A low-dimensional approach to nonlinear plane-Couette flow of viscoelastic fluids

Physics of Fluids

Volumn 12, Issue 2, 2000, Pages 345-365

  Ashrafi, Nariman ^a   Khayat, Roger E ^a  

^a UNIVERSITY OF WESTERN ONTARIO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

COUETTE FLOW; MATHEMATICAL ANALYSIS; SHEAR THINNING; VISCOELASTIC FLUID;

EID: 0034038534     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.870313     Document Type: Article

References (61)

1. G. R. Sell, C. Foias, and R. Temam, Turbulence in Fluid Flows: A Dynamical Systems Approach (Springer, New York, 1993).
2. P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, Cambridge, 1996).
3. E. N. Lorenz, "Deterministic nonperiodic flow," J. Atmos. Sci. 20, 130 (1963).
4. H. N. Shirer and R. Wells, Mathematical Structure of the Singularities at the Transition Between Steady States in Hydrodynamic Systems (Springer, Heidelberg, 1980).
5. G. Veronis, "Motions at subcritical values of the Rayleigh number in a rotating fluid," J. Fluid Mech. 24, 545 (1966).
6. J. B. McLaughlin and P. C. Martin, "Transition to turbulence in a statically stressed fluid system," Phys. Rev. A 12, 186 (1975).
7. J. H. Curry, "A generalized Lorenz system," Commun. Math. Phys. 60, 193 (1978).
8. J. A. Yorke and E. D. Yorke, in Hydrodynamic Instabilities and the Transition to Turbulence, edited by H. L. Swinney and J. P. Gollub (Springer, Berlin, 1981).
9. Y. Pomeau and P. Manneville, "Intermittent transition to turbulence in dissipative systems," Commun. Math. Phys. 74, 189 (1980).
10. C. Sparrow, The Lorenz Equations (Springer, New York, 1983).
11. H. Yahata, "Temporal development of the Taylor vortices in a rotating field. 2," Prog. Theor. Phys. 61, 791 (1979).
12. H. Yahata, "Temporal development of the Taylor vortices in a rotating field. 2," Prog. Theor. Phys. 61, 791 (1979).
13. H. Kuhlmann, "Model for Taylor-Couette flow," Phys. Rev. A 32, 1703 (1985).
14. H. Kuhlmann, D. Roth, and M. Luke, "Taylor flow and harmonic modulation of the deriving force," Phys. Rev. A 39, 745 (1988).
15. P. Chossat and G. Iooss, The Couette-Taylor Problem (Springer, New York, 1991).
16. P. R. Fenstermacher, H. L. Swinney, and J. P. Gollub, "Dynamical instabilities and the transition to chaotic Taylor vortex flow," J. Fluid Mech. 94, 103 (1979).
17. M. Gorman and H. L. Swinney, "Spatial and temporal characteristics of modulated waves in the circular Couette system," J. Fluid Mech. 117, 123 (1982).
18. A. E. Deane, I. G. Keverkidis, G. E. Karniakadis, and S. A. Orszag, "Low-dimensional models for complex flows: Application to grooved channels and circular cylinders," Phys. Fluids A 3, 2337 (1991).
19. E. Stone, "A study of low-dimensional models for the wall region of a turbulent layer," Ph.D. thesis, Cornell University, 1989.
20. X. Zheng and M. N. Glauser, "A low-dimensional description of the axisymmetric jet mixing layer," ASME Computers Eng. 2, 211 (1990).
21. R. E. Khayat, "Chaos and overstability in the thermal convection of viscoelastic fluids," J. Non-Newtonian Fluid Mech. 53, 227 (1994).
22. R. E. Khayat, "Nonlinear overstability in the thermal convection of viscoelastic fluids," J. Non-Newtonian Fluid Mech. 58, 331 (1995).
23. R. E. Khayat, "Fluid elasticity and transition to chaos in thermal convection," Phys. Rev. E 51, 380 (1995).
24. R. E. Khayat, "Onset of chaos in the thermal convection of weakly shear thinning fluids," J. Non-Newtonian Fluid Mech. 63, 153 (1996).
25. R. E. Khayat, "Onset of Taylor vortices and chaos in viscoelastic fluids," Phys. Fluids 7, 2191 (1995).
26. R. E. Khayat, "Low-dimensional approach to nonlinear overstability of purely elastic Taylor-vortex flow," Phys. Rev. Lett. 78, 4918 (1997).
27. N. Ashrafi and R. E. Khayat, "Finite amplitude Taylor-vortex flow of weakly shear-thinning fluids," Phys. Rev. E (in press).
28. R. E. Khayat, "Finite-amplitude Taylor-vortex flow of viscoelastic fluids," J. Fluid Mech. 400, 33 (1999).
29. R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, 2nd Ed. (Wiley, New York, 1987), Vol. 1.
30. S. J. Muller, E. S. J. Shaqfeh, and R. G. Larson, "Experimental study of the onset of oscillatory instability in viscoelastic Taylor-Couette flow," J. Non-Newtonian Fluid Mech. 46, 315 (1993).
31. G. V. Vinogradov, A. Ya Malkin, Yu G. Vanovskii, E. K. Borisenkova, B. V. Yarlykov, and G. v. Berezhneya, J. Polym. Sci. 10, 1061 (1972).
32. M. M. Denn, "Issues in viscoelastic fluid mechanics," Annu. Rev. Fluid Mech. 22, 13 (1990).
33. R. G. Larson, "Instabilities in viscoelastic flows," Rheol. Acta 31, 213 (1992).
34. R. W. Kolkka, D. S. Malkus, M. G. Hansen, and G. R. Ierley, "Spurt phenomena of the Johnson-Segalman fluid and related models," J. Non-Newtonian Fluid Mech. 29, 303 (1988).
35. D. S. Malkus, J. A. Nohel, and B. J. Plohr, "Dynamics of shear flow of a non-Newtonian fluid," J. Comput. Phys. 87, 464 (1990).
36. Y. Y. Renardy, "Spurt and instability in a two-layer Johnson-Segalman Liquid," Theor. Comput. Fluid Dyn. 7, 463 (1995).
37. G. C. Georgiou and D. Vlassopoulos, "On the stability of the simple shear flow of a Johnson-Segalman fluid," J. Non-Newtonian Fluid Mech. 75, 77 (1998).
38. M. D. Graham, "Effect of axial flow on viscoelastic Taylor-Couette instability," J. Fluid Mech. 360, 341 (1998).
39. V. V. Ramanan, K. A. Kumar, and M. D. Graham, "Stability of viscoelastic shear flows subjected to steady or oscillatory transverse flow," J. Fluid Mech. 379, 255 (1999).
40. R. G. Larson, E. S. G. Shaqfeh, and S. J. Muller, "A purely elastic instability in Taylor-Couette flow," J. Fluid Mech. 218, 573 (1990).
41. E. S. Shaqfeh, S. J. Muller, and R. G. Larson, "The effect of gap width and dilute solution properties on the viscoelastic Taylor-Couette instabilities," J. Fluid Mech. 235, 285 (1992).
42. P. J. Northey, R. C. Armstrong, and R. A. Brown, "Finite-amplitude time-periodic states in viscoelastic Taylor-Couette flow described by the UCM model," J. Non-Newtonian Fluid Mech. 42, 117 (1992).
43. M. Avgousti, B. Liu, and A. N. Beris, "Spectral methods for the viscoelastic time-dependent flow equations with applications to Taylor-Couette flow," Int. J. Numer. Methods Fluids 17, 49 (1993).
44. M. Avgousti and A. N. Beris, "Non-axisymmetric modes in viscoelastic Taylor-Couette flow," J. Non-Newtonian Fluid Mech. 50, 225 (1993).
45. R. Sureshkumar, A. N. Beris, and M. Avgousti, "Non-axisymmetric subcritical bifurcations in viscoelastic Taylor-Couette flow," Proc. R. Soc. London, Ser. A 447, 135 (1994).
46. M. Renardy, Y. Renardy, R. Sureshkumar, and A. N. Beris, "Hopf-Hopf and steady-Hopf mode interactions in Taylor-Couette flow of an upper convected Maxwell liquid," J. Non-Newtonian Fluid Mech. 63, 1 (1996).
47. B. M. Baumert and S. J. Muller, "Axisymmetric and non-axisymmetric elastic and inertio-elastic instabilities in Taylor-Couette flow," J. Non-Newtonian Fluid Mech. 83, 33 (1999).
48. M. W. Johnson and D. Segalman, "A model for viscoelastic fluid behavior which allows non-affine deformation" J. Non-Newtonian Fluid Mech. 2, 278 (1977).
49. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, 1983).
50. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, New York, 1990).
51. A. H. Nayfeh and B. Balachadran, Applied Nonlinear Dynamics (Wiley, New York, 1995).
52. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover, New York, 1961).
53. R. E. Khayat and A. Derdouri, "Inflation of hyperelastic cylindrical membranes as applied to blow moulding. I. Axisymmetric case," Int. J. Numer. Methods Eng. 37, 3773 (1994).
54. J. Carr, Applications of the Center Manifold Theory (Springer, New York, 1981).
55. A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981).
56. P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge University Press, Cambridge, 1981).
57. C. E. Grosch and S. Harold, "The stability of steady and time-dependent plane Poiseuille flow," J. Fluid Mech. 34, 177 (1968).
58. R. G. Larson, E. S. G. Shaqfeh, and S. J. Muller, "A purely elastic instability in Taylor-Couette flow," J. Fluid Mech. 218, 573 (1990).
59. R. M. Clever and F. H. Busse, "Tertiary and quaternary solutions for plane Couette flow," J. Fluid Mech. 344, 137 (1997).
60. M. Renardy and Y. Renardy, "Linear stability of plane Couette flow of an upper convected Maxwell fluid" J. Non-Newtonian Fluid Mech. 22, 23 (1986).
61. F. J. Lim and W. R. Schowalter, "Pseudo-spectral analysis of the stability of pressure-driven flow of a Giesekus fluid between parallel planes," J. Non-Newtonian Fluid Mech. 26, 135 (1987).