Biomechanics pp 266-332 | Cite as


  • Y. C. Fung


In the preceding chapters we studied the flow of blood in large blood vessels in which the main feature is a balance between the pressure forces, inertia forces, and forces of tissue and muscle. Only in the boundary layer are the viscous friction forces important. The boundary layer thickness grows with increasing distance from the entry section, and in a long tube the boundary layer on the wall eventually becomes so thick as to fill the entire tube. However, arteries divide and divide again. The vessel diameter decreases with each division, and soon the Reynolds and Womersley numbers become quite small, the entry length becomes only a small multiple of the vessel diameter, and the flow becomes fully developed even in relatively short vessels, and the analysis given in Section 3.2 becomes applicable.


Reynolds Number Apparent Viscosity Vessel Diameter Stoke Flow Entry Length 
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  1. Aroesty, J., and Gross, J.F. (1970). Convection and diffusion in the microcirculation. Microvasc. Res. 2: 247–267.PubMedCrossRefGoogle Scholar
  2. Atherton, A., and Born, G.V.R. (1972). Quantitative investigations of the adhesiveness of circulating polymorphonuclear leucocytes to blood vessel walls. J. Physiol. 222: 447–474.PubMedGoogle Scholar
  3. Baker, M., and Wayland, H. (1974). On-line volumetric flow rate and velocity profile measurement for blood microvessels. Microvasc. Res. 7: 131–143.PubMedCrossRefGoogle Scholar
  4. Bayliss, W.M. (1902). On the local reactions of the arterial wall to changes in internal pressure. J. Physiol. (Lond.) 28: 220–231.Google Scholar
  5. Bungay, P.M., and Brenner, H. (1973). The motion of a closely-fitting sphere in a fluid-filled tube. Int. J. Multiphase Flow 1: 25–56.CrossRefGoogle Scholar
  6. Caro, C.G., Pedley, T.J., Schroter, R.C., and Seed, W.A. (1978). The Mechanics of the Circulation. Oxford University Press, Oxford.Google Scholar
  7. Chen, T.C., and Skalak, R. (1970). Spheroidal particle flow in a cylindrical tube. Appl. Sci. Res. 22: 403–441.Google Scholar
  8. Desjardins, G, and Duling, B.R. (1987). Microvessel hematocrit: measurement and implications for capillary oxygen transport. Am. J. Physiol. 252: H494–H503.PubMedGoogle Scholar
  9. Desjardins, C., and Duling, B.R. (1990). Heparinase treatment suggests a role for the endothelial cell glycocalyx in regulation of capillary hematocrit. Am. J. Physiol. 258: H647–H654.PubMedGoogle Scholar
  10. Duling, B.R., and Desjardins, G (1987). Capillary hematocrit—what does it mean? News in Physiol. Sci. 2: 66–69.Google Scholar
  11. Fischer, T.M., Stohr-Liesen, M., and Schmid-Schönbein, H. (1978). The red cell as a fluid droplet: Tank tread-like motion of the human erythrocyte membrane in shear flow. Science 202: 894–896.PubMedCrossRefGoogle Scholar
  12. Folkow, B. (1964). Description of the myogenic hypothesis. Circ. Res. 15: I.279–I.285.Google Scholar
  13. Folkow, B., and Neil, E. (1971). Circulation. Oxford University Press, London.Google Scholar
  14. Fung, Y.C. (1965). Foundations of Solid Mechanics. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
  15. Fung, Y.C. (1973). Stochastic flow in capillary blood vessels. Microvasc. Res. 5: 34–48.PubMedCrossRefGoogle Scholar
  16. Fung, Y.C. (1990). Biomechanics: Motion, Flow, Stress, and Growth. Springer Verlag, New York.Google Scholar
  17. Fung, Y.C. (1993). Biomechanics: Mechanical Properties of Living Tissues. 2nd ed., Springer-Verlag, New York.Google Scholar
  18. Gaehtgens, P. (1980). Flow of blood through narrow capillaries: Rheological mechanisms determining capillary hematocrit and apparent viscosity. Biorheology J. 17: 183–189.Google Scholar
  19. Greenfield, A.D.M. (1964). Blood flow through the human forearm and digits as influenced by subatmospheric pressure and venous pressure. Circ. Res. 14:I.0–I.75.Google Scholar
  20. Hele-Shaw, H.S. (1897, 1898). Phil. Trans. Roy. Inst. Nav. Arch. 41: 21. 42: 49.Google Scholar
  21. Hele-Shaw, H.S., and Hag, A. (1898). Royal Soc. Phil. Trans. A 195: 303.Google Scholar
  22. Hochmuth, R.M., and Sutera, S.P. (1970). Spherical caps in low Reynolds-number tube flow. Chem. Eng. Sci. 25: 593–604.CrossRefGoogle Scholar
  23. Hyman, W.A., and Skalak, R. (1972). Non-Newtonian behavior of a suspension of liquid drops in fluid flow. Am. Inst. Chem. Eng. J. 18: 149–154.CrossRefGoogle Scholar
  24. Jendrucko, R.J., and Lee, J.S. (1973). The measurement of hematocrit of blood flowing in glass capillaries by microphotometry. Microvasc. Res. 6: 316–331.PubMedCrossRefGoogle Scholar
  25. Johnson, P.C. (1978). Peripheral Circulation. Wiley, New York.Google Scholar
  26. Johnson, P.C. (1980). The myogenic response. In Handbook of Physiology, Sec. 2. The Cardiovascular System, Vol. 2. Vascular Smooth Muscle (D.F. Bohr, A.P. Somlyo, and H.V. Sparks, Jr., eds.). American Physiological Society, Bethesda, MD, pp. 409–442.Google Scholar
  27. Johnson, P. G (1991). The myogenic response. News in Physiol. Sci. 6: 41–42.Google Scholar
  28. Johnson, P.C., and Intaglietta, M. (1976). Contributions of pressure and flow sensitivity to autoregulation in mesenteric arterioles. Am. J. Physiol. 231: 1686–1698.PubMedGoogle Scholar
  29. Kaley, G., and Altura, B.M. (1977). Microcirculation, Vols. 1 & 2. University Park Press, Baltimore, MD.Google Scholar
  30. Klitzman, B., and Duling, B.R. (1979). Microvascular hematocrit and red cell flow in resting and contracting striated muscle. Am. J. Physiol. 237: H481–H490.PubMedGoogle Scholar
  31. Lamb, H. (1932). Hydrodynamics, 6th ed. Cambridge University Press. Reprinted by Dover, New York.Google Scholar
  32. Lee, J.S., and Fung, Y.C. (1969). Stokes flow around a circular cylindrical post confined between two parallel plates. J. Fluid Mech. 37: 657–670.CrossRefGoogle Scholar
  33. Lee, T.Q., Schmid-Schönbein, G.W., and Zweifach, B.W. (1983). An application of an improved dual-slit photometric analyzer for volumetric flow rate measurements in microvessels. Microvasc. Res. 26: 351–361.PubMedCrossRefGoogle Scholar
  34. Lew, H.S., and Fung, Y.C. (1969a). On the low-Reynolds-number entry flow into a circular cylindrical tube. J. Biomech. 2: 105–119.PubMedCrossRefGoogle Scholar
  35. Lew, H.S., and Fung, Y.C. (1969b). The motion of the plasma between the red blood cells in the bolus flow. J. Biorheology 6: 109–119.Google Scholar
  36. Lew, H.S., and Fung, Y.C. (1969c). Flow in an occluded circular cylindrical tube with permeable wall. Z. Angew. Math. Physik 20: 750–766.CrossRefGoogle Scholar
  37. Lew, H.S., and Fung, Y.C. (1979a). Plug effect of erythrocytes in capillary blood vessels. Biophys. J. 10: 80–99.CrossRefGoogle Scholar
  38. Lew, H.S., and Fung, Y.C. (1970b). Entry flow into blood vessels at arbitrary Reynolds number. J. Biomech. 3: 23–38.PubMedCrossRefGoogle Scholar
  39. Lighthill, M.J. (1968). Pressure-forcing of tightly fitting pellets along fluid-filled elastic tubes. J. Fluid Mech. 34: 113–143.CrossRefGoogle Scholar
  40. Lipowsky, H.H., and Zweifach, B.W. (1977). Methods for the simultaneous measurement of pressure differentials and flow in single unbranched vessels of the microcirculation for rheological studies. Microvasc. Res. 14: 345–361.PubMedCrossRefGoogle Scholar
  41. Lipowsky, H.H., and Zweifach, B.W. (1978). Application of the “two-slit” photometric technique to the measurement of microvascular volumetric flow rates. Microvasc. Res. 15: 93–101.PubMedCrossRefGoogle Scholar
  42. Lipowsky, H.H., Usami, S., and Chien, S. (1980). In vivo measurements of “apparent viscosity” and microvessel hematocrit in the mesentery of the cat. Microvasc. Res. 19: 297–319.PubMedCrossRefGoogle Scholar
  43. Majno, G. (1965). Ultrastructure of the vascular membrane. In Handbook of Physiology, Sec. 2 Circulation, Vol. 3. (W.F. Hamilton, and P. Dow, eds.). American Physiological Society, Washington, D.C. pp. 2293–2375.Google Scholar
  44. Nellis, S.N., and Zweifach, B.W. (1977). A method for determining segmental resistances in the microcirculation from pressure-flow measurements. Circ. Res. 40: 546–556.PubMedCrossRefGoogle Scholar
  45. Norberg, K.A., and Hamberger, B. (1964). The sympathetic adrenergic neuron. Acta Physiol. Scand. 63 (Suppl. 238).Google Scholar
  46. Oseen, C.W (1910). Über die Stokessche Formel und liber die verwandte Aufgabe in der Hydrodynamik. Arkiv Mat. Astron. Fysik 6(29).Google Scholar
  47. Prothero, J., and Burton, A.C. (1961, 1962). The physics of blood flow in capillaries. I. The nature of the motion. Biophys. J., 1: 567–579. II. The capillary resistance to flow. Biophys. J. 2: 199-212.CrossRefGoogle Scholar
  48. Proudman, I., and Pearson, J.R.A. (1957). Expansions at small Reynolds number for the flow past a sphere and a circular cylinder. J. Fluid Mech. 2: 237–262.CrossRefGoogle Scholar
  49. Rothe, CF., Nash, F.D., and Thompson, D.E. (1971). Patterns in autoregulation of renal blood flow in the dog. Am. J. Physiol. 220: 1621–1626.PubMedGoogle Scholar
  50. Rouse, H. (1959). Advanced Fluid Mechanics. Wiley, New York.Google Scholar
  51. Sagawa, K., Kumoda, M., and Schramm, L.P. (1974). Nervous control of the circulation. In Cardiovascular Physiology, Vol. 1 (A.C. Guyton and C.E. Jones, eds). Butterworths, London, pp. 197–232.Google Scholar
  52. Schlichting, H. (1962). Boundary Layer Theory. McGraw-Hill, New York.Google Scholar
  53. Schmid-Schönbein, H., and Wells, R.E. (1969). Fluid drop-like transition of erythrocytes under shear. Science 165: 288–291.CrossRefGoogle Scholar
  54. Schmid-Schönbein, G.W., Fung, Y.C., and Zweifach, B. (1975). Vascular endothelium-leucocyte interaction: Sticking shear force in venules. Circ. Res. 36: 173–184.CrossRefGoogle Scholar
  55. Schmid-Schönbein, G.W., Skalak, R., Usami, S., and Chien, S. (1980a). Cell distribution in capillary networks. Microvasc. Res. 19: 18–44.PubMedCrossRefGoogle Scholar
  56. Schmid-Schönbein, G.W., Usami, S., Skalak, R., and Chien, S. (1980b). The interaction of leukocytes and erythrocytes in capillary and postcapillary vessels. Microvasc. Res. 19: 45–70.PubMedCrossRefGoogle Scholar
  57. Secomb, T.W, and Skalak, R. (1982). A two-dimensional model for capillary flow of an asymmetric cell. Microvasc. Res. 24: 194–203.PubMedCrossRefGoogle Scholar
  58. Skalak, R., Chen, P.H., and Chien, S. (1972). Effect of hematocrit and rouleaux on apparent viscosity in capillaries. Biorheology 9: 67–82.PubMedGoogle Scholar
  59. Skalak, R., Tozeren, A., Zarda, P.R., and Chien, S. (1973). Strain energy function of red cell membranes. Biophys. J. 13: 245–264.PubMedCrossRefGoogle Scholar
  60. Smaje, L., Zweifach, B.W., and Intaglietta, M. (1970). Micropressures and capillary filtration coefficients in single vessels of the cremaster muscle of the rat. Microvasc. Res. 2: 96–110.PubMedCrossRefGoogle Scholar
  61. Stokes, G.G. (1851). On the effect of the internal friction of fluids on the motion of pendulums. Trans. Cambridge Philosophical Soc. 9: 8; Mathematical and Physical Papers, Vol. 3, pp. 1-141.Google Scholar
  62. Stokes, G. (1898). Rep. British. Assoc. A 144 (also, papers, Vol. 3, p. 278).Google Scholar
  63. Targ, S.M. (1951). Basic Problems of the Theory of Laminar Flows (in Russian). Moscow.Google Scholar
  64. Tong, P., and Fung, Y.C. (1971). Slow viscous flow and its application to biomechanics. J. Appl. Mech. 38: 721–728.CrossRefGoogle Scholar
  65. Tözeren, H., and Skalak, R. (1978). The steady flow of closely fitting incompressible elastic spheres in a tube. J. Fluid Mech. 87: 1–16.CrossRefGoogle Scholar
  66. Tözeren, H., and Skalak, R. (1979). Flow of elastic compressible spheres in tubes. J. Fluid Mech. 95: 743–760.CrossRefGoogle Scholar
  67. Wang, H., and Skalak, R. (1969). Viscous flow in a cylindrical tube containing a line of spherical particles. J. Fluid Mech. 38: 75–96.CrossRefGoogle Scholar
  68. Wayland, H. (1982). A physicist looks at the microcirculation. Microvasc. Res. 23: 139–170.PubMedCrossRefGoogle Scholar
  69. Wiedeman, M.P., Tuma, R.E, and Mayrovitz, H.N. (1981). An Introduction to Microcirculation. Academic Press, New York.Google Scholar
  70. Wiederhielm, C.A., Woodbury, J.W, Kirk, S., and Rushmer, R.F. (1964). Pulsatile pressure in microcirculation of the frog’s mesentery. Am. J. Physiol. 207: 173–176.PubMedGoogle Scholar
  71. Yen, R.T., and Fung, Y.C. (1978). Effect of velocity distribution on red cell distribution in capillary blood vessels. Am. J. Physiol. 235: H251–H257.PubMedGoogle Scholar
  72. Yih, C.S. (1977). Fluid Mechanics. Corrected edition, West River Press, Ann Arbor, MI.Google Scholar
  73. Zarda, P.R., Chien, S., and Skalak, R. (1977). Interaction of a viscous incompressible fluid with an elastic body. In Computational Methods for Fluid-Structure Inter-action Problems (T. Belytschko and T.L. Geers, eds.), American Society of Mechanical Engineers, New York, pp. 65–82.Google Scholar
  74. Zweifach, B.W. (1974). Quantitative studies of microcirculatory structure and function. I. Analysis of pressure distribution in the terminal vascular bed. Circ. Res. 34: 843–857. II. Direct measurement of capillary pressure in splanchmic mesenteries. Circ. Res. 34: 858-868.PubMedCrossRefGoogle Scholar
  75. Zweifach, B.W., and Lipowsky, H.H. (1977). Quantitative studies of microcirculatory structure and function. III. Microvascular hemodynamics of cat mesentery and rabbit omentum. Circ. Res. 41: 380–390.PubMedCrossRefGoogle Scholar

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© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Y. C. Fung
    • 1
  1. 1.Department of BioengineeringUniversity of California, San DiegoLa JollaUSA

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