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Flow classification for viscoelastic materials

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Abstract

Flow classification is a broad field that can be approached in a variety of forms. In the present context, the basic idea is to search for a criterion that enables one to establish rheological material functions in steady and unsteady flows in the Lagrangean sense. This is still an open problem but there is some recent progress on the subject that is worthy of review. In this connection, the idea of persistence of straining has been used to understand and interpret simple flows and complex flow fields. We present a brief history of persistence of straining and the attempts to capture this concept in a rational definition.We also show some other criteria present in the literature to capture an analogous concept involving non-kinematic quantities as primitive for the flow classification. In this last type of criterion, special attention is given to persistence of stressing. New perspectives on anisotropic and history related measurers are also given.

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References

  1. Adrian R.J., Twenty years of particle image velocimetry, Exp. Fluids, 39, 159–169 (2005).

    Article  Google Scholar 

  2. Armfield S.W. and Street R.L., Fractional step methods for the navier-stokes equations on non-staggered grids, ANZIAM J., 42(E), C134–C156 (2000).

    MathSciNet  Google Scholar 

  3. Astarita G., Two dimensionless groups relevant in the analysis of steady flows of viscoelastic materials, Ind. Eng. Chem. Fund., 6, 257–(1967).

    Article  Google Scholar 

  4. Astarita G., Objective and generally applicable criteria for flow classification, J. Non-Newt. Fluid Mech., 6, 69–76 (1979).

    Article  Google Scholar 

  5. Astarita G., Reply to R. R. Huilgol, J. Non-Newt. Fluid Mech., 7, 97–99 (1980).

    Article  Google Scholar 

  6. Astarita G., Quasi-newtonian constitutive equations exhibiting flow-type sensitivity, J. Rheology, 35, 687–689 (1991).

    Article  Google Scholar 

  7. Astarita G. and Marrucci G., Elliptical flows of elastic liquids. In Proc. 1st Int. Congr. Theor. Appl. Mech., AIMETA, Udine (1971) Springer

    Google Scholar 

  8. Astarita G. and Marrucci G., Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill, Maidenhead (1974).

    Google Scholar 

  9. Astarita G. and Marucci P., Principles of Non-Newtonian fluid mechanics. McGraw-Hill, Boston (1974).

    Google Scholar 

  10. Bacchi R.D.A., Thompson R. and Machado F.J., Response of available criteria for vortex identification on some chaotic and turbulent flows. In COBEM, editor, Proc. Brazilian Congress on Mechanical Science and Engineering 2009, pages 56–63. ABCM (2009).

  11. Barnes H.A., The yield stress — a review, J. Non-Newt. Fluid Mech., 81, 133–178 (1999).

    Article  MATH  Google Scholar 

  12. Berger S.A. and Jou L-D., Flows in stenotic vessels, Annual Review of Fluid Mechanics, 32, 347–382 (2000).

    Article  MathSciNet  Google Scholar 

  13. Binding D.M., An approximate analysis for contraction and converging flows, J. Non-Newt. Fluid Mech., 27, 173–189 (1988).

    Article  MATH  Google Scholar 

  14. Binding D.M., Further considerations of axisymmetric contraction flows, J. Non-Newt. Fluid Mech., 41, 27–42 (1991).

    Article  MATH  Google Scholar 

  15. Binding D.M. and Walters K., On the use of flow through a contraction in estimating the extensional viscosity of mobile polymer solutions, J. Non-Newt. Fluid Mech., 30, 233–250 (1988).

    Article  Google Scholar 

  16. Bird R.B., Armstrong R.C. and Hassager O., Dynamics of Polymeric Liquids. John Wiley & Sons, New York, second edition, (1987) vol. 1.

    Google Scholar 

  17. Boulanger P., Hayes M., and Rajagopal K.R., Some unsteady exact solutions in the Navier-Stokes and the second grade fluid theories, SAACM, 1(2), 185–204 (1991).

    MathSciNet  Google Scholar 

  18. Boussinesq M.J., Imprimerie Nationale, Paris, XXIII (1877).

  19. Brunn P.O., Flow classification criteria and its use in general flows, Rheol. Acta, 46, 171–181 (2006).

    Article  Google Scholar 

  20. Brunn P.O. and Ryssel E., The w-D fluid: general theory with special emphasis on stationary two dimensional flows, Continuum Mech. Thermodyn., 9, 73–82 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  21. Brunn P.O. and Ryssel E., The giesekus fluid in w-d form for steady two-dimensional flows. part i. theory, Rheol. Acta, 38, 415–422 (1999).

    Article  Google Scholar 

  22. Carpenter M.H., Compact finite difference schemes with spectral-like resolution, J. Comp. Phys., 103(1), 16–42 (1992).

    Google Scholar 

  23. Carpenter M.H., The stability of numerical boundary treatments for compact high-order finite-difference schemes, J. Comp. Phys., 108(2), 272–295 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  24. Chauré A., Sur les mouvements à histoire de déformation constante, C. R. Acad. Sc. Paris, Série A, t. 274, 1839–1842 (1972).

    MATH  Google Scholar 

  25. Chella R. and Ottino J.M., Stretching in some classes of fluid motions and asymptotic mixing efficiencies as a measure of flow classification, Arch. Ration. Mech. Anal., 90, 15–42 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  26. Cintra J. and Tucker C.L., Microstructural evolution in polymer blends, J. Rheology, 34, 177–210 (1995).

    Google Scholar 

  27. Cogswell F.N., Converging flow of polymer melts in extrusion dies, Polym. Eng. Sci., 12, 64–73 (1972).

    Article  Google Scholar 

  28. Coleman B.D., Kinematical concepts with application in the mechanics and thermodynamics of incompressible viscoelastic fluids, Arch. Rat. Mech. Anal., 9, 273–300 (1962).

    MATH  Google Scholar 

  29. Coleman B.D., On the use of symmetry to simplify the constitutive equations of isotropic material with memory, Proc. Roy. Soc. A, 306, 449–476 (1968).

    Article  MATH  Google Scholar 

  30. Coleman B.D. and Noll W., Steady extension of incompressible simple fluids, Phys. Fluids, 5, 840–843 (1962).

    Article  MATH  MathSciNet  Google Scholar 

  31. Dash R.K., Jayaraman G. and Mehta K.N., Flow in a catheterized curved artery with stenosis, Journal of Biomechanics, 32, 49–61 (1999).

    Article  Google Scholar 

  32. Dealy J.M., Official nomenclature for material functions describing the response of a viscoelastic fluid to various shearing and extensional deformations, J. Rheol., 39(1), 253–265 (1995).

    Google Scholar 

  33. Dean L.W., Broadband noise generated by airfoils in turbulent flow. Number 71-587 in AIAA Paper (1971).

  34. Doi M., Molecular dynamics and rheological properties of concentrated-solutions of rodlike polymers in isotropic and liquid-crystalline phases, J. Polym. Sci. B: Polym. Phys., 19, 545–571 (1981).

    Google Scholar 

  35. Doshi S.R. and Dealy J.M., Exponential shear: a strong flow, J. Rheol., 31(7), 563–582 (1987).

    Article  Google Scholar 

  36. Dresselhaus E. and Tabor M., The persistence of strain in dynamical systems, J. Phys. A: Math. Gen., 22, 971–984 (1989).

    Article  MathSciNet  Google Scholar 

  37. Dresselhaus E. and Tabor M., The kinematics of stretching and alignment of material elements in general flow fields, J. Fluid Mech., 236, 415–444 (1991).

    Article  MathSciNet  Google Scholar 

  38. Drouot R. and Lucius M., Approximation du second ordre de la loi de comportement des fluides simples. Lois classiques déduites de l’introduction d’un nouveau tenseur objectif, Archiwum Mechaniki Stosowanej, 28(2), 189–198 (1976).

    MATH  MathSciNet  Google Scholar 

  39. Druot R., Definition du transport associe a un modele de fluide de deuxieme ordre, C.R. Acad. Sci. Paris, Serie A, 282, 923–929 (1976).

    Google Scholar 

  40. Engvall J., Ask P., Loyd D., and Wranne B.B., Coarctation of the aorta-a theoretical and experimental analysis of the effects of a centrally located arterial stenosis, Medical & Biological Engineering & Computing, 29, 291–296 (1991).

    Article  Google Scholar 

  41. Fattal R. and Kupferman R., Constitutive laws for matrixlogarithm of the conformation tensor, J. Non-Newt. Fluid Mech., 123, 281–285 (2004).

    Article  MATH  Google Scholar 

  42. Frank F.C. and Mackley M.R., Localized flow birefringence of polyethylene oxide solutions in a two roll mill, J. Polymer Science, 14, 1121–1131 (1976).

    Google Scholar 

  43. Fuller G.G., Optical Rheometry of Complex Fluids. Oxford University Press (1995).

  44. Gavrilakis S., Numerical simulation of low-reynolds-number turbulent flow through a straight square duct, J. Fluid Mech., 244, 101–116 (1992).

    Article  Google Scholar 

  45. Gavrilakis S., Turbulent velocity structures derived from pod analyses, Institut de Machines Hydrauliques et de Mécanique des Fluides, École Polytechnique Fédérale de Lausanne., Report T-93-30, C134–C156 (1993).

  46. Giesekus H., Die rheologische zustandsgleichung elasto-viskoser flüssigkeiten — insbesondere von weissenberg-flüssigkeiten — für allgemeine und stationäre fließvorgänge, Z. Angen. Math. Mech., 42, 32–61 (1962).

    Article  MATH  Google Scholar 

  47. Giesekus H., Stromungen mit konstantem geschwindigkeitsgradienten und die bewgung von darin suspendierten teilchen, teil ii. ebene strömungen und eine experimentelle anordnung zu ihrer realisierung, Rheol. Acta, 2, 32–61 (1962).

    Google Scholar 

  48. Gordon R.J. and Schowalter W.R., Anisotropic fluid theory: A different approach to the dumbbell theory of dilute polymer solutions, Trans. Soc. Rheol., 16(1), 79–97 (1972).

    Article  Google Scholar 

  49. Harnoy A., Stress relaxation effect in elastico-viscous lubricants in gears and rollers, J. Fluid Mech., 76(3), 501–517 (1976).

    Article  MATH  Google Scholar 

  50. Hazel A., An exciting theory, J. Fluid Mech. (submitted) (1998).

  51. Herbert T., Bertolotti F.T. and Santos G.R., Floquet analysis of secondary instability in shear flows. In Dwoyer D.L. and Hussaini M.Y., editors, Proc. ICASE/NASA Workshop on Stability of Time-dependent Spatially Varying Flows, pages 43–57. Springer (1987).

  52. Hinch E.J. and Leal L.G., Constitutive equations in suspension mechanics. part i. general formulation, J. Fluid Mech., 71, 481–495 (1975).

    Article  MATH  Google Scholar 

  53. Huilgol R.R., A class of motions with constant stretch history, Quart. Appl. Math., XXIX(1), 1–15 (1970).

    MathSciNet  Google Scholar 

  54. Huilgol R.R., Algorithms for motions with constant stretch history, Rheol. Acta, 15, 120–129 (1976).

    Article  MATH  Google Scholar 

  55. Huilgol R.R., Nearly extensional flows, J. Non-Newt. Fluid Mech., 5, 219–231 (1979).

    Article  MATH  Google Scholar 

  56. Huilgol R.R., Viscoelastic fluid theories based on the left cauchygreen tensor history, Rheol. Acta, 18, 451–455 (1979).

    Article  MATH  Google Scholar 

  57. Huilgol R.R., Comments on “Objective and generally applicable criteria for flow classification,” by Astarita G., J. Non-Newt. Fluid Mech., 7, 91–95 (1980).

    Article  Google Scholar 

  58. Huilgol R.R., On the zorawski’s condition for a velocity field in one frame to be steady in a second frame, Arch. Ration. Mech. Anal., 76, 183–191 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  59. Huilgol R.R., Intrinsically unsteady viscometric and quasiviscometric flows, J. Non-Newt. Fluid Mech., 87, 117–125 (1999).

    Article  MATH  Google Scholar 

  60. Huilgol R.R. and Phan-Thien N., Recent advances in continuum mechanics of viscoelastic liquids, Int. J. Engng. Sci, 24(2), 161–251 (1986).

    Article  MathSciNet  Google Scholar 

  61. Huilgol R.R. and Phan-Thien N., Fluid Mechanics of Viscoelasticity, Number 6 in Rheology Series. Elsevier (1997).

  62. Hunt J.C.R., Wray A.A. and Moin P., Eddies, streams and convergence zones in turbulent flows, Proc. 1988 Summer Program of the Center for Turbulence Research, NASA Ames/Stanford University: 193–207 (1988).

  63. Huser A. and Biringen S., Direct numerical simulation of turbulent flow in a square duct, J. Fluid Mech., 257, 65–77 (1993).

    Article  MATH  Google Scholar 

  64. Hussain A.K.M.F., Coherent structures-reality and myth, Physics of Fluids., 26(10), 2816–2850 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  65. Hussain A.K.M.F., Coherent structures and turbulence, Journal of Fluid Mech., 173, 303–356 (1986).

    Article  Google Scholar 

  66. James D.F. and Walters K., Measurement of the extensional viscosity of M1 in a converging channel rheometer, Journal of Non-Newtonian Fluid Mechanics, 35, 445–458 (1990).

    Article  Google Scholar 

  67. Johnson M.W. and Segalman D., A model for viscoelastic fluid behavior which allows non-affine deformation, J. Non-Newt. Fluid Mech., 2, 291–318 (1977).

    Article  Google Scholar 

  68. Joshi Y.M. and Denn M., Rupture of entangled polymeric liquids in elongational flow, J. Rheol., 47(1), 291–298 (2002).

    Google Scholar 

  69. Kachanov Y.S., Kozlov V.V., and Levchenko V.Y., Noninear development of a wave in a boundary layer, Izv. Akad. Nauk SSSR Mekh. Zhid. Gaza, 3, 49–58 (1977) in Russian.

    Google Scholar 

  70. Khakhar D.V. and Ottino J.M., A note on the linear vector model of Olbricht, Rallinson and Leal as applied to the breakup of slender axisymmetric drops, J. Non-Newt. Fluid Mech., 21, 127–131 (1986).

    Article  Google Scholar 

  71. Ku D.N., Blood flow in arteries, Annual Review of FluidMechanics, 29, 399–434 (1997).

    Article  MathSciNet  Google Scholar 

  72. Larson R., Constitutive equations for polymer melts and solutions. Butterworths, Boston (1988).

    Google Scholar 

  73. Larson R.G., A constitutive equation for polymer melts based on partially extending strand convection, J. Rheology, 28, 545–571 (1984).

    Article  MATH  Google Scholar 

  74. Laso M. and Ottinger H., Calculation of viscoelastic flow using molecular models — the connffessit approach, J. Non-Newt. Fluid Mech., 47, 1–20 (1993).

    Article  MATH  Google Scholar 

  75. Lim W.L., Chew Y.T., Chew T.C. and Low H.T., Steady flow dynamics of prosthetic aortic heart valves: a comparative evaluation with piv techniques, Journal of Biomechanics, 31, 411–421 (1998).

    Article  Google Scholar 

  76. Lodge A.S. and Meissner J., Comparison of network theory predictions with stress/time data in shear and elongation for a low-density polyethylene melt, Rheol. Acta, 12, 41–47 (1973).

    Article  Google Scholar 

  77. Lumley L., Drag reduction by additives, Annual Review of Fluid Mechanics, 1, 367–384 (1969).

    Article  Google Scholar 

  78. Luo X.Y. and Kuang Z.B., Non-Newtonian flow patterns associated with an arterial stenosis, Journal of Biomechanical Engineering, 114, 512–514 (1992).

    Article  Google Scholar 

  79. Mackay M.E. and Astarita G., Analysis of entry flow to determine elongational flow properties, J. Non-Newt. Fluid Mech., 70, 219–235 (1997).

    Article  Google Scholar 

  80. Malkin Ya A. and Petrie C.J.S., Some conditions for rupture of polymeric liquids in extension, J. Rheol., 41(7), 1–25 (1997).

    Article  Google Scholar 

  81. Masad J.A. and Nayfeh A.H., Effect of heat transfer on the instability of compressible boundary layers, Phys. Fluids A, 3, 2148–2163 (1991).

    Article  MATH  Google Scholar 

  82. Masad J.A. and Nayfeh A.H., Effect of suction on the instability of compressible boundary layers, Phys. Fluids A, 3, 2179–2190 (1991).

    Article  MATH  Google Scholar 

  83. Maslowe S.A., Shear flow instability and transition. In Swinney H.L. and Gollub J.T., editors, Hydrodynamic Instability and the Transition to Turbulence, chapter 7, pages 181–228. Springer, New York, second edition (1985).

    Google Scholar 

  84. McKinley G.H. and Sridhar T., Filament-stretching rheometry of complex fluids, Annual Reviews of Fluid Mechanics, 38, 375–415 (2002).

    Article  MathSciNet  Google Scholar 

  85. Mompean G., Thompson R.L. and Souza Mendes P.R., A general transformation procedure for differential constitutive equations, J. Non-Newt. Fluid Mech., 111, 151–174 (2003).

    Article  MATH  Google Scholar 

  86. Moser R.D., Kim J. and Mansour N.N., Direct numerical simulation of turbulent channel flow up to re τ = 590, Physics of Fluids., 11(4), 943–945 (1999).

    Article  Google Scholar 

  87. Naghdi P.M., Constitutive restrictions for idealized elasticviscoplastic materials, J. Appl. Mech., 51, 93–101 (1984).

    Article  MATH  Google Scholar 

  88. Noll W., Motions with constant stretch history, Arch. Ration. Mech. Anal., 2, 197–226 (1958).

    Article  MATH  Google Scholar 

  89. Noll W., Motions with constant stretch history, Arch. Rational Mech. Anal., 11, 97–104 (1962).

    Article  MATH  MathSciNet  Google Scholar 

  90. Olbricht W.L., Rallinson J.M. and Leal L.G., Strong flow criteria based on microstructure deformation, J. Non-Newt. Fluid Mech., 10, 255–270 (1982).

    Article  Google Scholar 

  91. Oldroyd J.G., On the formulation of rheological equations of state, Proc. Royal Soc. A, 200, 523–591 (1950).

    Article  MATH  MathSciNet  Google Scholar 

  92. Pasquali M. and Scriven L.E., Theoretical modeling of microstructured liquids: A simple thermodynamic approach, J. Non-Newt. Fluid Mech., 120, 101–135 (2004).

    Article  MATH  Google Scholar 

  93. Petrie C.J.S., One hundred years of extensional flow, J. Non-Newt. Fluid Mech., 137, 1–14 (2006a).

    Article  MATH  Google Scholar 

  94. Petrie C.J.S., Extensional viscosity: a critical discussion, J. Non-Newt. Fluid Mech., 137, 15–23 (2006b).

    Article  MATH  Google Scholar 

  95. Picart C., Piau J.M. and Galliard H., Human blood shear yield stress and its hematocrit dependence, J. Rheology, 42, 1–12 (1998).

    Article  Google Scholar 

  96. Pipkin A.C., Controllable viscometric flows, Quart. Appl. Math., 26, 87 (1968).

    MATH  MathSciNet  Google Scholar 

  97. Pipkin A.C. and Owen D.R., Nearly viscometric flows, Phys. Fluids, 10, 836 (1967).

    Article  Google Scholar 

  98. Pountney D.C. and Walters K., Nearly extensional flows, Phys. Fluids, 21, 1482–1484 (1978).

    Article  Google Scholar 

  99. Prandtl L., Attaining a steady air stream in wind tunnels, Technical Report 726, NACA (1933).

  100. Rajagopal K.R., On a flow of a simple fluid in an orthogonal rheometer, Arch. Rational Mech. Anal., 79, 39–47 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  101. Rajagopal K.R. and Carroll M.M., A note on an unsteady flow, Int. J. Engng. Sci, 26(7), 649–652 (1988).

    Article  Google Scholar 

  102. Rajagopal K.R., On implicit constitutive theories, Applic.Maths., 48, 279–319 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  103. Rajagopal K.R., On implicit constitutive theories for fluids, J. Fluid Mech., 550, 243–249 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  104. Rajagopal K.R., Private communications (2006).

  105. Rajagopal K.R. and Srinivasa A.R., Mechanics of the inelastic behavior of materials — part i, theoretical underpinnings, Int. J. Plasticity, 14, 945–967 (1998).

    Article  MATH  Google Scholar 

  106. Rajagopal K.R. and Srinivasa A.R., Mechanics of the inelastic behavior of materials — part ii, inelastic response, Int. J. Plasticity, 14, 969–995 (1998).

    Article  MATH  Google Scholar 

  107. Rajagopal K.R. and Srinivasa A.R., A thermodynamic framework for rate type fluid models, J. Non-Newt. Fluid Mech, 88, 207–227 (2000).

    Article  MATH  Google Scholar 

  108. Rajagopal K.R. and Srinivasa A.R., On thermomechanical restrictions of continua, Proc. R. Soc. Lond. A, 460, 631–651 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  109. Rajagopal K.R. and Srinivasa A.R., On the nature of constraints for continua undergoing dissipative processes, Proc. R. Soc. Lond. A, 461, 2785–2795 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  110. Ranganathan M., Mackley M.R. and Spitteler P.H.J., The application of the multipass rheometer to time-dependent capillary flow measurements of a polyethylene melt, J. Rheology, 43(2), 443–451 (1999).

    Article  Google Scholar 

  111. Romero O.J., Suszynski W.J., Scriven L.E. and Carvalho M.S., Low-flow limit in slot coating of dilute solutions of high molecular weight polymer, J. Non-Newt. Fluid Mech., 118, 137–156 (2004).

    Article  Google Scholar 

  112. Ryssel E. and Brunn P.O., Comparison of a quasi-newtonian fluid with a viscoelastic fluid in planar contraction flow, J. Non-Newt. Fluid Mech., 86, 309–335 (1999).

    Article  MATH  Google Scholar 

  113. Ryssel E. and Brunn P.O., Flow of a quasi-newtonian fluid through a planar contraction, J. Non-Newt. Fluid Mech., 85, 11–(1999).

    Article  MATH  Google Scholar 

  114. Ryssel E. and Brunn P.O., The giesekus fluid in w-d form for steady two-dimensional flows. part ii. numerical simulation, Rheol. Acta, 38, 423–436 (1999).

    Article  Google Scholar 

  115. Schunk P.R. and Scriven L.E., Constitutive equation for modeling mixed extension and shear in polymer solution processing, J. Rheology, 34(7), 1085–1117 (1990).

    Article  Google Scholar 

  116. Smith G.F., Isotropic functions of symmetric tensors, skew-symmetric tensors and vectors, Int. J. Engng. Sci., 9, 899–916 (1990).

    Article  Google Scholar 

  117. Llewellyn Smith S.G., Vortices and Rossby-wave radiation on the beta-plane. PhD thesis, University of Cambridge (1996).

  118. Soares E.J., Carvalho M.S. and Souza Mendes P.R., Gas displacement of non-newtonian liquids in capillary tubes, Int. J. Heat Fluid Flow, 27, 95–104 (2006).

    Article  Google Scholar 

  119. Soares M., Naccache M.F. and Souza Mendes P.R., Heat transfer to viscoplastic materials flowing laminarly in the entrance region of tubes, International Journal of Heat and Fluid Flow, 20, 60–67 (1999).

    Article  Google Scholar 

  120. Souza Mendes P.R., Padmanabhan M., Scriven L.E. and Macosko C.W., Inelastic constitutive equations for complex flows, Rheol. Acta, 34, 209–214 (1995).

    Article  Google Scholar 

  121. Souza Mendes P.R., Thompson R.L. and Naccache M.F., Simple viscoelastic constitutive equation for complex flows. In Aĭt-Kadi A., Dealy J.M., James D.F. and Williams M.C., editors, Proc. XIIth International Congress on Rheology, page 385, Quebec, Canada (1996) August 18–23.

  122. Srivastava V.P. Two-phase model of blood flow through stenosed tubes in the presence of a peripheral layer: Applications, Journal of Biomechanics, 29, 1377–1382 (1996).

    Article  Google Scholar 

  123. Srivastava V.P. and Saxena M., Two-layered model of casson fluid flow through stenotic blood vessels: Applications to the cardiovascular system, Journal of Biomechanics, 27, 921–928 (1994).

    Article  Google Scholar 

  124. Tabor M. and Kappler I., Stretching and alignment in chaotic and turbulent flows, Chaos, Solit. & Fractals, 4, 1031–1055 (1994).

    Article  MATH  Google Scholar 

  125. Tanner R.I. and Huilgol R.R., On a classification scheme for flow fields, Rheol. Acta, 14, 959–962 (1975).

    Article  MATH  Google Scholar 

  126. Tanner R.I., A test particle approach to flow classification for viscoelastic fluids, A. I. Ch. E. J., 22, 910–918 (1976).

    Google Scholar 

  127. Tejada-Martínez A.E. and Grosch C.E., Langmuir turbulence in shallow water: Part ii. large eddy simulations, J. Fluid Mech., 576, 63–108 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  128. Thompson R., Comments on “Intrinsically unsteady viscometric and quasi-viscometric flows” by Huilgol R.R., J. Non-Newt. Fluid Mech., 136, 179–180 (2006).

    Article  MATH  Google Scholar 

  129. Thompson R., Bacchi R.D.A. and Machado F.J., What is a vortex? In COBEM, editor, Proc. Brazilian Congress on Mechanical Science and Engineering 2009, pages 21–29. ABCM (2009).

  130. Thompson R.L., Performance of a new constitutive equation for non-Newtonian liquids. PhD thesis, Pontifícia Universidade Católica — RJ, Rio de Janeiro, Brazil (2001) in Portuguese.

    Google Scholar 

  131. Thompson R.L., Some perspectives on the dynamic history of a material element, Int. J. Engng. Sci, 46, 524–549 (2008).

    Article  Google Scholar 

  132. Thompson R.L. and de Souza Mendes P.R., Persistence of straining and flow classification, Int. J. Engng. Sci, 43(1–2), 79–105 (2005a).

    Article  MathSciNet  Google Scholar 

  133. Thompson R.L. and de Souza Mendes P.R., Considerations on kinematic flow classification criteria, J. Non-Newt. Fluid Mech., 128, 109–115 (2005b).

    Article  MATH  Google Scholar 

  134. Thompson R.L. and de Souza Mendes P.R., Further remarks on persistence of straining and flow classification, Int. J. Engng. Sci, 45, 504–508 (2007).

    Article  Google Scholar 

  135. Thompson R.L. and de Souza Mendes P.R., Comments on “Flow classification criteria and its use in general flows” by Brunn P.O., Rheol. Acta, 47, 959–961 (2008).

    Article  Google Scholar 

  136. Thompson R.L., Souza Mendes P.R. and Naccache M.F., A new constitutive equation and its performance in contraction flows, J. Non-Newt. Fluid Mech., 86, 375–388 (1999).

    Article  MATH  Google Scholar 

  137. Trouton F.T., On the coefficient of viscous traction and its relation to that of viscosity, Proc. Royal Soc. A, 77, 426–440 (1906).

    Google Scholar 

  138. Truesdell C., Two measures of vorticity, J. Rational Mech. and Analysis, 2, 173–217 (1953).

    MathSciNet  Google Scholar 

  139. Truesdell C. and Rajagopal K.R., An introduction to the mechanics of fluids. Birkhäuser, Boston (2000).

    Book  MATH  Google Scholar 

  140. Truesdell C. and Toupin R.A., The classical field theories of Mechanics, Number III/1 in Encyl. Phys. Springer, Berlin (1960).

  141. Truesdell C.A. and Noll W., The non-linear field theories of mechanics, Flügge’s Handbuch der Physik, III/3 (1965).

  142. Tucker C.L. and Moldenaers P., Microstructural evolution in polymer blends, Ann. Review Fluid Mech., 34, 177–210 (2002).

    Article  MathSciNet  Google Scholar 

  143. van de Konijnenberg J.A., Andersson H.I. and van Heijst G.J.F., Spin-up in a rectangular tank with a sloping bottom, Eur. J. Mech, B/Fluids, 16, 609–623 (1997).

    MATH  Google Scholar 

  144. VanArsdale W.E., Objective spin and the rivlin-ericksen model, Acta Mech., 162, 111–124 (2003).

    Article  MATH  Google Scholar 

  145. Venerus D.C., Exponential shear flow of branched polymer melts, Rheol. Acta, 39, 71–79 (2000).

    Article  Google Scholar 

  146. Wang C.C., A representation theorem for the constitutive equation of a simple material in motions with constant stretch history, Arch. Ration. Mech. Anal., 20, 329–340 (1965).

    MATH  Google Scholar 

  147. Weiss J., The dynamics of enstrophy transfer in two-dimensional hydrodynamics, Physica D, 48, 273–294 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  148. Winters K.B., MacKinnon J.A. and Mills B. A spectral model for process studies of rotating, density-stratified flows, J. of Atmospheric and Oceanic Technology., 21(1), 69–94 (2004).

    Article  Google Scholar 

  149. Yin W.L. and Pipkin A.C., Kinematics of viscometric flow, Arch. Rational Mech. Anal., 37, 111–135 (1970).

    Article  MATH  MathSciNet  Google Scholar 

  150. Zhang H. and Holden A.V., Estimation of the persistence of strain from a time series as a measure of the geometry of the attractor, Chaos, Solit. & Fractals, 2, 493–504 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  151. Ziabicki A., Studies on the orientation phenomenon by fiber formation from polymer melts ii, Journal of Appl. Polym. Sci., 2, 24–31 (1959)

    Article  Google Scholar 

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  1. LFTC-LMTA-Department of Mechanical Engineering (PGMEC), Universidade Federal Fluminense, Rua Passo da Pátria 156, Niteroi, RJ, 24210-240, Brazil

    Roney L. Thompson

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Thompson, R.L. Flow classification for viscoelastic materials. Int J Adv Eng Sci Appl Math 1, 69–83 (2009). https://doi.org/10.1007/s12572-010-0003-x

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