herschel-Bulkley constitutive model.

Computational Rheometry of Yielding and Viscoplastic Flow in Vane-and-Cup Rheometer Fixtures