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Secondary instabilities in Taylor–Couette flow of shear-thinning fluids

Published online by Cambridge University Press:  17 December 2021

S. Topayev ,
C. Nouar Open the ORCID record for C. Nouar [Opens in a new window]  and
J. Dusek Open the ORCID record for J. Dusek [Opens in a new window]
S. Topayev
Affiliation:
LEMTA, UMR 7563, CNRS – Université de Lorraine, 2 Avenue de la Forêt de Haye, BP 90161, 54505Vandoeuvre Lès Nancy, France
C. Nouar*
Affiliation:
LEMTA, UMR 7563, CNRS – Université de Lorraine, 2 Avenue de la Forêt de Haye, BP 90161, 54505Vandoeuvre Lès Nancy, France
J. Dusek
Affiliation:
ICUBE, UMR 7357, CNRS – Université de Strasbourg, 2 rue Boussingault, 67000Strasbourg, France
*
†Email address for correspondence: [email protected]
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Abstract

The stability of the Taylor vortex flow in Newtonian and shear-thinning fluids is investigated in the case of a wide gap Taylor–Couette system. The considered radius ratio is $\eta = R_1/R_2=0.4$. The aspect ratio (length over the gap width) of experimental configuration is 32. Flow visualization and measurements of two-dimensional flow fields with particle image velocimetry are performed in a glycerol aqueous solution (Newtonian fluid) and in xanthan gum aqueous solutions (shear-thinning fluids). The experiments are accompanied by axisymmetric numerical simulations of Taylor–Couette flow in the same gap of a Newtonian and a purely viscous shear-thinning fluid described by the Carreau model. The experimentally observed critical Reynolds and wavenumbers at the onset of Taylor vortices are in very good agreement with that obtained from a linear theory assuming a purely viscous shear-thinning fluid and infinitely long cylinders. They are not affected by the viscoelasticity of the used fluids. For the Newtonian fluid, the Taylor vortex flow (TVF) regime is found to bifurcate into a wavy vortex flow with a high frequency and low amplitude of axial oscillations of the vortices at ${Re} = 5.28 \, {Re}_c$. At ${Re} = 6.9 \, {Re}_c$, the frequency of oscillations decreases and the amplitude increases abruptly. For the shear-thinning fluids the secondary instability conserves axisymmetry. The latter is characterized by an instability of the array of vortices leading to a continuous sequence of creation and merging of vortex pairs. Axisymmetric numerical simulations reproduce qualitatively very well the experimentally observed flow behaviour.

JFM classification

Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 933 , 25 February 2022 , A4
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Agbessi, Y., Alibenyahia, B., Nouar, C. & Choplin, L. 2015 Linear stability of Taylor–Couette flow of shear-thinning fluids: modal and non-modal approaches. J. Fluid Mech. 775, 354–389.CrossRefGoogle Scholar
Ahlers, G., Cannell, D.S. & Lerma, M.A.D. 1983 Possible mechanism for transitions in wavy Taylor-vortex flow. Phys. Rev. A 27 (2), 1225.CrossRefGoogle Scholar
Alibenyahia, B., Lemaitre, C., Nouar, C. & Ait-Messaoudene, N. 2012 Revisiting the stability of circular Couette flow of shear-thinning fluids. J. Non-Newtonian Fluid Mech. 183, 37–51.CrossRefGoogle Scholar
Andereck, C.D., Liu, S.S. & Swinney, H.L. 1986 Flow regimes in a circular Couette system with independently rotating cylinders. J. Fluid Mech. 164, 155–183.CrossRefGoogle Scholar
Atkhen, K., Fontaine, J. & Wesfreid, J.E. 2000 Highly turbulent Couette–Taylor bubbly flow patterns. J. Fluid Mech. 422, 55–68.CrossRefGoogle Scholar
Bird, R.B., Amstrong, R. & Hassager, O. 1987 Dynamics of Polymeric Liquids. Wiley-Interscience.Google Scholar
Bot, P. & Mutabazi, I. 2000 Dynamics of spatio-temporal defects in the Taylor–Dean system. Eur. Phys. J. B 13 (1), 141–155.CrossRefGoogle Scholar
Bottaro, A. 1993 On longitudinal vortices in curved channel flow. J. Fluid Mech. 251, 627–660.CrossRefGoogle Scholar
Brandstäter, A., Swift, J., Swinney, H.L., Wolf, A., Farmer, J.D., Jen, E. & Crutchfield, P.J. 1983 Low-dimensional chaos in a hydrodynamic system. Phys. Rev Lett. 51 (16), 1442.CrossRefGoogle Scholar
Burkhalter, J.E. & Koschmieder, E.L. 1973 Steady supercritical Taylor vortex flow. J. Fluid Mech. 58 (3), 547–560.CrossRefGoogle Scholar
Cagney, N. & Balabani, S. 2019 a Influence of shear-thinning rheology on the mixing dynamics in Taylor–Couette flow. Chem. Engng Technol. 42 (8), 1680–1690.Google Scholar
Cagney, N. & Balabani, S. 2019 b Taylor–Couette flow of shear-thinning fluids. Phys. Fluids 31 (5), 053102.CrossRefGoogle Scholar
Cagney, N., Lacassagne, T. & Balabani, S. 2020 Taylor–Couette flow of polymer solutions with shear-thinning and viscoelastic rheology. J. Fluid Mech. 905, A28.CrossRefGoogle Scholar
Carreau, J.P. 1972 Rheological equations from molecular network theories. Trans. Soc. Rheol. 16 (1), 99–127.CrossRefGoogle Scholar
Cole, J.A. 1976 Taylor-vortex instability and annulus-length effects. J. Fluid Mech. 75 (1), 1–15.CrossRefGoogle Scholar
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21 (3), 385–425.CrossRefGoogle Scholar
Crawford, G.L., Park, K. & Donnelly, R.J. 1985 Vortex pair annihilation in Taylor wavy-vortex flow. Phys. Fluids 28 (1), 7–9.CrossRefGoogle Scholar
Crumeyrolle, O., Latrache, N., Mutabazi, I. & Ezersky, A.B. 2005 Instabilities with shear-thinning polymer solutions in the Couette–Taylor system. In Journal of Physics: Conference Series, vol. 14, p. 011. IOP Publishing.CrossRefGoogle Scholar
Crumeyrolle, O., Mutabazi, I. & Grisel, M. 2002 Experimental study of inertioelastic Couette–Taylor instability modes in dilute and semidilute polymer solutions. Phys. Fluids 14 (5), 1681–1688.CrossRefGoogle Scholar
Czarny, O., Serre, E., Bontoux, P. & Lueptow, R.M. 2003 Interaction between Ekman pumping and the centrifugal instability in Taylor–Couette flow. Phys. Fluids 15 (2), 467–477.CrossRefGoogle Scholar
Czarny, O., Serre, E., Bontoux, P. & Lueptow, R.M. 2004 Interaction of wavy cylindrical Couette flow with endwalls. Phys. Fluids 16 (4), 1140–1148.CrossRefGoogle Scholar
Daviaud, F., Hegseth, J. & Bergé, P. 1992 Subcritical transition to turbulence in plane Couette flow. Phys. Rev. Lett. 69 (17), 2511.CrossRefGoogle ScholarPubMed
Dennin, M., Cannell, D.S. & Ahlers, G. 1994 Measurement of a short-wavelength instability in Taylor vortex flow. Phys. Rev. E 49 (1), 462.CrossRefGoogle ScholarPubMed
Dessup, T., Tuckerman, L.S., Wesfreid, J.E., Barkley, D. & Willis, A.P. 2018 Self-sustaining process in Taylor–Couette flow. Phys. Rev. Fluids 3 (12), 123902.CrossRefGoogle Scholar
DiPrima, R.C., Eagles, P.M. & Ng, B.S. 1984 The effect of radius ratio on the stability of Couette flow and Taylor vortex flow. Phys. Fluids 27 (10), 2403–2411.CrossRefGoogle Scholar
Dominguez-Lerma, M.A., Ahlers, G. & Cannell, D.S. 1984 Marginal stability curve and linear growth rate for rotating Couette–Taylor flow and Rayleigh–Bénard convection. Phys. Fluids 27 (4), 856–860.CrossRefGoogle Scholar
Dutcher, C.S. & Muller, S.J. 2007 Explicit analytic formulas for newtonian Taylor–Couette primary instabilities. Phys. Rev. E 75 (4), 047301.CrossRefGoogle ScholarPubMed
Dutcher, C.S. & Muller, S.J. 2011 Effects of weak elasticity on the stability of high Reynolds number co-and counter-rotating Taylor–Couette flows. J. Rheol. 55 (6), 1271–1295.CrossRefGoogle Scholar
Dutcher, C.S. & Muller, S.J. 2013 Effects of moderate elasticity on the stability of co-and counter-rotating Taylor–Couette flows. J. Rheol. 57 (3), 791–812.CrossRefGoogle Scholar
Dutcher, C.S. & Muller, S.J. 2009 Spatio-temporal mode dynamics and higher order transitions in high aspect ratio newtonian Taylor–Couette flows. J. Fluid Mech. 641, 85–113.CrossRefGoogle Scholar
Elçiçek, H. & Güzel, B. 2020 Effect of shear-thinning behaviour on flow regimes in Taylor–Couette flows. J. Non-Newtonian Fluid Mech. 279, 104277.CrossRefGoogle Scholar
Escudier, M.P., Gouldson, I.W. & Jones, D.M. 1995 Taylor vortices in Newtonian and shear-thinning liquids. Proc. R. Soc. Lond. A 449, 155–176.Google Scholar
Esser, A. & Grossmann, S. 1996 Analytic expression for Taylor–Couette stability boundary. Phys. Fluids 8 (7), 1814–1819.CrossRefGoogle Scholar
Fasel, H. & Booz, O. 1984 Numerical investigation of supercritical Taylor-vortex flow for a wide gap. J. Fluid Mech. 138, 21–52.CrossRefGoogle Scholar
Gillissen, J.J.J., Cagney, N., Lacassagne, T., Papadopoulou, A., Balabani, S. & Wilson, H.J. 2020 Taylor–Couette instability in disk suspensions: experimental observation and theory. Phys. Rev. Fluids 5 (8), 083302.CrossRefGoogle Scholar
Gillissen, J.J.J. & Wilson, H.J. 2018 Taylor–Couette instability in disk suspensions. Phys. Rev. Fluids 3 (11), 113903.CrossRefGoogle Scholar
Groisman, A. & Steinberg, V. 1998 Mechanism of elastic instability in Couette flow of polymer solutions: experiment. Phys. Fluids 10 (10), 2451–2463.CrossRefGoogle Scholar
Guo, Y. & Finlay, W.H. 1991 Splitting, merging and wavelength selection of vortices in curved and/or rotating channel flow due to Eckhaus instability. J. Fluid Mech. 228, 661–691.Google Scholar
Hecht, F. 2012 New development in freefem++. J. Numer. Maths 20 (3–4), 251–266.Google Scholar
Hegseth, J.J. 1996 Turbulent spots in plane Couette flow. Phys. Rev. E 54 (5), 4915.CrossRefGoogle ScholarPubMed
Hoffmann, C., Altmeyer, S., Heise, M., Abshagen, J. & Pfister, G. 2013 Axisymmetric propagating vortices in centrifugally stable Taylor–Couette flow. J. Fluid Mech. 728, 458–470.CrossRefGoogle Scholar
Jones, C.A. 1985 The transition to wavy Taylor vortices. J. Fluid Mech. 157, 135–162.CrossRefGoogle Scholar
King, G.P., Li, Y., Lee, W., Swinney, H.L. & Marcus, P.S. 1984 Wave speeds in wavy Taylor-vortex flow. J. Fluid Mech. 141, 365–390.CrossRefGoogle Scholar
Kogelman, S. & DiPrima, R.Ct. 1970 Stability of spatially periodic supercritical flows in hydrodynamics. Phys. Fluids 13 (1), 1–11.CrossRefGoogle Scholar
Koschmieder, E.L. 1993 Bénard cells and Taylor vortices. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, vol. 844, pp. 37–51.Google Scholar
Lacassagne, T., Cagney, N. & Balabani, S. 2021 Shear-thinning mediation of elasto-inertial Taylor–Couette flow. J. Fluid Mech. 915.CrossRefGoogle Scholar
Li, Z. & Khayat, R.E. 2004 A non-linear dynamical system approach to finite amplitude Taylor-vortex flow of shear-thinning fluids. Intl J. Numer. Meth. Fluids 45 (3), 321–340.CrossRefGoogle Scholar
Lindner, A., Bonn, D. & Meunier, J. 2000 Viscous fingering in a shear-thinning fluid. Phys. Fluids 12 (2), 256–261.CrossRefGoogle Scholar
Linek, M. & Ahlers, G. 1998 Boundary limitation of wave numbers in Taylor-vortex flow. Phys. Rev. E 58 (3), 3168.CrossRefGoogle Scholar
Lorenzen, A., Pfister, G. & Mullin, T. 1983 End effects on the transition to time-dependent motion in the Taylor experiment. Phys. Fluids 26 (1), 10–13.CrossRefGoogle Scholar
Majji, M.V., Banerjee, S. & Morris, J.F. 2018 Inertial flow transitions of a suspension in Taylor–Couette geometry. J. Fluid Mech. 835, 936–969.CrossRefGoogle Scholar
Martinand, D., Serre, E. & Lueptow, R.M. 2014 Mechanisms for the transition to waviness for Taylor vortices. Phys. Fluids 26 (9), 094102.CrossRefGoogle Scholar
Masuda, H., Horie, T., Hubacz, R., Ohta, M. & Ohmura, N. 2017 Prediction of onset of Taylor–Couette instability for shear-thinning fluids. Rheol. Acta 56, 73–84.CrossRefGoogle Scholar
Meincke, O. & Egbers, C. 1999 Routes into chaos in small and wide gap Taylor–Couette flow. Phys. Chem. Earth Pt B 24 (5), 467–471.CrossRefGoogle Scholar
Meyer-Spasche, R. & Keller, H.B. 1985 Some bifurcation diagrams for Taylor vortex flows. Phys. Fluids 28 (5), 1248–1252.CrossRefGoogle Scholar
Mullin, T. 1985 Onset of time dependence in Taylor–Couette flow. Phys. Rev. A 31 (2), 1216.CrossRefGoogle ScholarPubMed
Ng, J.H., Jaiman, R.K. & Lim, T.T. 2018 Interaction dynamics of longitudinal corrugations in Taylor–Couette flows. Phys. Fluids 30 (9), 093601.CrossRefGoogle Scholar
Nore, C., Moisy, F. & Quartier, L. 2005 Experimental observation of near-heteroclinic cycles in the von Kármán swirling flow. Phys. Fluids 17 (6), 064103.CrossRefGoogle Scholar
Paap, H.-G. & Riecke, H. 1990 Wave-number restriction and mode interaction in Taylor vortex flow: appearance of a short-wavelength instability. Phys. Rev. A 41 (4), 1943.CrossRefGoogle ScholarPubMed
Park, K., Crawford, G.L. & Donnelly, R.J. 1983 Characteristic lengths in the wavy vortex state of Taylor–Couette flow. Phys. Rev. Lett. 51 (15), 1352.CrossRefGoogle Scholar
Razzak, M.A., Khoo, B.C. & Lua, K.B. 2019 Numerical study on wide gap Taylor–Couette flow with flow transition. Phys. Fluids 31 (11), 113606.CrossRefGoogle Scholar
Riecke, H. & Paap, H.-G. 1986 Stability and wave-vector restriction of axisymmetric Taylor vortex flow. Phys. Rev. A 33 (1), 547.CrossRefGoogle ScholarPubMed
Schwarz, K.W. 1990 Phase slip and turbulence in superfluid $^4\textrm {He}$: a vortex mill that works. Phys. Rev. Lett. 64 (10), 1130.CrossRefGoogle Scholar
Sinevic, V., Kuboi, R. & Nienow, A.W. 1986 Power numbers, Taylor numbers and Taylor vortices in viscous Newtonian and non-Newtonian fluids. Chem. Engng Sci. 41 (11), 2915–2923.CrossRefGoogle Scholar
Smieszek, M., Crumeyrolle, O., Mutabazi, I. & Egbers, C. 2008 Instabilities with polyacrylamide solution in small and large aspect ratios Taylor–Couette systems. In Journal of Physics: Conference Series, vol. 137, p. 012021. IOP Publishing.CrossRefGoogle Scholar
Snyder, H.A. & Lambert, R.B. 1966 Harmonic generation in Taylor vortices between rotating cylinders. J. Fluid Mech. 26 (3), 545–562.CrossRefGoogle Scholar
Tagg, R. 1994 The Couette–Taylor problem. Nonlinear Sci. Today 4, 1–25.Google Scholar
Tanner, R. 2000 Engineering Rheology. Oxford University Press.Google Scholar
Taylor, G.I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Trans. R. Soc. Lond. A 223, 289–343.Google Scholar
Teng, H., Liu, N., Lu, X. & Khomami, B. 2015 Direct numerical simulation of Taylor–Couette flow subjected to a radial temperature gradient. Phys. Fluids 27 (12), 125101.CrossRefGoogle Scholar
Topayev, S., Nouar, C., Bernardin, D., Neveu, A. & Bahrani, S.A. 2019 Taylor-vortex flow in shear-thinning fluids. Phys. Rev. E 100 (2), 023117.CrossRefGoogle ScholarPubMed
Watanabe, T. & Toya, Y. 2012 Vertical Taylor–Couette flow with free surface at small aspect ratio. Acta Mechanica 223 (2), 347–353.CrossRefGoogle Scholar
Wereley, S.T. & Lueptow, R.M. 1998 Spatio-temporal character of non-wavy and wavy Taylor–Couette flow. J. Fluid Mech. 364, 59–80.CrossRefGoogle Scholar
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