Mathematical Problems in Classical and Non-Newtonian Fluid Mechanics

  • Giovanni P. Galdi
Part of the Oberwolfach Seminars book series (OWS, volume 37)


Blood flow per se is a very complicated subject. Thus, it is not surprising that the mathematics involved in the study of its properties can be, often, extremely complex and challenging.


Shear Rate Weak Solution Mathematical Problem Fluid Mechanics Piping System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Advani, A.S., Flow and Rheology in Polymer Composites Manufacturing, Elsevier, Amsterdam (1994).Google Scholar
  2. [2]
    Arada, N. and Sequeira, A., Strong Steady Solutions for a Generalized Oldroy-B Model with Shear-Dependent Viscosity in a Bounded Domain, Math. Mod. and Meth. in Appl. Sci. 13 (2003), 1303–1323.CrossRefMathSciNetMATHGoogle Scholar
  3. [3]
    Arada, N. and Sequeira, A., Steady Flows of Shear-Dependent Oldroyd-B Fluids around an Obstacle, J. Math. Fluid Mech. 7 (2005), 451–483.CrossRefMathSciNetMATHGoogle Scholar
  4. [4]
    Astarita, G. and Marucci, G., Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill (1974).Google Scholar
  5. [5]
    Becker, L.E., McKinley, G.H., and Stone, H.A., Sedimentation of a Sphere Near a Plane Wall: Weak Non-Newtonian and Inertial Effects, J. Non-Newtonian Fluid Mech. 63 (1996), 201–233.CrossRefGoogle Scholar
  6. [6]
    Beirão da Veiga, H., On the Existence of Strong Solutions to a Coupled Fluid-Structure Evolution Problem, J. Math. Fluid Mech. 6 (2004), 21–52.CrossRefMathSciNetMATHGoogle Scholar
  7. [7]
    Beirão da Veiga, H., Time periodic solutions of the Navier-Stokes equations in unbounded cylindrical domains—Leray’s problem for periodic flows. Arch. Ration. Mech. Anal. 178 (2005), 301–325.CrossRefMathSciNetMATHGoogle Scholar
  8. [8]
    Beiroãa Veiga, H., On the Regularity of Flows with Ladyzhenskaya Shear-Dependent Viscosity and Slip or Nonslip Boundary Conditions, Comm. Pure Appl. Math. 58 (2005), 552–577.CrossRefMathSciNetGoogle Scholar
  9. [9]
    Beirão da Veiga, H., On Some Boundary Value Problems for Incompressible Viscous Flows with Shear Dependent Viscosity, Elliptic and Parabolic Problems, Progr. Nonlinear Differential Equations Appl., Vol. 63, Birkhäuser, Basel, 2005, 23–32.Google Scholar
  10. [10]
    Beirão da Veiga, H., On Some Boundary Value Problems for Flows with Shear Dependent Viscosity, Variational Analysis and Applications Nonconvex Optim. Appl., Vol. 79, Springer, New York, 2005, 161–172.CrossRefGoogle Scholar
  11. [11]
    Beirão da Veiga, H., Navier-Stokes Equations with Shear-Dependent Viscosity. Regularity up to the Boundary, J. Math. Fluid Mech., in press.Google Scholar
  12. [12]
    Berker, R., 1964, Contrainte sur un Paroi en Contact avec un Fluide Visqueux Classique, un Fluide de Stokes, un Fluide de Coleman-Noll, C.R. Acad. Sci. Paris, 285, 5144–5147.Google Scholar
  13. [13]
    R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Volume I, John Wiley & Sons, second ed. (1987).Google Scholar
  14. [14]
    Bretherton, F.P.,The motion of a Rigid Particle in a Shear Flow at Low Reynolds Number. J. Fluid Mech. 14 (1962), 284–304.Google Scholar
  15. [15]
    Bitbol, M., Red Blood Cell Orientation in Orbit C = 0, Biophys. J. 49 (1986), 1055–1068.Google Scholar
  16. [16]
    Blavier, E._and Mikelić, A., On the Stationary Quasi-Newtonian Flow Obeying a Power Law, Math. Meth. Appl. Sci. 18 (1995), 927–948.CrossRefMATHGoogle Scholar
  17. [17]
    Bönisch, S. and Galdi, G.P., Lift and Migration of Spheres in a Two-Dimensional Channel, in progress.Google Scholar
  18. [18]
    Browder, F.E., Existence and Uniqueness Theorems for Solutions of Nonlinear Boundary Value Problems, Proc. Sympos. Appl. Math. 17 Amer. Math. Soc..Providence, R.I., 1965, 24–49.Google Scholar
  19. [19]
    Carreau, P.J., Rheological equations from molecular network theories, Ph.D. thesis, University of. Wisconsin (1968).Google Scholar
  20. [20]
    Chambolle, A., Desjardins, B., Esteban, M.J., Grandmont, C., Existence of Weak Solutions for an Unsteady Fluid-Plate Interaction Problem, J. Math. Fluid Mech. 7 (2005), 368–404.CrossRefMathSciNetMATHGoogle Scholar
  21. [21]
    Chang, W., Trebotich, D., Lee, L.P., and Liepmann, D., Blood Flow in Simple Microchannels, Proceedings of the 1st Annual International IEEE-EMBS Special Topic Conference on Microtechnologies in Medicine & Biology, Lyon, France (2000).Google Scholar
  22. [22]
    Chhabra R.P., Bubbles, Drops and Particles in Non-Newtonian Fluids, CRC Press (1993).Google Scholar
  23. [23]
    Cheng, C.H.A, Cutand, D., and Shkoller, D., Navier-Stokes Equations Interacting with a Nonlinear Elastic Shell (2006), preprint.Google Scholar
  24. [24]
    Coleman, B.D. and Noll, W., On Certain Steady Flows of General Fluids, Arch. Rational Mech. Anal. 3 (1959), 289–303.CrossRefMathSciNetMATHGoogle Scholar
  25. [25]
    Coleman, B.D. and Noll, W., An Approximation Theorem for Functionals with Applications in Continuum Mechanics, Arch. Rational Mech. Anal. 6 (1960), 55–70.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Coleman, B.D. and Noll, W., Simple Fluids with Fading Memory, Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics, Oxford, Pergamon Press, 1962, 530–552.Google Scholar
  27. [27]
    Coscia, V., and Galdi, G.P., Existence, Uniqueness and Stability of Regular Steady Motions of a Second Grade Fluid, Int. J. Nonl. Mech. 29 (1994), 493–516.CrossRefMathSciNetMATHGoogle Scholar
  28. [28]
    Coutand, D., and Shkoller, S., Motion of an Elastic Solid Inside of an Incompressible Viscous Fluid, Arch. Rational Mech. Anal. 176 (2005), 25–102.CrossRefMathSciNetMATHGoogle Scholar
  29. [29]
    Coutand, D., and Shkoller, S., On the Interaction Between Quasilinear Elastodynamics and the Navier-Stokes Equations, Arch. Rational Mech. Anal. 179 (2006), 303–352.CrossRefMathSciNetMATHGoogle Scholar
  30. [30]
    Ebmeyer, C., Steady Flow of Fluids with Shear-Dependent Viscosity under Mixed Boundary Value Conditions in Polyhedral Domains, Math. Models Methods Appl. Sci. 10 (2000), 629–650.MathSciNetMATHGoogle Scholar
  31. [31]
    Feireisl, E., On the Motion of Rigid Bodies in a Viscous Incompressible Fluid, J. Evol. Equ. 3 (2003), 419–441.CrossRefMathSciNetMATHGoogle Scholar
  32. [32]
    Fontelos, M.A. and Friedman, A., Stationary non-Newtonian Fluid Flows in Channel-Like and Pipe-Like Domains, Arch. Ration. Mech. Anal. 151 (2000), 1–43.CrossRefMathSciNetMATHGoogle Scholar
  33. [33]
    Frehse, J., Málek, J., and Steinhauer, M., An Existence Result for Fluids with Shear Dependent Viscosity-Steady Flows, Nonlinear Anal. Th. Methods Appl. 30 (1997), 3041–3049.CrossRefMATHGoogle Scholar
  34. [34]
    Frehse, J., Málek, J., and Steinhauer, M., On Analysis of Steady Flows of Fluids with Shear-Dependent Viscosity Based on the Lipschitz Truncation Method, SIAM J. Math. Anal. 34 (2003), 1064–1083.CrossRefMathSciNetMATHGoogle Scholar
  35. [35]
    Galdi, G.P., Mathematical Theory of Second-Grade Fluids, Stability and Wave Propagation in Fluids and Solids, G.P. Galdi ed., Springer-Verlag, Berlin, 1995, 67–104.Google Scholar
  36. [36]
    Galdi, G.P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Linearised Steady Problems, Springer Tracts in Natural Philosophy, Vol. 38, Springer-Verlag, 2nd Corrected Edition (1998).Google Scholar
  37. [37]
    Galdi, G.P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Nonlinear Steady Problems, Springer Tracts in Natural Philosophy, Vol. 39, Springer-Verlag, 2nd Corrected Edition (1998).Google Scholar
  38. [38]
    Galdi, G.P., 2002, On the Motion of a Rigid Body in a Viscous Liquid: A Mathematical Analysis with Applications, Handbook of Mathematical Fluid Mechanics, North-Holland Elsevier Science, Vol. 1 (2002), 653–792.Google Scholar
  39. [39]
    Galdi, G.P., Grobbelaar, M. and Sauer, N., Existence and Uniqueness of Classical Solutions of the Equations of Motion for Second-Grade Fluids, Arch. Rational Mech. Anal. 124 (1993), 221–237.CrossRefMathSciNetMATHGoogle Scholar
  40. [40]
    Galdi, G.P, and Heuveline, V., Lift and Sedimentation of Particles in the Flow of a Viscoelastic Liquid in a Channel, Free and Moving Boundarie Analysis, Simulation and Control, R. Glowinski and J.-P. Zolesio Eds, CRC Publ., in press.Google Scholar
  41. [41]
    Galdi, G.P., Pileckas, K. and Silvestre, A.L, Relation Between Pressure-Drop and Flow Rate in Unsteady Poiseuille Flow, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), submitted.Google Scholar
  42. [42]
    Galdi, G.P., Pokorný, M., Vaidya, A., Joseph, D.D., and Feng, J., Orientation of Symmetric Bodies Falling in a Second-Order Liquid at Non-Zero Reynolds Number, Math. Models Methods Appl. Sci. 12 (2002), 1653–1690.CrossRefMathSciNetMATHGoogle Scholar
  43. [43]
    Galdi, G.P., and Robertson, A.M., The Relation Between Flow Rate and Axial Pressure Gradient for Time-Periodic Poiseuille Flow in a Pipe, J. Math. Fluid Mech. 7suppl. 2 (2005), 215–223.CrossRefMathSciNetGoogle Scholar
  44. [44]
    Galdi, G.P., Sequeira, A. and Videman, J., Steady Motions of a Second-Grade Fluid in an Exterior Domain, Adv. Math. Sci. Appl. 7 (1997), 977–995.MathSciNetMATHGoogle Scholar
  45. [45]
    Galdi, G.P. and Silvestre, A.L., Existence of Time-Periodic Solutions to the Navier-Stokes Equations Around a Moving Body Pacific J. Math. 223 (2006), 251–268.MathSciNetMATHGoogle Scholar
  46. [46]
    Galdi G.P., and Vaidya A., Translational Steady Fall of Symmetric Bodies in a Navier-Stokes Liquid, with Application to Particle Sedimentation, J. Math. Fluid Mech. 3 (2001), 183–211.CrossRefMathSciNetMATHGoogle Scholar
  47. [47]
    Grandmont, C., Existence for a Three-Dimensional Steady State Fluid-Structure Interaction Problem, J. Math. Fluid Mech 4 (2002), 76–94.CrossRefMathSciNetMATHGoogle Scholar
  48. [48]
    Gresho, P.M., Some Current CFD Issues Relevant to the Incompressible Navier-Stokes Equations, Comput. Methods Appl. Mech. Eng. 87 (1991), 201–252.CrossRefMathSciNetMATHGoogle Scholar
  49. [49]
    Grossman, P.D., and Soane, D.S., Orientation Effects on the Electrophoretic Mobility of Rod-Shaped Molecules in Free Solution, Anal. Chem. 62, (1990), 1592–1596.CrossRefGoogle Scholar
  50. [50]
    Guillopé, G., Hakim, A., and Talhouk, R., Existence of Steady Flows of Slightly Compressible Viscoelastic Fluids of White-Metzner Type Around an Obstacle, Comm. Pure Appl. Anal. 4 (2005), 23–43.Google Scholar
  51. [51]
    Guillopé C. and J.-C. Saut, Existence Results for the Flow of Viscoelastic Fluids with a Differential Constitutive Law, Nonlinear Anal. Th. Methods Appl. 15 (1990), 849–869.CrossRefMATHGoogle Scholar
  52. [52]
    Guillopé, C. and J.-C. Saut, Existence and Stability of Steady Flows of Weakly Viscoelastic Fluids, Proc. Roy. Soc. Edinburgh A119 (1991), 137–158.Google Scholar
  53. [53]
    Guillopé, G. and Talhouk, R., Steady Flows of Slightly Compressible Viscoelastic Fluids of Jeffreys’ Type Around an Obstacle, Diff. Int. Eq. 16 (2003), 1293–1320.MATHGoogle Scholar
  54. [54]
    Hagen, G. On the Motion of Water in Narrow Cylindrical Tubes, Pogg. Ann., 46 (1839), 423–442.CrossRefGoogle Scholar
  55. [55]
    Hames, B.D., and Rickwood, D., Eds., Gel Electrophoresis of Proteins, IRL Press, Washington, D.C. (1984).Google Scholar
  56. [56]
    Hakim, A., Mathematical Analysis of Viscoelastic Fluids of White-Metzner Type, J. Math. Anal. Appl. 185 (1994), 675–705.CrossRefMathSciNetMATHGoogle Scholar
  57. [57]
    Heywood, J.G., The Navier-Stokes Equations: On the Existence, Regularity and Decay of Solutions, Indiana U. Math. J., 29 (1980), 639–681.CrossRefMathSciNetMATHGoogle Scholar
  58. [58]
    Heywood, J.G., Rannacher, R., and Turek, S., Artificial Boundaries and Flux and Pressure Conditions for the Incompressible Navier-Stokes Equations, Int. J. Numer. Meth. in Fluids 22 (1996), 325–352.CrossRefMathSciNetMATHGoogle Scholar
  59. [59]
    Horgan, C.O., and Wheeler, L.T., Spatial Decay Estimates for the Navier-Stokes Equations with Application to the Problem of Entry Flow, SIAM J. Appl. Math. 35 (1978), 97–116.CrossRefMathSciNetMATHGoogle Scholar
  60. [60]
    Joseph, D.D., Instability of the Rest State of Fluids of Arbitrary Grade Larger than One, Arch. Rational Mech. Anal. 75 (1980), 251–256.MathSciNetGoogle Scholar
  61. [61]
    Joseph, D.D., Fluid Dynamics of Viscoelastic Liquids, Applied Mathematical Sciences, 84, Springer-Verlag (1990).Google Scholar
  62. [62]
    Joseph, D.D., 2000, Interrogations of Direct Numerical Simulation of Solid-Liquid Flow, Web Site:
  63. [63]
    Joseph, D.D., and Feng, J., A Note on the Forces that Move Particles in a Second-Order Fluid, J. Non-Newtonian Fluid Mech. 64 (1996), 299–302.CrossRefGoogle Scholar
  64. [64]
    Joseph, D.D., and Fosdick, R.L., The Free Surface on a Liquid Between Cylinders Rotating at Different Speeds. Part I, Arch. Rational Mech. Anal. 49 (1973), 321–380.MathSciNetMATHGoogle Scholar
  65. [65]
    Joseph, D.D., Beavers, G.S., and and Fosdick, R.L., The Free Surface on a Liquid Between Cylinders Rotating at Different Speeds. Part II, Arch. Rational Mech. Anal. 49 (1973), 381–401.MathSciNetMATHGoogle Scholar
  66. [66]
    Kučera, P. and Skalák, Z., Local Solutions to the Navier-Stokes Equations with Mixed Boundary Conditions, Acta Appl. Math. 54 (1998), 275–288.CrossRefMathSciNetGoogle Scholar
  67. [67]
    Ladyzhenskaya, O.A., On Some New Equations Describing Dynamics of Incompressible Fluids and on Global Solvability of Boundary Value Problems to These Equations, Trudy Steklov Math. Inst. 102 (1967), 85–104.MATHGoogle Scholar
  68. [68]
    Ladyzhenskaya, O.A., On Some Modifications of the Navier-Stokes Equations for Large Gradients of Velocity, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 7 (1968), 126–154.MathSciNetMATHGoogle Scholar
  69. [69]
    Ladyzhenskaya, O.A., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York (1969).MATHGoogle Scholar
  70. [70]
    Ladyzhenskaya, O.A., Boundary Value Problems of Mathematical Physics, Springer-Verlag (1985).Google Scholar
  71. [71]
    Ladyzhenskaya, O.A., and Solonnikov, V.A., Determination of Solutions of Boundary Value Problems for Steady-State Stokes and Navier-Stokes Equations in Domains Having an Unbounded Dirichlet Integral, Zap. Nauchn. Sem. Leningrad Ot-del. Mat. Inst. Steklov (LOMI) 96 (1980), 117–160; English Transl.: J.Soviet Math., 2 no. 1 (1983), 728-761.MathSciNetMATHGoogle Scholar
  72. [72]
    Ladyzhenskaya, O.A. and Ural’ceva, N.N., Linear and Quasilinear Elliptic Equations, Academic Press, New York, (1968).MATHGoogle Scholar
  73. [73]
    Lee, S.C., Yang, D.Y., Ko, J., and You, J.R., Effect of compressibility on flow field and fiber orientation during the filling stage of injection molding J Mater. Process. Tech. 70 (1997), 83–92.CrossRefGoogle Scholar
  74. [74]
    Leigh, D.C., Non-Newtonian Fluids and the Second Law of Thermodynamics, Phys. Fluids 5 (1962), 501–502.CrossRefMATHGoogle Scholar
  75. [75]
    Lieberman, G.M., Boundary Regularity for Solutions of Degenerate Elliptic Equations, Nonlinear Anal. Th. Methods Appl. 12 (1988), 1203–1219.CrossRefMathSciNetMATHGoogle Scholar
  76. [76]
    Lions, J.-L., Quelques Methodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris (1969).MATHGoogle Scholar
  77. [77]
    Liu, W.B. and Barrett, W., A Remark on the Regularity of the Solutions of the p-Laplacian and its Application to their Finite Element Approximation, J. Math. Anal. Appl. 178 (1993), 470–487.CrossRefMathSciNetMATHGoogle Scholar
  78. [78]
    Liu, Y.J., and Joseph, D.D., Sedimentation of Particles in Polymer Solutions, J. Fluid Mech. 255 (1993), 565–595.CrossRefGoogle Scholar
  79. [79]
    Malek, J., Nečas, J., Rokyta, M., and Růžička, M., Weak and Measure-Valued Solutions to Evolutionary PDEs, Vol. 13 Chapman & Hall, London (1996).MATHGoogle Scholar
  80. [80]
    Marušić-Paloka, E., Steady Flow of a Non-Newtonian Fluid in Unbounded Channels and Pipes, Math. Mod. Meth. Appl. Sci. 10 (2000), 1425–1445.CrossRefGoogle Scholar
  81. [81]
    Miller, R.K., Feldstein, A., Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM J. Math. Anal. 2 (1971), 242–258.CrossRefMathSciNetMATHGoogle Scholar
  82. [82]
    Minty, G.J., On a ‘Monotonicity’ Method for the Solution of Nonlinear Equations in Banach Spaces, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 1038–1041.CrossRefMathSciNetMATHGoogle Scholar
  83. [83]
    Noll, W., A Mathematical Theory of the Mechanical Behavior of Continuous Media, Arch. Rational Mech. Anal. 2 (1958), 197–226.CrossRefMATHGoogle Scholar
  84. [84]
    Novotný, A. Sequeira, A. Videman, J.H., Steady Motions of Viscoelastic Fluids in Three-Dimensional Exterior Domains. Existence, Uniqueness and Asymptotic Behaviour, Arch. Ration. Mech. Anal. 149 (1999), 49–67.CrossRefMathSciNetMATHGoogle Scholar
  85. [85]
    Ostwald, K., Uber die Geschwindigkeitsfunktion der Viskosität disperser Systeme I, Kolloid Zeit. 36 (1925), 99–117.CrossRefGoogle Scholar
  86. [86]
    Pileckas, K., Navier-Stokes System in Domains with Cylindrical Outlets to Infnity. Leray’s Problem, Handbook of Mathematical Fluid Mechanics, North-Holland Elsevier Science, in press.Google Scholar
  87. [87]
    Pileckas, K., Sequeira, A., and Videman, J.H., Steady Flows of Viscoelastic Fluids in Domains with Outlets to Infinity J. Math. Fluid Mech. 2 (2000), 185–218.CrossRefMathSciNetMATHGoogle Scholar
  88. [88]
    Poiseuille, J.L.M., Recherches Experimentales sur le Mouvement des Liquides dans les Tubes de Tres Petits Diameters, C. R. Acad. Sci. Paris 11 (1840), 961–967.Google Scholar
  89. [89]
    Prilepko, A.I., Orlovsky, D.G., Vasin, I.A., Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York, Basel (1999).Google Scholar
  90. [90]
    Prouse, G., Soluzioni Periodiche delle Equazioni di Navier-Stokes, Rend. Atti. Accad. Naz. Lincei 35 (1963), 403–409.MathSciNetGoogle Scholar
  91. [91]
    Rannacher, R., Methods for Numerical Flow Simulation, article in this volume.Google Scholar
  92. [92]
    Renardy, M., Existence of Slow Steady Flows of Viscoelastic Fluids with Differential Constitutive Equations Z. Angew. Math. Mech. 65 (1985), 449–451.CrossRefMathSciNetMATHGoogle Scholar
  93. [93]
    Roco, M.C., (Ed.), Particulate Two-Phase Flow, Butterworth-Heinemann Publ., Series in Chemical Engineering (1993).Google Scholar
  94. [94]
    Robertson, A.M., Review of Relevant Continuum Mechanics, article in this volume.Google Scholar
  95. [95]
    Růžička, M., A Note on Steady Flow of Fluids with Shear Dependent Viscosity, Nonlinear Anal. Th. Methods Appl. 30 (1997), 3029–3039.CrossRefMATHGoogle Scholar
  96. [96]
    Sandri, D., Sur L’Approximation Numérique des Écoulements Quasi-Newtoniens dont la Viscosité suit la loi Puissance ou la loi de Carreau, Math. Mod. Numer. Anal. 27 (1993), 131–155.MathSciNetMATHGoogle Scholar
  97. [97]
    Schmid-Schonbein, H., and Wells, R., Fluid Drop-Like Transition of Erythrocytes under shear, Science 165 (1969), 288–291.CrossRefGoogle Scholar
  98. [98]
    Segrè, G., and Silberberg, A., Radial Poiseuille Flow of Suspensions, Nature 189 (1961), 209–210.Google Scholar
  99. [99]
    Segrè G., and Silberberg, A., Behaviour of Macroscopic Rigid Spheres in Poiseuille Flow, Part I, J. Fluid Mech. 14 (1962), 115–135.CrossRefGoogle Scholar
  100. [100]
    Sequeira, A., and Baía, M., A Finite Element Approximation for the Steady Solution of a Second-Grade Fluid Model, J. Comput. Appl. Math. 111 (1999), 281–295.CrossRefMathSciNetMATHGoogle Scholar
  101. [101]
    Sequeira, A., and Videman, J.-H. Mathematical Results and Numerical Methods for Steady Incompressible Viscoelastic Fluid Flows, Math. Appl. 528 (2001), 339–365.MathSciNetGoogle Scholar
  102. [102]
    Serrin, J.B, Poiseuille and Couette Flow of Non-Newtonian Fluids, Z. Angew. Math. Mech. 39 (1959), 295–299.CrossRefMathSciNetMATHGoogle Scholar
  103. [103]
    Simader, C.G., and Sohr, H., 1997, The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains, Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Vol. 360.Google Scholar
  104. [104]
    Talhouk, R., Existence Results for Steady Flow of Weakly Compressible Viscoelastic Fluids with Differential Constitutive Law, Diff. and Integral Eq. 12 (1999), 741–742.MathSciNetMATHGoogle Scholar
  105. [105]
    Tinland, B., Meistermann, L., Weill, G., Simultaneous Measurements of Mobility, Dispersion, and Orientation of DNA During Steady-Field Gel Electrophoresis Cou-pling a Fluorescence Recovery After Photobleaching Apparatus with a Fluorescence Detected Linear Dichroism Setup, Phys. Rev. E 61 (2000), 6993–6998.CrossRefGoogle Scholar
  106. [106]
    Thomson, W., and Tait, P.G., Natural Philosophy, Vols 1, 2, Cambridge University Press (1879).Google Scholar
  107. [107]
    Trainor, G.L., DNA Sequencing, Automation and Human Genome, Anal. Chem. 62 (1990), 418–426.CrossRefGoogle Scholar
  108. [108]
    Tricomi, F.G., Integral Equations, Intersience, New York (1957).Google Scholar
  109. [109]
    Truesdell, C.A., The Meaning of Viscometry in Fluid Dynamics, Ann. Rev. Fluid Mech. 6 (1974), 111–146.CrossRefGoogle Scholar
  110. [110]
    Uijttewaal, W.S.J., Nijhof, E.-J., and Heethaar R.M., Lateral Migration of Blood Cells and Microspheres in Two-Dimensional Poiseuille Flow: a Laser-Doppler Study, J. Biomech. 27 (1994), 35–42.CrossRefGoogle Scholar
  111. [111]
    Vaidya, A., A Note on the Orientation of Symmetric Rigid Bodies Sedimenting in a Power-Law Fluid, Appl. Math. Letters 18 (2005), 1332–1338.CrossRefMathSciNetMATHGoogle Scholar
  112. [112]
    Vaidya, A., Observations on the Transient Nature of Shape-Tilting Bodies Sedimenting in Polymeric Liquids, J. Fluids and Struct. 22 (2006), 253–259.CrossRefGoogle Scholar
  113. [113]
    Vejvoda, O., Herrmann, L., Lovicar, V., Sova, M.; Straškraba, I., and Štědrý, M., Partial Differential Equations: Time-Periodic Solutions, Martinus Nijhoff Publishers, The Hague (1981).Google Scholar
  114. [114]
    Wang, J., Bail, R.-Y., Lewandowski, C., Galdi, G.P. and Joseph, D.D., Sedimntation of Cylindrical Particles in a Viscoelastic Liquid: Shape-Tilting, China Particuology 2 (2004), 13–18.CrossRefGoogle Scholar
  115. [115]
    Wolf, J., Existence of Weak Solutions to the Equations of Non-Stationary Motion of Non-Newtonian Fluids with Shear-Rate Dependent Viscosity, J. Math. Fluid Mech., in press.Google Scholar
  116. [116]
    Womersley, J.R., Method for the Calculation of Velocity, Rate of Flow and Viscous Drag in Arteries when the Pressure Gradient is Known, J. Physiol. 127 (1955), 553–556.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Giovanni P. Galdi
    • 1
  1. 1.Department of Mechanical Engineering and Materials ScienceUniversity of PittsburghPittsburghUSA

Personalised recommendations