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Flow of viscoelastic fluids between rotating disks

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Abstract

Few boundary-value problems in fluid mechanics can match the attention that has been accorded to the flow of fluids, Newtonian and non-Newtonian, between parallel rotating disks rotating about a common axis or about distinct axes. An interesting feature which has been recently observed is the existence of solutions that are not axially symmetric even in the case of flow due to the rotation of disks about a common axis. In this article we review the recent efforts that have been expended in the study of both symmetric and asymmetric solutions in the case of both the classical linearly viscous fluid and viscoelastic fluids.

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References

  1. T.N.G. Abbot and K. Walters, Rheometrical flow systems, Part 2. Theory for the orthogonal rheometer, including an exact solution of the Navier-Stokes equations, J. Fluid Mech., 40, 205 (1970).

    Google Scholar 

  2. M.L. Adams and A.Z. Szeri, Incompressible flow between finite disks, Trans. ASME J. Appl. Mech., 49, 1 (1982).

    Google Scholar 

  3. R.C. Arora and V.K. Stokes, On the heat transfer between two rotating disks, Internat. J. Heat Mass Transfer, 25, 2119 (1972).

    Google Scholar 

  4. R.J. Atkin and R.E. Craine, Continuum theories of mixtures: applications, J. Internat. Math. Appl., 17, 153 (1976).

    Google Scholar 

  5. M. Balaram and B.R. Luthra, A numerical study of rotationally symmetric flow of second order fluid, J. Appl. Mech., 40, 685 (1973).

    Google Scholar 

  6. M. Balaram and K.S. Sastri, Rotational symmetric flow of non-Newtonian liquid in presence of infinite rotating disc, Arch. Mech. Stos., 17, 359 (1965).

    Google Scholar 

  7. B. Banerjee and A.K. Borkakati, Heat transfer in a hydrodynamic flow between eccentric disks rotating at different speeds, Z. Angew. Math. Phys., 33, 414 (1982).

    Google Scholar 

  8. G.K. Batchelor, Note on a class of solutions of the Navier-Stokes equations representing steady rotationally-symmetric flow, Quart. J. Mech. Appl. Math., 4, 29 (1951).

    Google Scholar 

  9. E.R. Benton, On the flow due to a rotating disk, J. Fluid Mech., 24, 781 (1966).

    Google Scholar 

  10. R. Berker, Sur quelques cas d'integration des equations due mouvement d'un fluide incompressible, Lille, Paris (1936).

  11. R. Berker, Integration des equations du mouvement d'um fluide visquex, incompressible, Handbuch der Physik, Vol. VIII/2, Springer-Verlag, Berlin (1963).

    Google Scholar 

  12. R. Berker, A new solution of the Navier-Stokes equation for the motion of a fluid contained between two parallel planes rotating about the same axis, Arch. Mech. Stos., 31, 265 (1979).

    Google Scholar 

  13. R. Berker, An exact solution of the Navier-Stokes equation, the vortex with curvilinear axis, Internat. J. Engrg. Sci., 20, 217 (1982).

    Google Scholar 

  14. B. Bernstein, E.A. Kearsley and L.J. Zapas, A study of stress relaxation with finite strain, Trans. Soc. Rheol., 7, 391 (1963).

    Google Scholar 

  15. P.L. Bhatnagar, On two-dimensional boundary layer in non-Newtonian fluids in constant coefficients of viscosity and cross-viscosity, Indian J. Math., 3, 95 (1961).

    Google Scholar 

  16. R.K. Bhatnagar, Secondary flow of an elastico-viscous fluid between two coaxial cones having the same vortex and rotating about a common axis, Proc. Nat. Acad. Sci. India, P.L. Bhatnagar Commemoration Volume, 39, 107 (1969).

    Google Scholar 

  17. R.K. Bhatnagar, Proc. Indian Acad. Sci., 58, 279 (1973).

    Google Scholar 

  18. R.K. Bhatnagar, Flow between coaxial rotating disks: with and without externally applied magnetic field, Internat. J. Math. Sci., 4, 181 (1981).

    Google Scholar 

  19. R.K. Bhatnagar, Numerical solutions for flow of an Oldroyd fluid confined between coaxial rotating disks, J. Rheol., 26, 19 (1983).

    Google Scholar 

  20. R.K. Bhatnagar and M.G.N. Perera Numerical solutions for flow of an Oldroyd fluid confined between coaxial rotating disks, J. Rheol., 26, 19 (1982).

    Google Scholar 

  21. R.K. Bhatnagar and J.V. Zago, Numerical investigations of flow of a viscoelastic fluid between rotating coaxial disks, Rheol. Acta, 17, 557 (1978).

    Google Scholar 

  22. J.L. Bleustein and A.E. Green, Dipolar fluids, Internat. J. Engrg. Sci., 5, 323 (1967).

    Google Scholar 

  23. L.L. Blyler and S.J. Kurtz, Analysis of the Maxwell orthogonal rheometer, J. Appl. Polym. Sci., 11, 127 (1967).

    Google Scholar 

  24. R.M. Bowen, Theory of Mixtures in Continuum Physics, Vol. III (ed. A.C. Eringen), p. 1 (1976).

    Google Scholar 

  25. M.V. Bower, K.R. Rajagopal, and A.S. Wineman, Flow of K-BKZ fluids between parallel plates rotating about distinct axes: shear thinning and inertial effects, J. Non-Newtonian Fluid Mech., 22, 289 (1987).

    Google Scholar 

  26. D.R. Caldwell and C.W. van Atta, Characteristics of Ekman boundary layer instabilities, J. Fluid Mech., 44, 79 (1970).

    Google Scholar 

  27. J.H. Cerutti, Collocation methods for systems of ordinary differential equations and parabolic partial differential equations, Ph.D. thesis, University of Wisconsin (1975).

  28. B.D. Coleman, Substantially stagnant motions, Trans. Soc. Rheol., 6, 293 (1962).

    Google Scholar 

  29. B.D. Coleman, A representation theorem for one constitutive equation of a simple material in motions with constant stretch history, Arch. Rational Mech. Anal., 20, 329 (1965).

    Google Scholar 

  30. R.E. Craine, Oscillations of a plate in a binary mixture of incomparable Newtonian fluids, Internat. J. Engrg. Sci., 9, 1177 (1971).

    Google Scholar 

  31. I. Crewther, R.R. Huilgol, and R. Josza, Axisymmetric and non-axisymmetric flows of a non-Newtonian fluid between co-axial rotating disks, Pre-print (personal communication).

  32. P.K. Currie, Constitutive equations for polymer melts predicted by Doi-Edwards and Curtiss-Bird kinetic theory models, J. Non-Newtonian Fluid Mech., 11, 53 (1982).

    Google Scholar 

  33. R.X. Dai, K.R. Rajagopal, and A.Z. Szeri, A numerical study of the flow of a K-BKZ fluid between plates rotating about non-coincident axes, J. Non-Newtonian Fluid Mech., 38, 289 (1991).

    Google Scholar 

  34. C. Devanathan, Flow of finitely conducting fluid between torsionally oscillating parallel infinite plane disks in the presence of uniform magnetic field perpendicular to the disks, Proc. Nat. Inst. Sci. India, 28, 747 (1962).

    Google Scholar 

  35. D. Dijkstra and G.J.F. van Heijst, The flow between finite rotating disks enclosed by a cylinder, J. Fluid Mech., 128, 123 (1983).

    Google Scholar 

  36. N. Dohara, The flow and heat transfer between a torsionally oscillating and a stationary disk, J. Engrg. Math., 15, 1 (1981).

    Google Scholar 

  37. N. Dohara, Bull. Iron Steel Tech. College, 15, 11 (1981).

    Google Scholar 

  38. R. Drouot, Sur un cas d'integration des equations du mouvement d'un fluide incompresseible du deuxieme ordre, C. R. Acad. Sci. Paris, 265A, 300 (1967).

    Google Scholar 

  39. J.E. Dunn and R.L. Fosdick, Thermodynamics, stability and boundedness of fluids of complexity two and fluids of second grade, Arch. Rational Mech. Anal., 56, 191 (1974).

    Google Scholar 

  40. J.E. Dunn and K.R. Rajagopal, A critical review and thermodynamic analysis of fluids of the differential type (to appear).

  41. A.R. Elcrat, On the swirling flow between rotating coaxial disks, J. Differential Equations, 18, 423 (1975).

    Google Scholar 

  42. A.R. Elcrat, On one flow between a rotating disk and a porous disk, Arch. Rational Mech. Anal., 73, 63 (1980).

    Google Scholar 

  43. M.E. Erdogan, Non-Newtonian flow due to non-coaxially rotations of a disk and a fluid at infinity, Z. Angew. Math. Mech., 56, 141 (1967).

    Google Scholar 

  44. A.C. Eringen, Simple microfluids, Internat. J. Engrg. Sci, 2, 205 (1964).

    Google Scholar 

  45. A.C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16, 1 (1966).

    Google Scholar 

  46. A.J. Faller, An experimental study of the instability of the laminar Ekman boundary layer, J. Fluid Mech., 15, 560 (1963).

    Google Scholar 

  47. A.J. Faller and R.E. Kaylor, Dynamics of Fluids and Plasmas (ed. S.I. Pai), Academic Press, New York (1966).

    Google Scholar 

  48. R.L. Fosdick and K.R. Rajagopal, Anomalous features in the model of second order fluids, Arch Rational Mech. Anal., 70, 145 (1978).

    Google Scholar 

  49. R.L. Fosdick and K.R. Rajagopal, Uniqueness and drag for fluids of second grade in steady motion, Internat J. Non-Linear Mech., 13, 131 (1979).

    Google Scholar 

  50. R.L. Fosdick and K.R. Rajagopal, Thermodynamics and stability of fluids of third grade, Proc. Roy. Soc. London Ser. A, 339, 351 (1980).

    Google Scholar 

  51. J.D. Goddard, The dynamics of simple fluids in steady circular shear, Quart. Appl. Math., 31, 107 (1983).

    Google Scholar 

  52. J.D. Goddard, J.B. Melville, and K. Zhang, Similarity solutions for stratified rotating-disk flow, presented at the Annual Meeting of the A.I.Ch.E., Chicago (1985) and the Amer. Phys. Soc. Div. of Fluid Dynamics, Tuscon (1985).

  53. S. Gogus, The steady flow of a binary mixture between two rotating non-coaxial disks, Internat. J. Engrg. Sci., (to appear).

  54. C. Goldstein and W.R. Schowalter, Further studies of fluid non-linearity: the orthogonal rheometer and the oscillating sphere, Trans. Soc. Rheol., 19, 1 (1975).

    Google Scholar 

  55. P.S. Granville, J. Slip Res., 17, 181 (1973).

    Google Scholar 

  56. D. Greenspan, Numerical studies of flow between rotating coaxial disks, J. Inst. Math. Applic., 9, 370 (1972).

    Google Scholar 

  57. H.P. Greenspan, Theory of Rotating Fluids, Cambridge University Press, Cambridge (1968).

    Google Scholar 

  58. H.P. Greenspan and L.N. Howard, On a time dependent motion of a rotating fluid, J. Fluid Mech., 17, 385 (1963).

    Google Scholar 

  59. N. Gregory, J.J. Stuart and W.S. Walker, On the stability of three dimensional boundary layers with application to the flow due to a rotating disk, Philos. Trans. Roy. Soc. London Ser. A, 248, 155 (1955).

    Google Scholar 

  60. S.P. Hastings, On existence theorems for some problems from boundary layer theory, Arch. Rational Mech. Anal., 38, 308 (1970).

    Google Scholar 

  61. G.H. Hoffman, Extension of perturbation series by computer: viscous flow between two infinite rotating disks, J. Comput. Phys., 16, 240 (1974).

    Google Scholar 

  62. M. Holodniok, M. Kubicek, and V. Halvacek, Computation of flow between two rotating coaxial disks, J. Fluid Mech., 81, 680 (1978).

    Google Scholar 

  63. M. Holodniok, M. Kubicek, and V. Hlavacek, Computation of flow between two rotating coaxial disks, J. Fluid Mech., 108, 227 (1981).

    Google Scholar 

  64. R.R. Huilgol, On the properties of the motion with constant stretch history occurring in the Maxwell rheometers, Trans. Soc. Rheol., 13, 513 (1969).

    Google Scholar 

  65. R.R. Huilgol and J.B. Keller, Flow of viscoelastic fluids between rotating disks, Part 1, J. Non-Newtonian Fluid Mech., 18, 107 (1975).

    Google Scholar 

  66. R.R. Huilgol and K.R. Rajagopal, Non-axisymmetric flow of a viscoelastic fluid bertween rotating disks, J. Non-Newtonian Fluid Mech., 23, 423 (1987).

    Google Scholar 

  67. Z.H. Ji, K.R. Rajagopal, and A.Z. Szeri, Multiplicity of solutions in von Karman flows of viscoelastic fluids, J. Non-Newtonian Fluid Mech., 36, 1 (1990).

    Google Scholar 

  68. P.N. Kaloni, Several comments on the paper “Some remarks on useful theorems for second order fluid,” J. Non-Newtonian Fluid Mech., 36, 70 (1990).

    Google Scholar 

  69. P.N. Kaloni and A.M. Siddiqui, A note on the flow of a viscoelastic fluid between eccentric disks, J. Non-Newtonian Fluids, 26, 125 (1987).

    Google Scholar 

  70. S.R. Kasivishwanathan and M.V. Gandhi, A class of exact solution for the magnetohydrodynamic flow of a micropolar fluid (to appear).

  71. S.R. Kasivishwanathan and A.R. Rao, Exact solutions for one unsteady flow and heat transfer between eccentrically rotating disks, Internat. J. Engrg. Sci., 27, 731 (1989).

    Google Scholar 

  72. A. Kaye, Note No. 134, College of Aeronautics, Cranfield Institute of Technology (1962).

  73. H.B. Keller, Applications of Bifurcation Theory (ed. P. Rabinowitz), Academic Press, New York (1977).

    Google Scholar 

  74. H.B. Keller and R.K.-H. Szeto, Calculations of flow between rotating disks, Computing Methods in Applied Sciences and Engineering (ed. R. Glowinski and J.L. Lions), North-Holland, Amsterdam, p. 51 (1980).

    Google Scholar 

  75. D.G. Knight, Flow between eccentric disks rotating at different speeds: inertia effects, Z. Angew. Math. Phys., 31, 309 (1980).

    Google Scholar 

  76. R. Kobayashi, Y. Kohama, and C. Takamadate, Spiral vortices in boundary layer transition regime on rotating disk, Acta Mech., 35, 71 (1980).

    Google Scholar 

  77. H.O. Kreiss and S.V. Parter, On the swirling flow between rotating coaxial disks, asymptotic behavior, I & II, Proc. Roy. Soc. Edinburgh Sect. A, 90 (3–4), 293 (1981).

    Google Scholar 

  78. H.O. Kreiss and S.V. Parter, On the swirling flow between rotating coaxial disks: existence and non-uniqueness, Comm. Pure Appl. Math., 36, 35 (1983).

    Google Scholar 

  79. C.Y. Lai, K.R. Rajagopal and A.Z. Szeri, Asymmetric flow between parallel rotating disks, J. Fluid Mech., 146, 203 (1984).

    Google Scholar 

  80. C.Y. Lai, K.R. Rajagopal, and A.Z. Szeri, Asymmetric flow above a rotating disk, J. Fluid Mech., 157, 471 (1985).

    Google Scholar 

  81. G.N. Lance and M.H. Rogers, The axially symmetric flow of a viscous fluid between two infinite rotating disks, Proc. Roy. Soc. London Ser. A, 266, 109 (1961).

    Google Scholar 

  82. G. MacSithigh, R.L. Fosdick, and K.R. Rajagopal, A plane non-linear shear for an elastic layer with a non-convex stored energy function, Internat. J. Solids and Structures, 22, 1129 (1986).

    Google Scholar 

  83. B.J. Matkowsky and W.L. Siegmann, The flow between counter-rotating disks at high Reynolds number, SIAM J. Appl. Math., 30, 720 (1976).

    Google Scholar 

  84. B. Maxwell and R.P. Chartoff, Studies of a polymer melt in an orthogonal rheometer, Trans. Soc. Rheol., 9, 51 (1965).

    Google Scholar 

  85. J.B. McLeod and S.V. Parter, On the flow between two counter-rotating infinite plane disks, Arch. Rational. Mech. Anal., 54, 301 (1974).

    Google Scholar 

  86. G.L. Mellor, P.J. Chapple, and V.K. Stokes, On the flow between a rotating and a stationary disk, J. Fluid Mech., 31, 95 (1968).

    Google Scholar 

  87. S.N. Murthy and R.K.P. Ram, MHD and head transfer due to eccentric rotations of a porous disk and a fluid at infinity, Internat. J. Engrg. Sci., 16, 943 (1978).

    Google Scholar 

  88. N.D. Nguyen, J.P. Ribault, P. Florent, Multiple solutions for the flow between coaxial disks, J. Fluid Mech., 68, 369 (1975).

    Google Scholar 

  89. W. Noll, Motions with constant stretch history, Arch. Rational Mech. Anal., 11, 97 (1962).

    Google Scholar 

  90. J.G. Oldroyd, On the formulation of rheological equations of state, Proc. Roy. Soc. London Ser. A, 200, 523 (1950).

    Google Scholar 

  91. M.R. Osborne, On shooting methods for boundary value problems, J. Math. Anal. Appl., 27, 417 (1969).

    Google Scholar 

  92. S.V. Parter, On the swirling flow between rotating coaxial disks: a survey, Theory and Applications of Singular Perturbations, Proc. of a Conf., Oberwolfach, 1981 (ed. W. Eckhaus and E.M. Dejager). Lecture Notes in Mathematics, Vol. 942, Springer-Verlag, Berlin, pp. 258–280 (1982).

    Google Scholar 

  93. S.V. Parter and K.R. Rajagopal, Swirling flow between rotating plates, Arch. Rational Mech. Anal., 86, 305 (1984).

    Google Scholar 

  94. C.E. Pearson, Numerical solutions for time-dependent viscous flow between two rotating coaxial disks, J. Fluid Mech., 21, 623 (1965).

    Google Scholar 

  95. H.J. Pesch and P. Rentrop, Numerical solutions of the flow between two-counter rotating coaxial disks, J. Fluid Mech., 21, 623 (1965).

    Google Scholar 

  96. P.J. Peucheux and C. Boutin, Study of a laminar flow of two non-mixable liquids between two coaxial disks rotating in opposite senses, Z. Angew. Math. Phys., 36, 238 (1985).

    Google Scholar 

  97. N. Phan-Thien, Coaxial disk flow and flow about a rotating disk of a Maxwellian fluid, J. Fluid Mech., 128, 427 (1983).

    Google Scholar 

  98. N. Phan-Thien, Coaxial disk flow of an Oldroyd B-fluid: exact solution and stability, J. Non-Newtonian Fluid Mech, 13, 325 (1983).

    Google Scholar 

  99. K.G. Picha and E.R.C. Eckert, Study of the air flow between coaxial disks rotating with arbitrary velocities in an open or enclosed space, Proc. U.S. 3rd National Congress on Applied Mechanics, p. 791 (1958).

  100. K.R. Rajagopal, The flow of a second order fluid between rotating parallel plates. J. Non-Newtonian Fluid Mech., 9, 185 (1981).

    Google Scholar 

  101. K.R. Rajagopal, On the flow of a simple fluid in an orthogonal rheometer, Arch. Rational Mech. Anal., 79, 29 (1982).

    Google Scholar 

  102. K.R. Rajagopal, On the creeping flow of the second order fluid, J. Non-Newtonian Fluid Mech., 15, 239 (1984).

    Google Scholar 

  103. K.R. Rajagopal, A class of exact solutions to the Navier-Stokes equations, Internat. J. Engrg. Sci., 22, 451 (1984).

    Google Scholar 

  104. K.R. Rajagopal and A.S. Gupta, Flow and stability of second grade fluids between two parallel rotating plates, Arch. Mech. Stos., 33, 663 (1981).

    Google Scholar 

  105. K.R. Rajagopal and A.S. Gupta, Flow and stability of a second grade fluid between two parallel rotating plates about non-coincident axes, Internat. J. Engrg. Sci., 19, 1401 (1985).

    Google Scholar 

  106. K.R. Rajagopal and P.N. Kaloni, Continuum Mechanics and Its Applications, Hemisphere Press, Washington, DC (1989).

    Google Scholar 

  107. K.R. Rajagopal, P.N. Kaloni, and L. Tao, Longitudinal and torsional oscillations of a solid cylinder in a simple fluid, J. Math. Phys. Sci, 23, 445 (1989).

    Google Scholar 

  108. K.R. Rajagopal, M. Renardy, Y. Renardy, and A.S. Wineman, Flow of viscoelastic fluids between plates rotating about distinct axes, Rheol. Acta, 25, 459 (1986).

    Google Scholar 

  109. K.R. Rajagopal and A.S. Wineman, A class of exact solutions for the flow of a viscoelastic fluid, Arch. Mech. Stos., 35, 747 (1983).

    Google Scholar 

  110. K.R. Rajagopal and A.S. Wineman, Flow of a BKZ fluid in an orthogonal rheometer, J. Rheol., 27, 509 (1983).

    Google Scholar 

  111. K.R. Rajagopal and A.S. Wineman, New exact solutions in non-linear elasticity, Internat. J. Engrg. Sci., 23, 217 (1985).

    Google Scholar 

  112. K.R. Rajagopal and A.S. Wineman, On a class of deformation of a material with non-convex stored energy function, J. Structural Mech., 12, 471 (1984–85).

    Google Scholar 

  113. A.R. Rao and S.R. Kasivishwanathan, On exact solution of the unsteady Navier-Stokes equation—the vortex with instantaneous curvilinear axis, Internat. J. Engrg. Sci, 25, 337 (1987).

    Google Scholar 

  114. A.R. Rao and S.R. Kasivishwanathan, A class of exact solutions for the flow of a micropolar fluid, Internat. J. Engrg. Sci., 25, 443 (1987).

    Google Scholar 

  115. P.R. Rao and A.R. Rao, Heat transfer in a MHD flow between eccentric disks rotating at different speeds, Z. Angew. Math. Phys., 34, 549 (1983).

    Google Scholar 

  116. H. Rasmussen, High Reynolds number flow between two infinite rotating disks, J. Anst. Math. Soc., 12, 483 (1971).

    Google Scholar 

  117. M. Reiner, A mathematical theory of dilatancy, Amer. J. Math., 67, 350 (1945).

    Google Scholar 

  118. E. Reshotko and R.L. Rosenthal, Laminar flow between two infinite disks, one rotating and one stationary, Israel J. Tech., 9, 93 (1971).

    Google Scholar 

  119. W. Rheinboldt, Solutions fields of non-linear equations and continuation methods, SIAM J. Numer. Anal., 17, 221 (1982).

    Google Scholar 

  120. W. Rheinboldt and J.V. Burkhardt, A locally parameterized continuation process, ACM Trans. Math. Softward, 9, 215 (1983).

    Google Scholar 

  121. R.S. Rivlin and J.L. Ericksen, Stress deformation relations for isotropic materials, J. Rational Mech. Anal., 4, 323 (1955).

    Google Scholar 

  122. S.M. Roberts and J.S. Shipman, Computation of the flow between a rotating and stationary disk, J. Fluid Mech., 73, 53 (1976).

    Google Scholar 

  123. H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York (1979)

    Google Scholar 

  124. D. Schultz and D. Greenspan, Simplification and improvement of a numerical method for Navier-Stokes problems, Proc. Colloquium on Differential Equations, Kesthaly, Hungary, p. 201 (1974).

    Google Scholar 

  125. V.P. Sharma, Flow and heat transfer due to small torsional oscillations of a disk about a constant mean, Acta Mech., 32, 19 (1979).

    Google Scholar 

  126. A. Sirivat, An experimental study of stability and transition of flow between rotating disks, Fluid Dynamics Res., (to appear).

  127. A. Sirivat, Stability of flow between stationary and rotating disks, Phys. Fluids, (to appear).

  128. A. Sirivat, K.R. Rajagopal, and A.Z. Szeri, An experimental investigation of one flow of non-Newtonian fluids between rotating disks, J. Fluid Mech., 186, 243 (1988).

    Google Scholar 

  129. A.C. Srivatsava, The effect of magnetic field on the flow between two non-parallel planes, Quart. J. Math. Appl. Mech., 14, 353 (1961).

    Google Scholar 

  130. K. Stewartson, On the flow between two rotating coaxial disks, Proc. Cambridge Philos. Soc., 49, 333 (1953).

    Google Scholar 

  131. J.T. Stuart, On the effects of uniform suction on the steady flow due to a rotating disk, Quart. J. Mech. Appl. Math. 7, 446 (1954).

    Google Scholar 

  132. A.Z. Szeri, A. Giron, S.J. Schneider, and H.N. Kaufmann, Flow between rotating disks, Part I. Basic flow, J. Fluid Mech., 134, 133 (1983).

    Google Scholar 

  133. A.Z. Szeri, C. Lai, and A. Kayhan, Rotating disk with uniform suction in streaming flow, Internat. J. Numer. Methods Fluids, 6, 175 (1986).

    Google Scholar 

  134. A.Z. Szeri, S.J. Schneider, F. Labbe, and H.N. Kaufmann, Flow between rotating disks, Part I. Basic flow, J. Fluid Mech., 134, 103 (1983).

    Google Scholar 

  135. K.K. Tam, A note on the asymptotic solution of the flow between two oppositely rotating infinite plane disks, SIAM J. Appl. Math., 17, 1305 (1969).

    Google Scholar 

  136. P.R. Tatro and E.L. Mollo-Christensen, Experiments on Ekman layer instability, J. Fluid Mech., 28, 531 (1967).

    Google Scholar 

  137. C. Thornley, On Stokes and Rayleigh layers in rotating system, Quart. J. Mech. Appl. Math., 21, 451 (1968).

    Google Scholar 

  138. C. Truesdell, A First Course in Rational Continuum Nechanics, Academic Press, New York (1977).

    Google Scholar 

  139. C. Truesdell, Rational Thermodynamics, Springer-Verlag, New York, (1984).

    Google Scholar 

  140. C. Truesdell and W. Noll, The non-linear field theories of mechanics, Handbuch der Physik, Vol. III/3 (ed. Flugge), Springer-Verlag, Berlin (1975).

    Google Scholar 

  141. T. von Karman, Uber laminare und turbulente Reibung, Z. Angew. Math. Mech., 1, 232 (1921).

    Google Scholar 

  142. M.H. Wagner, Analysis of time dependent non-linear stress growth data for shear and elongational flow of a low density branched polyethylene line melt, Rheol. Acta, 15, 133 (1976).

    Google Scholar 

  143. W.P. Walsh, On the flow of a non-Newtonian fluid between rotating, coaxial disks, Z. Angew. Math. Phys., 38, 495 (1987).

    Google Scholar 

  144. C.C. Wang, A representation theorem for the constitutive equation of a simple material in motions with constant stretch history, Arch. Rational Mech. Anal., 20, 329 (1965).

    Google Scholar 

  145. A.M. Watts, On the von Karman equations for axi-symmetric flow, Appl. Math. Preprint No. 74, University of Queensland (1974).

  146. P.D. Weidman, On the spin-up and spin-down of a rotating fluid, Part 2. Measurements and instability, J. Fluid Mech., 77, 709 (1976).

    Google Scholar 

  147. K.H. Well, Note on a problem by Lanee and a problem by Bellman, J. Math. Anal. Appl., 40, 258 (1972).

    Google Scholar 

  148. L.O. Wilson and N.L. Schryer, Flow between a stationary and rotating disk with suction, J. Fluid Mech., 85, 479 (1978).

    Google Scholar 

  149. P.J. Zandbergen and D. Dijkstra, Von Karman swirling flows, Annual Rev. Fluid Mech., 19, 465 (1987).

    Google Scholar 

  150. Zhang and J.D. Goddard, Inertial and elastic effects in circular shear (ERD) flow of a visoelastic fluids, J. Non-Newtonian Fluid Mech., 33, 233 (1989).

    Google Scholar 

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Communicated by M.Y. Hussaini

The support of the Air Force Office of Scientific Research is gratefully acknowledged.

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Rajagopal, K.R. Flow of viscoelastic fluids between rotating disks. Theoret. Comput. Fluid Dynamics 3, 185–206 (1992). https://doi.org/10.1007/BF00417912

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