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Hydrodynamic stability of plane Poiseuille flow in Maxwell fluid with cross-flow

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Abstract

This paper deals with the linear stability analysis of a plane Poiseuille in a Maxwell fluid in the presence of a uniform cross-flow with respect to two-dimensional wave disturbances. The physical problem is reduced to a modified Orr–Sommerfeld equation with nonlinear eigenvalues and solved numerically using the Chebyshev spectral collocation method. Attention is focused on the combined effects of uniform cross-flow and the relaxation time of the fluid (Deborah number) on the flow stability. Results obtained in this framework show that for a given Reynolds number the cross-flow can either delay or advances the stability of this system depending on the magnitude of the wave number. This result is observed in the case of a Newtonian fluid as well as in the case of a Maxwell one. An important feature concerns the existence of bicritical states that occur when two different instability modes with incommensurate wave numbers bifurcate simultaneously from the basis state is observed. In addition, it is attempted to determine the critical parameters of instability when one varies the cross-flow in the presence of fluid’s elasticity. It turns out that the cross-flow has a stabilizing effect which becomes less pronounced when the fluid’s elasticity is present.

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References

  1. A.S. Monin, A.M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 1 (Gidrometeoizdat, St. Petersburg, 1992; MIT, Cambridge, MA, 1971)

  2. S.A. Orszag, Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50(4), 689–703 (1971)

    Google Scholar 

  3. S.A. Orszag, L.C. Kells, J. Fluid Mech. 96, 159 (1980). https://doi.org/10.1017/S0022112080002066

    Google Scholar 

  4. S.J. Davies, C.M. White, Proc. R. Soc. Lond. A 119, 92 (1928). https://doi.org/10.1098/rspa.1928.0086

    Google Scholar 

  5. S.G. Chefranov, A.G. Chefranov, Linear instability of plane Couette and Poiseuille flows. (JETP) J. Exp. Theor. Phys. 122, 925–931 (2016)

    Google Scholar 

  6. S.G. Chefranov, A.G. Chefranov. arXiv:1007.1097v1 [physics.flu-dyn]

  7. S.G. Chefranov, A.G. Chefranov, J. Exp. Theor. Phys. 119, 331 (2014). https://doi.org/10.1134/S1063776114070127

    Google Scholar 

  8. S.G. Chefranov, A.G. Chefranov, Dokl. Phys. 60, 327 (2015). https://doi.org/10.1134/S1028335815070083

    Google Scholar 

  9. S. Bottin, O. Dauchot, F. Daviaud, Phys. Rev. Lett. 79, 4377 (1997). https://doi.org/10.1103/PhysRevLett.79.4377

    Google Scholar 

  10. M. Avila, M.J. Belisle, J.M. Lopez, F. Marques, W.S. Saric, Mode competition in modulated Taylor–Couette flow. J. Fluid Mech. 601, 381–406 (2008)

    Google Scholar 

  11. M. Riahi, S. Aniss, M.O. Touhami, Families of reversing and non-reversing Taylor vortex flows between two co-oscillating cylinders with different amplitudes. Phys. Fluids 31, 014101 (2019)

    Google Scholar 

  12. A.S. Berman, Concerning laminar flow in channels with porous walls. J. Appl. Phys. 27, 1557 (1956)

    Google Scholar 

  13. D. Hains, Stability of plan Couette–Poiseuille flow with uniform cross flow. Phys. Fluids 14, 1620–1623 (1971)

    Google Scholar 

  14. D.M. Sheppard, Hydrodynamic stability of the flow between parallel porous walls. Phys. Fluids 15, 241–244 (1972)

    Google Scholar 

  15. J.H.M. Fransson, P.H. Alfredsson, On the hydrodynamics stability of channel flow with cross flow. Phys. Fluids 15(2), 436–441 (2003)

    Google Scholar 

  16. M. Lamine, A. Hifdi, Stability analysis of plane Couette–Poiseuille flow with porous walls under transverse magnetic field. J. Therm. Eng. 2(4), 807–813 (2016)

    Google Scholar 

  17. T. Annable, R. Buscall, R. Ettelaie, P. Shepherd, D. Whittlestone, Influence of surfactants on the rheology of associating polymers in solution. Langmuir 10(4), 1060–1070 (1994)

    Google Scholar 

  18. N. Lahlou, M. Riahi, S. Aniss, M.O. Touhami, Control of the effects of the concentration on the viscoelastic behavior of the aqueous polyethylene-oxide solution. J. Phys. Conf. Ser. 1045, 012025 (2018)

    Google Scholar 

  19. P. Fischer, H. Rehage, Rheological-master curves of viscoelastic surfactant solutions by varying the solvent viscosity and temperature. Langmuir 13(26), 7012–7020 (1997)

    Google Scholar 

  20. M.E. Cates, Flow behavior of entangled surfactant micelles. J. Phys. Condens. Matter 8, 9167 (1996)

    Google Scholar 

  21. H. Hoffmann, Structure and flow in Surfactant Solutions 578, 2–31 (1994). 2003

  22. M. Riahi, S. Aniss, M.O. Touhami, S.S. Lami, Centrifugal instability of pulsed Taylor–Couette flow in a Maxwell fluid. Eur. Phys. J. E 39, 82 (2016)

    Google Scholar 

  23. M. Riahi, S. Aniss, M.O. Touhami, S.S. Lami, Stability of a pulsed Taylor–Couette flow in a viscoelastic fluid. J. Soc. Rheol. Jpn. 42(5), 321–327 (2014)

    Google Scholar 

  24. M. Riahi, S. Aniss, M.O. Touhami, S.S. Lami, Pulsed Taylor–Couette flow in a viscoelastic fluid under inner cylinder modulation. Eur. Phys. J. Plus 130, 253 (2015)

    Google Scholar 

  25. A.C. Weideman, S.C. Reddy, ACM Trans. Math. Softw. 26, 465 (2000)

    Google Scholar 

  26. A. Hifdi, M. Ouazzani-Touhami, J. Khalid-Naciri, Channel entrance flow and its linear stability. J. Stat. Mech.: Theor. Exp. P06003 (stacks.iop.org/JSTAT /2004/P06003) (2004)

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  1. Laboratory of Mechanics, Faculty of Sciences Aïn-Chock, University Hassan II, Casablanca, Morocco

    M. Lamine, M. Riahi & A. Hifdi

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  1. M. Lamine
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  2. M. Riahi
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Lamine, M., Riahi, M. & Hifdi, A. Hydrodynamic stability of plane Poiseuille flow in Maxwell fluid with cross-flow. Eur. Phys. J. Plus 135, 189 (2020). https://doi.org/10.1140/epjp/s13360-020-00165-6

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  • DOI: https://doi.org/10.1140/epjp/s13360-020-00165-6

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