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Unsteady flow of Carreau fluid in a pipe

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Abstract

The Carreau fluid flow in an infinite pipe, caused by a time-dependent pressure gradient, is studied theoretically. The problem is axisymmetric in space and reduces to a nonlinear parabolic equation for the axial velocity. This equation has no analytical solution in the general case of different Carreau numbers. In the present paper, it is proved that the velocity and its gradient, as well as their differences from the similar Newtonian solution, are bounded by explicitly found constants, which depend on the Carreau number and on the other parameters of the viscosity rheological model. These estimates are also discussed for the special case of oscillatory pressure gradient using some numerical examples at different Carreau numbers.

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References

  1. Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena. Wiley, New York (2002)

    Google Scholar 

  2. Chhabra, R.P.: Bubbles, Drops, and Particles in Non-Newtonian Fluids. CRC Press, Taylor & Francis Group, Boca Raton (2007)

    Google Scholar 

  3. de Souza Mendes, P.R.: Dimensionless non-Newtonian fluid mechanics. J. Non-Newton. Fluid Mech. 147, 109–116 (2007)

    Article  Google Scholar 

  4. Womersley, J.R.: Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol. 127, 553–563 (1955)

    Article  Google Scholar 

  5. Loudon, C., Tordesillas, C.: The use of the dimensionless Womersley number to characterize the unsteady nature of internal flow. J. Theor. Biol. 191, 63–78 (1998)

    Article  Google Scholar 

  6. Amaratunga, M., Rabenjafimanantsoa, H.A., Time, R.W.: Estimation of shear rate change in vertically oscillating non-Newtonian fluids: predictions on particle settling. J. Non-Newton. Fluid Mech. 277, 104236 (2020)

    Article  MathSciNet  Google Scholar 

  7. Tabakova, S., Kutev, N., Radev, S.: Oscillatory Carreau flows in straight channels. R. Soc. Open Sci. 7, 191305 (2020)

    Article  Google Scholar 

  8. Tabakova, S., Kutev, N., Radev, S.: Estimates of the oscillatory flow in a straight channel at small Carreau numbers. AIP Conf. Proc. 2159, 030036 (2019)

    Article  Google Scholar 

  9. Kutev, N., Tabakova, S., Radev, S.: Velocity and shear rate estimates of some non-Newtonian oscillatory flows in tubes. AIP Conf. Proc. 1773, 080002 (2016)

    Article  Google Scholar 

  10. Radev, S., Tabakova, S., Kutev, N.: Asymptotic study of the nonlinear velocity problem for the oscillatory non-Newtonian flow in a straight channel. In: Advanced Computing in Industrial Mathematics, in Studies in Computational Intelligence, vol. 728, pp. 159–168. Springer (2018)

  11. Tabakova, S., Nikolova, E., Radev, S.: Carreau model for oscillatory blood flow in a tube. AIP Conf. Proc. 1629, 336–343 (2014)

    Article  Google Scholar 

  12. Scotti, Ch.M., Shkolnik, A.D., Muluk, S.C., Finol, E.A.: Fluid-structure interaction in abdominal aortic aneurysms: effects of asymmetry and wall thickness. BioMed. Eng. OnLine 4, 64–85 (2005)

    Article  Google Scholar 

  13. Lamata, P., Pitcher, A., Krittian, S., Nordsletten, D., Bissell, M.M., Cassar, T., Barker, A.J., Markl, M., Neubauer, S., Smith, N.P.: Aortic relative pressure components derived from four-dimensional flow cardiovascular magnetic resonance. Magn. Reson. Med. 72, 1162–1169 (2014)

    Article  Google Scholar 

  14. Schlichting, H., Gersten, K.: Boundary Layer Theory. Springer, Berlin (2000)

    Book  Google Scholar 

  15. Myers, T.G.: Application of non-Newtonian models to thin film flow. Phys. Rev. E 72, 066302-1–066302-11 (2005)

  16. Kawohl, B., Kutev, N.: Global behaviour of solutions to a parabolic mean curvature equation. Differ. Integral Equ. 8, 1923–1946 (1995)

    MathSciNet  MATH  Google Scholar 

  17. Dammers, R., Stifft, F., Tordoir, J.H., Hameleers, J.M., Hoeks, A.P., Kitslaar, P.J.: Shear stress depends on vascular territory: comparison between common carotid and brachial artery. J. Appl. Physiol. 94, 485–489 (2003)

    Article  Google Scholar 

  18. Pinho, F.T., Whitelaw, J.H.: Flow of non-Newtonian fluids in a pipe. J. Non-Newton. Fluid Mech. 34(2), 129–144 (1990)

    Article  Google Scholar 

  19. Xu, D., Warnecke, S., Song, B., Ma, X., Hof, B.: Transition to turbulence in pulsating pipe flow. J. Fluid Mech. 831, 418–432 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

N.K. was partially supported by the National Scientific Program “Information and Communication Technologies for a Single Digital Market in Science, Education and Security (ICTinSES),” contract no. D01–205/23.11.2018, financed by the Ministry of Education and Science in Bulgaria and by the Grant No BG05M2OP001-1.001-0003-C01, financed by the Science and Education for Smart Growth Operational Program (2018-2023).

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bontchev str., bl. 9, 1113, Sofia, Bulgaria

    N. Kutev

  2. Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bontchev str., bl. 4, 1113, Sofia, Bulgaria

    S. Tabakova & St. Radev

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  1. N. Kutev
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  2. S. Tabakova
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  3. St. Radev
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Correspondence to S. Tabakova.

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Kutev, N., Tabakova, S. & Radev, S. Unsteady flow of Carreau fluid in a pipe. Z. Angew. Math. Phys. 72, 196 (2021). https://doi.org/10.1007/s00033-021-01624-5

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