Current issues in non-Newtonian flows: A mathematical perspective

Journal of Non-Newtonian Fluid Mechanics

Volumn 90, Issue 2-3, 2000, Pages 243-259

  Renardy, Michael ^a  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

Author keywords

Non Newtonian fluid; Viscoelastic flows; Weissenberg number

Indexed keywords

EQUATIONS OF MOTION; MATHEMATICAL MODELS; NON NEWTONIAN FLOW; VISCOELASTICITY;

WEISSENBERG NUMBER;

NON NEWTONIAN LIQUIDS;

MATHEMATICAL ANALYSIS; NON-NEWTONIAN FLOW; VISCOELASTIC FLOW;

EID: 0034029356     PISSN: 03770257     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0257(99)00081-6     Document Type: Review

References (105)

1. Akbay U., Frischmann G., Wasner J. Stabilität der ebenen Couette Strömung eines Maxwell-Fluids und ihre kritische Weissenbergzahl. Rheol. Acta. 26:1987;119-126.
2. Avgousti M., Beris A.N. Non-axisymmetric modes in viscoelastic Taylor-Couette flow. J. Non-Newton Fluid Mech. 50:1993;225-251.
3. Avgousti M., Beris A.N. Viscoelastic Taylor-Conette flow: bifurcation analysis in the presence of symmetries. Proc. Roy. Soc. London A. 443:1993;17-37.
4. Avgousti M., Liu B., Beris A.N. Spectral methods for the viscoelastic time-dependent flow equations with applications to Taylor-Conette flow. Int. J. Numer. Meth. Fluids. 17:1993;49-74.
5. Beard D.W., Davies M.H., Walters K. The stability of elastico-viscous flow between rotating cylinders, Part 3. J. Fluid Mech. 24:1966;321-334.
6. Beris A.N., Avgousti M., Souvaliotis A. Spectral calculations of viscoelastic flows: evaluation of the Giesekus constitutive equation in model flow problems. J. Non-Newton Fluid Mech. 44:1992;197-228.
7. Byars J.A., Oztekin A., Brown R.A., McKinley G.H. Spiral instabilities in the flow of highly elastic fluids between parallel disks. J. Fluid Mech. 271:1994;173-218.
8. Chen K. Interfacial instability due to elastic stratification in concentric coextrusion of two viscoelastic fluids. J. Non-Newton Fluid Mech. 40:1991;155-175.
9. Chen K., Joseph D.D. Elastic shortwave instability in extrusion flows of viscoelastic liquids. J. Non-Newton Fluid Mech. 42:1992;189-211.
10. Chen K. Elastic instability of the interface in Couette flow of viscoelastic liquids. J. Non-Newton Fluid Mech. 40:1991;261-267.
11. Chen K., Zhang Y. Stability of the interface in co-extrusion flow of two viscoelastic fluids through a pipe. J. Fluid Mech. 247:1993;489-502.
12. P. Chossat and G. Iooss, The Couette-Taylor Problem, Springer 1994.
13. Coates P.J., Armstrong R.C., Brown R.A. Calculation of the steady state viscoelastic flow through axisymmetric contraction with the EEME formulation. J. Non-Newton Fluid Mech. 42:1992;141-188.
14. Crewther I., Huilgol R.R., Josza R. Axisymmetric and non-axisymmetric flows of a non-Newtonian fluid between coaxial rotating disks. Phil. Trans. Roy. Soc. London A. 337:1991;467-495.
15. Davies A.R., Devlin J. On corner flows of Oldroyd-B fluids. J. Non-Newton Fluid Mech. 50:1993;173-191.
16. Dean W.R., Montagnon P.E. On the steady motion of viscous liquid in a corner. Proc. Cambridge Phil. Soc. 45:1949;389-394.
17. Denn M.M. Issues in viscoelastic fluid mechanics. Ann. Rev. Fluid Mech. 22:1990;13-34.
18. Espafiol P., Yuan X.F., Ball R.C. Shear banding flow in the Johnson-Segalman fluid. J. Non-Newton Fluid Mech. 65:1996;93-109.
19. Graham M.D. Wall slip and the nonlinear dynamics of large amplitude oscillatory shear flows. J. Rheol. 39:1995;697-712.
20. Green III T. Oscillating convection in an elasticoviscous liquid. Phys. Fluids. 11:1968;1410-1412.
21. Guillopé C., Saut J.C. Existence and stability of steady flows of weakly viscoelastic fluids. Proc. Roy. Soc. Edinburg. 119A:1991;137-158.
22. Hagen T., Renardy M. Boundary layer analysis of the Phan-Thien-Tanner and Giesekus model in high Weissenberg number flow. J. Non-Newton Fluid Mech. 73:1997;181-189.
23. Hatzikiriakos S.G., Dealy J.M. Wall slip of molten high density polyethylene I. Sliding plate rheometer studies. J. Rheol. 35:1991;497-523.
24. E. Hille, R.S. Phillips, Functional Analysis and Semigroups, American Mathematical Society, 1957.
25. Hinch E.J. The flow of an Oldroyd B fluid around a sharp corner. J. Non-Newton Fluid Mech. 50:1993;161-171.
26. Hunter J., Slemrod M. Viscoelastic flow exhibiting hysteretic phase changes. Phys. Fluids. 26:1983;2345-2351.
27. Ji Z., Rajagopal K.R., Szeri A.Z. Multiplicity of solutions in von Karman flows of viscoelastic fluids. J. Non-Newton Fluid Mech. 36:1990;1-25.
28. Joo Y.L., Shaqfeh E.S.G. The effects of inertia on the viscoelastic Dean, The effects of inertia on the viscoelastic Dean and Taylor-Couette flow instabilities with application to coating flows. Phys. Fluids A. 4:1992;2415-2431.
29. Joo Y.L., Shaqfeh E.S.G. A purely elastic instability in Dean and Taylor-Dean flow. Phys. Fluids A. 4:1992;524-543.
30. Joseph D.D., Renardy M., Saut J.C. Hyperbolicity, Hyperbolicity and change of type in the flow of viscoelastic fluids. Arch. Rat. Mech. Anal. 87:1985;213-251.
31. D.D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer, 1990.
32. D.D. Joseph, Y.Y. Renardy, Fundamentals of Two-Fluid Dynamics, vol. 2, Springer, 1993.
33. Khayat R.E. Non-linear overstability in the thermal convection of viscoelastic fluids. J. Non-Newton Fluid Mech. 58:1995;331-356.
34. Khayat R.E. Chaos and overstability in the thermal convection of viscoelastic fluids. J. Non-Newton Fluid Mech. 53:1994;227-255.
35. Khomami B., Wilson G.M. An experimental investigation of interfacial instability in superposed flow of viscoelastic fluids in a converging/diverging channel geometry. J. Non-Newton Fluid Mech. 58:1995;47-65.
36. C. Kim, G. Baek, Overstability of viscoelastic fluid heated from below, in: P. Moldenaers, R. Keunings, Theoretical and Applied Rheology, Elsevier 1992, pp. 189-191.
37. King R.C., Apelian M.R., Armstrong R.C., Brown R.A. Numerically stable finite element techniques for viscoelastic calculations in smooth and singular geometries. J. Non-Newton Fluid Mech. 29:1988;147-216.
38. Koch H., Tataru D. On the spectrum of hyperbolic semigroups. Comm. Part. Duff. Equat. 20:1995;901-937.
39. Kolkka R.W., lerley G.R. On the convected linear stability of a viscoelastic Oldroyd-B fluid heated from below. J. Non-Newton Fluid Mech. 25:1987;209-237.
40. Kolkka R.W., lerley G.R. Spurt phenomena for the Giesekus viscoelastic liquid model. J. Non-Newton Fluid Mech. 33:1989;305-323.
41. Kolkka R.W. Spurt phenomena of the Johnson-Segalman fluid and related models. J. Non-Newton Fluid Mech. 29:1988;303-335.
42. Kwon Y., Leonov A.I. Stability constraints in the formulation of viscoelastic constitutive equations. J. Non-Newton Fluid Mech. 58:1995;25-46.
43. Larson R.G., Shaqfeh E.S.G., Muller S.J. A purely elastic instability in Taylor-Couette flow. J. Fluid Mech. 218:1990;573-600.
44. Larson R.G. Instabilities in viscoelastic flows. Rheol. Acta. 31:1992;213-263.
45. Lee K.C., Finlayson B.A. Stability of plane Poisenille and Couette flow of a Maxwell fluid. J. Non-Newton Fluid Mech. 21:1986;65-78.
46. Liang S.F., Acrivos A. Experiments on bouyancy driven convection in non-Newtonian fluids. Rheol. Acta. 9:1970;447-455.
47. Lunsmann W.J., Genieser L., Armstrong R.C., Brown R.A. Finite element analysis of steady viscoelastic flow around a sphere in a tube: calculations with constant viscosity models. J. Non-Newton Fluid Mech. 48:1993;63-99.
48. Malkus D.S., Nohel J.A., Plohr B.J. Dynamics of shear flow of a non-Newtonian fluid. J. Comp. Phys. 87:1990;464-487.
49. Malkus D.S., Nohel J.A., Plohr B.J. Analysis of new phenomena in shear flow of non-Newtonian fluids. SIAM J. Appl. Math. 51:1991;899-929.
50. D.S. Malkus, J.A. Nohel, B.J. Plohr, Oscillations in piston-driven shear flow of a non-Newtonian fluid, in: J.F. Dijksman and G.D.C. Kuiken (Eds.), IUTAM Symposium on Numerical Simulation of Non-Isothermal Flow of Viscoelastic Liquids, Kluwer, 1995.
51. McKinley G.H., Byars J.A., Brown R.A., Armstrong R.C. Observations on the elastic instability in cone and plate and parallel plate flows of a polyisobutylene Boger fluid. J. Non-Newton Fluid Mech. 40:1991;201-229.
52. McKinley G.H., Oztekin A., Byars J.A., Brown R.A. Self-similar spiral instabilities in elastic flows between a cone and a plate. J. Fluid Mech. 285:1995;123-164.
53. Muller S.J., Larson R.G., Shaqfeh E.S.G. A purely elastic transition in Taylor-Couette flow. Rheol. Acta. 28:1989;499-503.
54. Muller S.J., Shaqfeh E.S.G., Larson R.G. Experimental studies of the onset of instability in viscoelastic Taylor-Couette flow. J. Non-Newton Fluid Mech. 46:1993;315-330.
55. Nohel J.A., Pego R.L., Tzavaras A.E. Stability of discontinuous steady states in shearing motions of a non-Newtonian fluid. Proc. Roy. Soc. Edinburgh. 115:1990;39-59.
56. Northey P.J., Armstrong R.C., Brown R.A. Finite-amplitude time-periodic states in viscoelastic Taylor-Couette flow described by the UCM model. J. Non-Newton Fluid Mech. 42:1992;117-139.
57. Olagunju D.O. Instabilities and bifurcations of von-Karman similarity solutions in swirling viscoelastic flow. Z. angew. Math. Phys. 46:1995;224-238.
58. Olagunju D.O. Elastic instabilities in cone-and-plate flow: small gap theory. Z. Angew. Math. Phys. 46:1995;946-959.
59. Olagunin D.O. Hopf bifurcation in creeping cone-and-plate flow of a viscoelastic fluid. Z. Angew. Math. Phys. 48:1997;361-369.
60. Olagunju D.O., Cook L.P. Linear stability analysis of cone and plate flow of an Oldroyd-B fluid. J. Non-Newton Fluid Mech. 47:1993;93-105.
61. Oztekin A., Brown R.A. Instability of a viscoelastic fluid between rotating parallel disks. J. Fluid Mech. 255:1993;473-502.
62. Park H.M., Lee H.S. Nonlinear hydrodynamic stability of viscoelastic fluids heated from below. J. Non-Newton Fluid Mech. 60:1995;1-26.
63. Park H.M., Lee H.S. Hopf bifurcations of viscoelastic fluids heated from below. J. Non-Newton Fluid Mech. 66:1996;1-34.
64. J.R.A. Pearson, C.J.S. Petrie, On melt flow instability of extruded polymers, in: R.E. Wetton, R.H. Whorlow (Eds.), Polymer Systems: Deformation and Flow, Macmillan, 1968, pp. 163-187.
65. Phan-Thien N. Coaxial-disk flow of an Oldroyd-B fluid: exact solution and stability. J. Non-Newton Fluid Mech. 13:1983;325-340.
66. Phan-Thien N. Cone and plate flow of an Oldroyd-B fluid is unstable. J. Non-Newton Fluid Mech. 17:1985;37-44.
67. 0-semigroups. Trans. Am. Math. Soc. 284:1984;847-857.
68. M. Renardy, W.J. Hrusa, J.A. Nohel, Mathematical Problems in Viscoelasticity, Lougman, 1987.
69. Renardy M. The high Weissenberg number limit of the UCM model and the Euler equations. J. Non-Newton Fluid Mech. 69:1997;293-301.
70. Renardy M. High Weissenberg number boundary layers for the upper convected Maxwell fluid. J. Non-Newton Fluid Mech. 68:1997;125-132.
71. Renardy M. The stresses of an upper convected Maxwell fluid in a Newtonian velocity field near a reentrant corner. J. Non-Newton Fluid Mech. 50:1993;127-134.
72. Renardy M. How to integrate the upper convected Maxwell (UCM) stresses near a reentrant corner (and maybe elsewhere, too). J. Non-Newton Fluid Mech. 52:1994;91-95.
73. Renardy M. A matched solution for corner flow of the upper convected Maxwell fluid. J. Non-Newton Fluid Mech. 58:1995;83-89.
74. Renardy M. Reentrant corner behavior of the PTT fluid. J. Non-Newton Fluid Mech. 69:1997;99-104.
75. Renardy M., Renardy Y. Linear stability of plane Couette flow of an upper convected Maxwell fluid. J. Non-Newton Fluid Mech. 22:1986;23-33.
76. Renardy M., Renardy Y. Pattern selection in the Bénard problem for a viscoelastic fluid. Z. Angew. Math. Phys. 43:1992;154-180.
77. Renardy M. A centre manifold theorem for hyperbolic PDEs. Proc. Roy. Soc. Edinburgh. 122A:1992;363-377.
78. o-semigroups. Comm. Part. Diff. Eq. 18:1993;1299-1307.
79. Renardy M. On the linear stability of hyperbolic PDEs and viscoelastic flows. Z. Angew. Math. Phys. 45:1994;854-865.
80. Renardy M. Nonlinear stability of Jeffreys fluids at low Weissenberg numbers. Arch. Rat. Mech. Anal. 132:1995;37-48.
81. Renardy M. Spectrally determined growth is generic. Proc. Am. Math. Soc. 124:1996;2451-2453.
82. Renardy M., Renardy Y., Sureshkumar R., Beris A.N. Hopf-Hopf and steady-Hopf mode interactions in Taylor-Couette flow of an upper convected Maxwell liquid. J. Non-Newton Fluid Mech. 63:1996;1-31.
83. Renardy M. Short wave instabilities resulting from memory slip. J. Non-Newton Fluid Mech. 35:1990;73-76.
84. M. Renardy, Asymptotic structure of the stress field in flow past a cylinder at high Weissenberg number, J. Non-Newton Fluid Mech, to appear.
85. Renardy M., Renardy Y. Instability due to second normal stress jump in two-layer shear flow of a Giesekus fluid. J. Non-Newtonian Fluid Mech. 81:1999;215-234.
86. Renardy Y. Spurt and instability in a two-layer Johnson-Segalman liquid. Theor. Comp. Fluid Dyn. 7:1995;463-475.
87. Renardy Y. Weakly nonlinear behavior of periodic disturbances in two-layer plane channel flow of upper convected Maxwell liquids. J. Non-Newton Fluid Mech. 56:1995;101-126.
88. Renardy Y. Stability of the interface in two-layer Couette flow of upper convected Maxwell liquids. J. Non-Newton Fluid Mech. 28:1988;99-115.
89. Roseublat S. Thermal convection in a viscoelastic liquid. J. Non-Newton Fluid Mech. 21:1986;201-223.
90. Shaqfeh E.S.G., Muller S.J., Larson R.G. The effects of gap width and dilute solution properties on the viscoelastic Taylor-Conette instability. J. Fluid Mech. 235:1992;285-317.
91. Singh P., Leal L.G. Finite element simulation of flow around a 3π/2 corner using the FENE dumbbell model. J. Non-Newton Fluid Mech. 58:1995;279-313.
92. Sokolov M., Tanner R.I. Convective stability of a general viscoelastic fluid heated from below. Phys. Fluids. 15:1972;534-539.
93. Spenley N.A., Cates M.E., McLeish T.C.B. Nonlinear rheology of wormlike micelles. Phys. Rev. Lett. 71:1993;939-942.
94. Su Y.-Y., Khomami Y.B. Stability of multilayer power law and second-order fluids in plane Poisenille flow. Chem. Eng. Comm. 109:1991;209-223.
95. Su Y.-Y., Khomami Y.B. Interfacial stability of multilayer viscoelastic fluids in slit and converging channel die geometries. J. Rheol. 36:1992;357-387.
96. Su Y.-Y., Khomami Y.B. Purely elastic interfacial instabilities in superposed flow of polymeric fluid. Rheol. Acta. 31:1992;413-420.
97. Sureshkumar R., Beris A.N., Avgousti M. Non-axisymmetric subcritical bifurcations in viscoelastic Taylor-Couette flow. Proc. Roy. Soc. London A. 447:1994;135-153.
98. R.I. Tanner, Constitutive equations for the computing person, in: A.S. Lodge, M. Renardy, J.A. Nohel (Eds.), Viscoelasticity and Rheology, Academic Press, 1985, pp. 421-439.
99. Thomas R.H., Walters K. The stability of elastico-viscous flow between rotating cylinders, Part I. J. Fluid Mech. 18:1964;33-43.
100. Thomas R.H., Walters K. The stability of elastico-viscous flow between rotating cylinders, Part 2. J. Fluid Mech. 19:1964;557-560.
101. Trefethen L.N., Trefethen A.E., Reddy S.C., Driscoll T.A. Hydrodynamic stability without eigenvalues. Science. 261:1993;578-584.
102. Vest C.M., Arpaci V.A. Overstability of a viscoelastic fluid layer heated from below. J. Fluid Mech. 36:1969;613-623.
103. A.L. Yarin, Free Liquid Jets and Films: Hydrodynamics and Rheology, Lougman, 1993.
104. 0-semigroups. Bull. Acad. Polon. Sci., Ser. Sci. Math. 23:1975;895-898.
105. Zieliuska B.J., Demay Y. Couette-Taylor instability in viscoelastic fluids. Phys. Rev. A. 38:1988;897-903.