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Mechanics of Time-Dependent Materials

, Volume 20, Issue 1, pp 95–122 | Cite as

Shear-thinning and constant viscosity predictions for rotating sphere flows

  • Isaías E. Garduño
  • Hamid R. Tamaddon-Jahromi
  • Michael F. Webster
Article

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.

References

  1. Aboubacar, M., Webster, M.F.: A cell-vertex finite volume/element method on triangles for abrupt contraction viscoelastic flows. J. Non-Newton. Fluid Mech. 98, 83–106 (2001) CrossRefMATHGoogle Scholar
  2. Acharya, A., Maaskant, P.: The measurement of the material parameters of viscoelastic fluids using a rotating sphere and a rheogoniometer. Rheol. Acta 17, 377–382 (1978) CrossRefGoogle Scholar
  3. Belblidia, F., Matallah, H., Puangkird, B., Webster, M.F.: Alternative subcell discretisations for viscoelastic flow: Stress interpolation. J. Non-Newton. Fluid Mech. 146, 59–78 (2007) CrossRefMATHGoogle Scholar
  4. Belblidia, F., Matallah, H., Webster, M.F.: Alternative subcell discretisations for viscoelastic flow: Velocity-gradient approximation. J. Non-Newton. Fluid Mech. 151, 69–88 (2008) CrossRefMATHGoogle Scholar
  5. Boger, D.V., Walters, K.: Rheological Phenomena in Focus. Elsevier, Amsterdam (1993) MATHGoogle Scholar
  6. Chilcott, M., Rallison, J.: Creeping flow of dilute polymer solutions past cylinders and spheres. J. Non-Newton. Fluid Mech. 29, 381–432 (1988) CrossRefMATHGoogle Scholar
  7. Donea, J.: Taylor–Galerkin method for convective transport problems. Int. J. Numer. Methods Eng. 20, 101–119 (1984) CrossRefMATHGoogle Scholar
  8. Fosdick, R., Kao, B.: Steady flow of a simple fluid around a rotating sphere. Rheol. Acta 19, 675–697 (1980) CrossRefMathSciNetMATHGoogle Scholar
  9. 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) CrossRefGoogle Scholar
  10. Giesekus, H.: Mass and heat transfer at low flow of viscoelastic fluids around a rotating sphere. Rheol. Acta 9, 30–38 (1970) CrossRefGoogle Scholar
  11. Giesekus, H.: Phenomenological Rheology. Springer, New York (1994) CrossRefMATHGoogle Scholar
  12. Hermes, R.: Measurement of the limiting viscosity with a rotating sphere viscometer. J. Appl. Polym. Sci. 10, 1793–1799 (1966) CrossRefGoogle Scholar
  13. Huilgol, R.R., Phan-Thien, N.: Fluid Mechanics of Viscoelasticity. Elsevier, Amsterdam (1997) Google Scholar
  14. Kelkar, J., Mashelkar, R., Ulbrecht, J.: A rotating sphere viscometer. J. Appl. Polym. Sci. 17, 3069–3083 (1973) CrossRefGoogle Scholar
  15. Manero, O., Mena, B.: On the measurement of second normal stresses using a rotating-sphere viscometer. Chem. Eng. J. 15, 159–163 (1978) CrossRefGoogle Scholar
  16. Mashelkar, R., Kale, D., Kelkar, J., Ulbrecht, J.: Determination of material parameters of viscoelastic fluids by rotational non-viscometric flows. Chem. Eng. Sci. 27, 973–985 (1972) CrossRefGoogle Scholar
  17. Matallah, H., Townsend, P., Webster, M.F.: Recovery and stress-splitting schemes for viscoelastic flows. J. Non-Newton. Fluid Mech. 75, 139–166 (1998) CrossRefMATHGoogle Scholar
  18. Mena, B., Levinson, E., Caswell, B.: Torque on a sphere inside a rotating cylinder. Z. Angew. Math. Phys. 23, 173–181 (1972) CrossRefGoogle Scholar
  19. Phan-Thien, N., Tanner, R.I.: A new constitutive equation derived from network theory. J. Non-Newton. Fluid Mech. 2, 353–365 (1977) CrossRefGoogle Scholar
  20. Tamaddon-Jahromi, H.R., Webster, M.F., Williams, P.R.: Excess pressure drop and drag calculations for strain-hardening fluids with mild shear-thinning: Contraction and falling sphere problems. J. Non-Newton. Fluid Mech. 166, 939–950 (2011) CrossRefGoogle Scholar
  21. Thomas, R.H., Walters, K.: The motion of an elastico-viscous liquid due to a sphere rotating about its diameter. Q. J. Mech. Appl. Math. 17, 39–53 (1964) CrossRefMathSciNetMATHGoogle Scholar
  22. Walters, K., Savins, J.: A Rotating-Sphere elastoviscometer. J. Rheol. 9, 407–416 (1965) CrossRefGoogle Scholar
  23. Walters, K., Waters, N.D.: On the use of a rotating sphere in the measurement of elastico-viscous parameters. Br. J. Appl. Phys. 14, 667 (1963) CrossRefMathSciNetGoogle Scholar
  24. Walters, K., Waters, N.D.: The interpretation of experimental results obtained from a rotating-sphere elastoviscometer. Br. J. Appl. Phys. 15, 989 (1964) CrossRefGoogle Scholar
  25. Walters, K., Webster, M.F., Tamaddon-Jahromi, H.R.: The numerical simulation of some contraction flows of highly elastic liquids and their impact on the relevance of the Couette correction in extensional rheology. Chem. Eng. Sci. 64, 4632–4639 (2009) CrossRefGoogle Scholar
  26. Wapperom, P., Webster, M.F.: A second-order hybrid finite-element/volume method for viscoelastic flows. J. Non-Newton. Fluid Mech. 79, 405–431 (1998) CrossRefMATHGoogle Scholar
  27. Wapperom, P., Webster, M.F.: Simulation for viscoelastic flow by a finite volume/element method. Comput. Methods Appl. Mech. Eng. 180, 281–304 (1999) CrossRefMATHGoogle Scholar
  28. Webster, M.F., Tamaddon-Jahromi, H.R., Aboubacar, M.: Time-dependent algorithm for viscoelastic flow-finite element/volume schemes. Numer. Methods Partial Differ. Equ. 21, 272–296 (2005) CrossRefMathSciNetMATHGoogle Scholar
  29. Zienkiewicz, O.C., Morgan, K., Peraire, J., Vandati, M., Löhner, R.: Finite elements for compressible gas flow and similar systems. In: 7th Int. Conf. Comput. Meth. Appl. Sci. Eng. Versailles, France (1985) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Isaías E. Garduño
    • 1
  • Hamid R. Tamaddon-Jahromi
    • 1
  • Michael F. Webster
    • 1
  1. 1.Institute of Non-Newtonian Fluid Mechanics, College of EngineeringSwansea UniversitySwanseaUK

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