Abstract
This study investigates the numerical solution of a viscoelastic flow for an Oldroyd-B model, due to the rotation of a sphere about its diameter. Analysis of the elastic-viscous problem had been reported by Thomas and Walters (Q J Mech Appl Math 17:39–53, 1964), Walters and Savins (J Rheol 9:407–416, 1965) and Giesekus (Rheol Acta 9:30–38, 1970). In this respect, three different flow patterns (types 1–3) predicted by Thomas and Walters (Q J Mech Appl Math 17:39–53, 1964) have been successfully reproduced when using an Oldroyd-B fluid to represent a Boger fluid. Initially, solutions for the Oldroyd-B model were calibrated in the second-order regime against the analytical solution. Then, the work is extended to cover three different flows regimes (second-order regime, transitional and general flow) and two settings of polymeric solvent-fraction. Analysis based on the bounding sphere-radius, associated with type 2 flow, and through different flow regimes revealed that the distinctive symmetrical shape formed in the second-order regime was not preserved, but an elliptical shape was acquired. Moreover, for general and transitional flow regimes, a new and third vortex was identified in the polar region of the sphere. The adjustment of this feature between two different fluid compositions, with solutions for high-solvent and high-polymeric versions (low-high polymeric contributions), was contrasted. The second normal stress difference (N 2) on the field was increased as the m parameter developed across the different flow regimes. Different torque values for several m values were compared against the theory, demonstrating the expected linear behaviour. The numerical algorithm involved a hybrid sub-cell finite-element/finite volume discretisation (fe/fv), which solved the system of momentum-continuity-stress equations. It is based on a semi-implicit time-stepping Taylor-Galerkin/pressure-correction parent-cell finite element method for momentum continuity, whilst invoking a sub-cell cell-vertex fluctuation distribution finite volume scheme for the stress. The hyperbolic aspects of the constitutive equation were addressed discretely through finite volume upwind Fluctuation Distribution techniques and inhomogeneity calls upon Median Dual Cell approximation.
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Notes
There was some doubt expressed as to the correct physical nature in the sign of some of the material parameters used.
Nb: Use of different molecular weight fluids used experimentally by Giesekus (1970)
The equation for the force over the surface of the sphere (Tamaddon-Jahromi et al. (2011)) is
$$ {F}_w=2\pi {a}^2{\displaystyle {\int}_0^{\pi}\left\{{T}_{rz} \sin \left(\theta \right)+\left({T}_{zz}-p\right) \cos \left(\theta \right)\right\}} \sin \left(\theta \right)d\theta $$
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Acknowledgments
The authors are grateful to Professor Ken Walters, FRS for first suggesting this problem to the authors and his contributions throughout the work in many fruitful discussions. I. Garduño gratefully acknowledges financial support of Consejo Nacional de Ciencia y Tecnología (Mexico) through the scholarship no. 310618.
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Garduño, I.E., Tamaddon-Jahromi, H.R. & Webster, M.F. Oldroyd-B numerical solutions about a rotating sphere at low Reynolds number. Rheol Acta 54, 235–251 (2015). https://doi.org/10.1007/s00397-014-0831-x
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DOI: https://doi.org/10.1007/s00397-014-0831-x