# Shear-thinning and constant viscosity predictions for rotating sphere flows

## Abstract

The steady motion of a rotating sphere is analysed through two contrasting viscoelastic models, a constant viscosity (FENE-CR) model and a shear-thinning (LPTT) model. Giesekus (Rheol. Acta 9:30–38, 1970) presented an intriguing rotating viscoelastic flow, which to date has not been completely explained. In order to investigate this flow, sets of parameters have been explored to analyse the significant differences introduced with the proposed models, while the momentum-continuity-stress equations are solved through a hybrid finite-element/finite volume numerical scheme. Solutions are discussed for first, sphere angular velocity increase (\(\varOmega\)), and second, through material velocity-scale increase (\(\alpha\)). Numerical predictions for different solvent-ratios (\(\beta\)) show significant differences as the sphere angular velocity increases. It is demonstrated that an emerging equatorial anticlockwise vortex emerges in a specific range of \(\varOmega\). As such, this solution matches closely with the Giesekus experimental findings. Additionally, inside the emerging inertial vortex, a contrasting positive second normal stress-difference (\(N_{2} ( \dot{\gamma} ) = \tau_{rr} - \tau_{\theta\theta}\)) region is found compared against the negative \(N_{2}\)-enveloping layer.

## Keywords

Rotating sphere Secondary flow field FENE-CR model LPTT model## Notes

### Acknowledgement

I.E. Garduño gratefully acknowledges financial support from Consejo Nacional de Ciencia y Tecnología (Mexico) through the scholarship No. 310618.

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