Abstract
Order is found within the chaotic nonlinear flow between rotating coaxial cylinders. The flow stability analysis is carried out for a pseudoplastic fluid through bifurcation diagram and Lyapunov exponent histogram. The fluid is assumed to follow the Carreau–Bird model, and mixed boundary conditions are imposed. The low-order dynamical system, resulted from Galerkin projection of the conservation of mass and momentum equations, includes additional nonlinear terms in the velocity components originated from the shear-dependent viscosity. It is observed that the base flow loses its radial flow stability to the vortex structure at a lower critical Taylor number, as the shear-thinning effects increase. The emergence of the vortices corresponds to the onset of a supercritical bifurcation, which is also seen in the flow of a linear fluid. However, unlike the Newtonian case, shear-thinning Taylor vortices lose their stability as the Taylor number reaches a second critical number corresponding to the onset of a Hopf bifurcation. Complete flow field together with viscosity maps are given for different scenarios in the bifurcation diagram.
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References
Taylor G.I.: Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. Royal Soc. Lond. Ser. A 223, 289–343 (1923)
Ashrafi N., Khayat R.: Shear-thinning-induced chaos in Taylor-Couette flow. Phys. Rev. E 61(2), 1455–1467 (2000)
Kuhlmann H.: Model for Taylor Couette flow. Phys. Rev. A 32(3), 1703–1707 (1985)
Hoffmann C., Altmeyer S., Pinter A., Lücke M.: Transitions between Taylor vortices and spirals via wavy Taylor vortices and wavy spirals. New J. Phys. 11, 1–24 (2009)
Lorenz E.N.: Deterministic nonperiodic flows. J. Atmos. Sci. 20, 130 (1963)
Berger H.R.: Mode analysis of Taylor-Couette flow in finite gaps. ZAMM Z. Angew. Math. Mech. 79(2), 91–96 (1999)
Kuhlmann H., Roth D., Lücke M.: Taylor flow and harmonic modulation of the driving force. Phys. Rev. A 39, 745 (1988)
Sparrow C.: The Lorenz Equations. Springer, New York (1983)
Li Z., Khayat R.: A non-linear dynamical system approach to finite amplitude Taylor-Vortex flow of shear-thinning fluids. Int. J. Numer. Meth. Fluids. 45, 321–340 (2004)
Baumert B.M., Muller S.J.: Flow visualization of the elastic Taylor-Couette flow in Boger fluids. Rheol. Acta 34, 147 (1995)
Ashrafi N., Binding D.M., Walters K.: Cavitation effects in eccentric-cylinder flows of Newtonian and non-Newtonian Fluids. Chem. Eng. Sci. 56, 5565–5574 (2001)
Dusting J., Balbani S.: Mixing in a Taylor-Couette reactor in the non-wavy regime. Chem. Eng. Sci 64, 3103–3111 (2009)
Larson R.G.: Instabilities in viscoelastic flows. Rheol. Acta 31, 213 (1992)
Khayat R., Ashrafi N.: A Low-dimensional approach to nonlinear plane-Poiseulle flow of viscoelastic fluids. Phys. Fluids 14(5), 1757–1767 (2002)
Escudier M.P., Gouldson I.W., Jonset D.M.: Taylor vortices in Newtonian and shear-thinning liquids. Proc. R. Soc. Lond. A 449, 155–176 (1995)
Bird R.B., Curtiss C.F., Armstrong R.C., Hassager O.: Dynamics of Polymeric Liquids, vol. 1. 2nd edn. Wiley, New York (1987)
Larson R.G., Shaqfeh E.S.G., Muller S.J.: A purely elastic instability in Taylor-Couette flow. J. Fluid Mech. 218, 573 (1990)
Khellaf K., Lauriat G.: Numerical study of heat transfer in a non-Newtonian Carreau-fluid between rotating concentric vertical cylinders. J. Non-Newtonian Fluid Mech. 89, 45–61 (2000)
Pascal J.P., Rasmussen H.: Stability of power law fluid flow between rotating cylinders. Dyn. Syst 10, 65–93 (1995)
Veronis G.: Motions at subcritical values of the Rayleigh number in a rotating fluid. J. Fluid Mech 24, 545 (1966)
Drazin P.G., Reid W.H.: Hydrodynamic Stability. Cambridge University press, Cambridge (1981)
Criminale W.O., Jackson T.L., Joslin R.D.: Theory and Computation in Hydrodynamic Stability. Cambridge University press, Cambridge (2003)
Yorke, J.A., Yorke, E.D.: Hydrodynamic Instabilities and the Transition to turbulence. In: Swinney, H.L., Gollub, J.P. (eds.) Springer, Berlin (1981)
Yahata H.: Temporal development of the Taylor vortices in a rotating field 1. Prog. Theor. Phys. 59, 1755 (1978)
Thomas R.H., Walters K.: The stability of elastico-viscous flow between rotating cylinders. Part 1. J. Fluid Mech. 18, 33 (1964)
Coronado-Matutti O., Souza Mendes P.R., Carvalho M.S.: Instability of inelastic shear-thinning liquids in a Couette flow between concentric cylinders. J. Fluid Eng. 126, 385–390 (2004)
Pirro D., Quadrio M.: Direct numerical simulation of turbulent Taylor-Couette flow. Eur. J Mech. B Fluids 27, 552–566 (2008)
Altmeyer S., Hoffmann Ch., Heise M., Abshagen J., Pinter A., Lucke M., Pfister G.: Wall effects on the transitions between Taylor vortices and spiral vortices. Phys. Rev. E 81, 066313 (2010)
Argyris J., Faust G., Haase M.: An Exploration of Chaos. Elsevier Science B.V., Amsterdam (1994)
Wiggins S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. 2nd edn. Springer, New York (2003)
Berge P., Pomeau Y., Vidal C.: Order within Chaos. Hermann and Wiley, Paris (1984)
Hilborn R.C.: Chaos and Nonlinear Dynamics. Oxford University Press, Oxford (2000)
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Ashrafi, N. Order in chaotic pseudoplastic flow between coaxial cylinders. Arch Appl Mech 82, 809–825 (2012). https://doi.org/10.1007/s00419-011-0594-0
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DOI: https://doi.org/10.1007/s00419-011-0594-0