Biodynamics pp 224-289 | 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 and inertial forces (due to transient acceleration and convective acceleration). 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. The flow is then said to be fully developed. In a fully developed flow, there is an interplay of inertial forces, pressure forces, and viscous forces. In the aorta of man, the length is not sufficient to allow full development of boundary layer ; hence the whole aorta may be considered an entrance region, and the pulse wave can be analyzed approximately by neglecting the viscous stresses. However, arteries divide and divide again. The vessel diameter decreases with each division, and soon the Reynolds number becomes quite small, the entry length becomes only a small multiple of the vessel diameter, and the flow becomes fully developed over most of the length of the vessel. At the same time, the frequency parameter, or the Womersley number, also decreases, so the transient boundary layer also becomes as thick as the tube radius, and the flow becomes in phase with the pressure gradient. Hence, in the smaller arteries the anslysis given in Sec. 3.2 is applicable.


Reynolds Number White Blood Cell Apparent Viscosity Vessel Diameter Blood Vessel Wall 
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. Aroesty, J. and Gross, J. F. (1970). Convection and diffusion in the microcirculation Microvascular Res. 2: 247–267.CrossRefGoogle 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.Google Scholar
  3. Baker, M. and Wayland, H. (1974). On-line volumetric flow rate and velocity profile measurement for blood microvessels Microvascular Res. 7: 131–143.CrossRefGoogle Scholar
  4. Bayliss, W. M. (1902). On the local reactions of the arterial wall to changes in internal pressure J. Physiol. (London) 28: 220–231.Google Scholar
  5. Bülbring, E. (1955). Correlation between membrane potential, spike discharge and tension in smooth muscle J. Physiol. (London) 128: 200–221.Google Scholar
  6. 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.zbMATHCrossRefGoogle Scholar
  7. Burton, A. C. (1951). On the physical equilibrium of small blood vessels Am. J. Physiol. 164: 319–329.Google Scholar
  8. Caro, C. G., Pedley, T. J., Schroter, R. C. and Seed, W. A. (1978) The Mechanics of the Circulation, Oxford University Press, Oxford, New York.zbMATHGoogle Scholar
  9. Chen, T. C. and Skalak, R. (1970). Spheroidal particle flow in a cylindrical tube Appl. Sci. Res. 22 : 403–441.zbMATHGoogle Scholar
  10. Fischer, T. M., Stohr-Liesen, M., and Schmid-Schoenbein, 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.ADSCrossRefGoogle Scholar
  11. Folkow, B. (1964). Description of the myogenic hypothesis Supplements to Circ. Res. 15: pp. I.279-I.285.Google Scholar
  12. Folkow, B. and Neil, E. (1971) Circulation. Oxford University Press, London.Google Scholar
  13. Fronek, K. and Zweifach, B. W. (1974). The effect of vasodilatator agents on microvascular pressures in skeletal muscle Angiologia, 3: 35–39, Unione Intern. di Angiologia.Google Scholar
  14. Fronek, K. and Zweifach, B. W. (1975). Microvascular pressure distribution in skeletal muscle and the effect of vasodilation Amer. J. Physiol. 228: 791–796.Google Scholar
  15. Fry, D. L. (1968). Acute vascular endothelial changes associated with increased blood velocity gradients Circulation Res. 22: 165–197.CrossRefGoogle Scholar
  16. Fry, D. L. (1969). Certain histological and chemical responses of the vascular interface of acutely induced mechanical stress in an aorta of the dog Circulation Res. 24: 93–108.CrossRefGoogle Scholar
  17. Fung, Y. C. (1973). Stochastic flow in capillary blood vessels Microvascular Res. 5: 34–48.CrossRefGoogle Scholar
  18. Fung, Y. C. (1981) Biomechanics: Mechanical Properties of Living Tissues. SpringerVerlag, New York, Heidelberg, Berlin.Google Scholar
  19. Gaehtgens, P. (1980). Flow of blood through narrow capillaries : Rheological mechanisms determining capillary hematocrit and apparent viscosity Biorheology J. 17: 183–189.Google Scholar
  20. Greenfield, A. D. M. (1964). Blood flow through the human forearm and digits as influenced by subatmospheric pressure and venous pressure Supplements to Circ. Res. 14 : pp. I.70-I.75.Google Scholar
  21. Hochmuth, R. M. and Sutera, S. P. (1970). Spherical caps in low Reynolds-number tube flow Chemical Eng. Sci. 25 : 593–604.CrossRefGoogle Scholar
  22. Hyman, W. A. and Skalak, R. (1972). Non-Newtonian behavior of a suspension of liquid drops in fluid flow Amer. Inst. Chem. Eng. J. 18 : 149–154.CrossRefGoogle Scholar
  23. Jendrucko, R. J. and Lee, J. S. (1973). The measurement of hematocrit of blood flowing in glass capillaries by microphotometry Microvascular Res. 6: 316–331.CrossRefGoogle Scholar
  24. Johnson, P.C. (1978) Peripheral Circulation. Wiley, New York.Google Scholar
  25. 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.) Amer. Physiological Society, Bethesda, Md., pp. 409–442.Google Scholar
  26. Johnson, P. C. and Intaglietta, M. (1976). Contributions of pressure and flow sensitivity to autoregulation in mesenteric arterioles Am. J. Physiol. 231: 1686–1698.Google Scholar
  27. Kaley, G. and Altura, B. M. (1977) Microcirculation, Vols. 1 & 2. University Park, Baltimore, MD.Google Scholar
  28. Lamb, H. (1932) Hydrodynamics. 6th ed. Cambridge Univ. Press. Reprinted by Dover, New York.zbMATHGoogle Scholar
  29. Lee, T. Q., Schmid-Schoenbein, G. W., and Zweifach, B. W. (1983). An application of an improved dual-slit photometric analyzer for volumetric flow rate measurements in microvessels Microvascular Res. 26 : 351–361.CrossRefGoogle Scholar
  30. Lew, H. S. and Fung, Y. C. (1969a). On the low-Reynolds-number entry flow into a circular cylindrical tube J. Biomechanics 2: 105–119.CrossRefGoogle Scholar
  31. 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
  32. Lew, H. S. and Fung, Y. C. (1969c). Flow in an occluded circular cylindrical tube with permeable wall Zeit angew. Math. Physik 20 (5) : 750–766.zbMATHCrossRefGoogle Scholar
  33. Lew, H. S. and Fung, Y. C. (1970a). Plug effect of erythrocytes in capillary blood vessels Biophysical J. 10: 80–99.ADSCrossRefGoogle Scholar
  34. Lew, H. S. and Fung, Y. C. (1970b). Entry flow into blood vessels at arbitrary Reynolds number J. Biomechanics 3: 23–38.CrossRefGoogle Scholar
  35. Lighthill, M. J. (1968). Pressure-forcing of tightly fitting pellets along fluid-filled elastic tubes J. Fluid Mech. 34: 113–143.MathSciNetADSzbMATHCrossRefGoogle Scholar
  36. Ling, S. C., Atabek, H. B., Fry, D. L., Patel, D. J. and Janicki, J. S. (1968). Application of heated-film velocity and shear probes to hemodynamics studies Circulation Res. 23: 789–801.CrossRefGoogle Scholar
  37. 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 Microvascular Res. 14: 345–361.CrossRefGoogle Scholar
  38. Lipowsky, H. H. and Zweifach, B. W. (1978). Application of the “two-slit” photometric technique to the measurement of microvascular volumetric flow rates Microvascular Res. 15: 93–101.CrossRefGoogle Scholar
  39. Lipowsky, H. H., Usami, S. and Chien, S. (1980). In vivo measurements of “apparent viscosity” and microvessel hematocrit in the mesentery of the cat Microvascular Res. 19: 297–319.CrossRefGoogle Scholar
  40. Majno, G. (1965). Ultrastructure of the vascular membrane. In W. F. Hamilton and P. Dow (eds.) Handbook of Physiology, Sec. 2 Circulation, Vol. 3. American Physiological Soc., Washington, D.C. pp. 2293–2375.Google Scholar
  41. Nellis, S. N. and Zweifach, B. W. (1977). A method for determining segmental resistances in the microcirculation from pressure-flow measurements Circulation Res. 40 (6) : 546–556.CrossRefGoogle Scholar
  42. Norberg, K. A. and Hamberger, B. (1964). The sympathetic adrenergic neuron Acta Physiol. Scandinay. 63 (Suppl. 238).Google Scholar
  43. Ogawa, Y. (1976). A morphological study of microvascular beds in the cutaneous area J. of Yokohama City Univ. Ser. Sport Sci. and Med. 5: 1–37.Google Scholar
  44. Oseen, C. W. (1910). Über die Stokessche Formel und über die verwandte Aufgabe in der Hydrodynamik Arkiv. Mat. Astron. Fysik. 6(29).Google Scholar
  45. Øien, A. H. and Aukland, K. (1983). A mathematical analysis of the myogenic hypothesis with special reference to autoregulation of renal blood flow Circ. Res. 52: 241–252.CrossRefGoogle Scholar
  46. Prothero, J. and Burton, A. C. (1961, 1962). The physics of blood flow in capillaries. I. The nature of the motion Biophysical J. ,1: 567–579. II. The capillary resistance to flow ibid, 2: 199–212.Google Scholar
  47. 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 Mechanics 2: 237–262.MathSciNetADSzbMATHCrossRefGoogle Scholar
  48. Rothe, C. F., Nash, F. D. and Thompson, D. E. (1971). Patterns in autoregulation of renal blood flow in the dog Am. J. Physiol. 220: 1621–1626.Google Scholar
  49. Rouse, H. (1959) Advanced Fluid Mechanics. Wiley, New York.Google Scholar
  50. Sagawa, K., Kumoda, M. and Schramm, L. P. (1974). Nervous control of the circulation. In Cardiovascular Physiology (A. C. Guyton and C. E. Jones, eds.). Butterworths, London, University Park Press, Baltimore, Vol. 1, p. 197–232.Google Scholar
  51. Schlichting, H. (1962) Boundary Layer Theory. McGraw-Hill, New York.Google Scholar
  52. Schmid-Schoenbein, G. W., Fung, Y. C. and Zweifach, B. (1975). Vascular endothelium-leucocyte interaction : Sticking shear force in venules Circulation Res. 36: 173–184.CrossRefGoogle Scholar
  53. Schmid-Schoenbein, G. W., Skalak, R., Usami, S. and Chien, S. (1980a). Cell distribution in capillary networks Microvascular Res. 19: 18–44.CrossRefGoogle Scholar
  54. Schmid-Schoenbein, G. W., Usami, S., Skalak, R. and Chien, S. (1980b). The interaction of leukocytes and erythrocytes in capillary and postacpillary vessels Microvascular Res. 19 : 45–70.CrossRefGoogle Scholar
  55. Schmid-Schoenbein, H. and Wells, R. E. (1969). Fluid drop-like transition of erythrocytes under shear Science 165: 288–291.ADSCrossRefGoogle Scholar
  56. Secomb, T. W. and Skalak, R. (1982). A two-dimensional model for capillary flow of an asymmetric cell Microvas. Res. 24: 194–203.CrossRefGoogle Scholar
  57. Skalak, R., Chen, P. H. and Chien, S. (1972). Effect of hematocrit and rouleaux on apparent viscosity in capillaries Biorheology 9: 67–82.Google Scholar
  58. Skalak, R., Tozeren, A., Zarda, P. R. and Chien, S. (1973). Strain energy function of red cell membranes Biophysical J. 13 : 245–264.ADSCrossRefGoogle Scholar
  59. Smaje, L., Zweifach, B. W. and Intaglietta, M. (1970). Micropressures and capillary filtration coefficients in single vessels of the cremaster muscle of the rat Microvascular Res. 2: 96–110.CrossRefGoogle Scholar
  60. Stokes, G. G. (1851). On the effect of the internal friction of fluids on the motion of pendulums Trans. Cambridge Philosophical Soc. 9:p. 8 Mathematical and Physical Papers, Vol 3, pp. 1–141.ADSGoogle Scholar
  61. Svanes, K. and Zweifach, B. W. (1968). Variations in small blood vessel hematocrits produced in hypothermic rats by microocclusion Microvascular Res. 1: 210–220.CrossRefGoogle Scholar
  62. Targ, S. M. (1951) Basic Problems of the Theory ofLaminar Flows (in Russian). Moskva.Google Scholar
  63. Tong, P. and Fung, Y. C. (1971). Slow viscous flow and its application to biomechanics J. Appl. Mechanics 38: 721–728.ADSzbMATHCrossRefGoogle Scholar
  64. Tözeren, H. and Skalak, R. (1978). The steady flow of closely fitting incompressible elastic spheres in a tube J. Fluid Mechanics 87 : 1–16.ADSzbMATHCrossRefGoogle Scholar
  65. Tözeren, H. and Skalak, R. (1979). Flow of elastic compressible spheres in tubes J. Fluid Mechanics 95(4): 743–760.ADSzbMATHCrossRefGoogle Scholar
  66. Wang, H. and Skalak, R. (1969). Viscous flow in a cylindrical tube containing a line of spherical particles J. Fluid Mechanics 38: 75–96.ADSzbMATHCrossRefGoogle Scholar
  67. Wayland, H. (1982). A physicist looks at the microcriculation Microvascular Res. 23: 139–170.CrossRefGoogle Scholar
  68. Wiedeman, M. P., Tuma, R. F. and Mayrovitz, H. N. (1981) An Introduction to Microcirculation. Academic Press, New York.Google Scholar
  69. Wiederhielm, C. A., Woodbury, J. W., Kirk, S. and Rushmer, R. F. (1964). Pulsatile pressure in microcirculation of the frog’s mesentery Amer. J. Physiol. 207: 173–176.Google Scholar
  70. Yen, R. T. and Fung, Y. C. (1978). Effect of velocity distribution on red cell distribution in capillary blood vessels Amer. J. Physiol. 235(2): H251-H257.Google Scholar
  71. Yih, C. S. (1969) Fluid Mechanics. McGraw—Hill. New edn., West River Press, Ann Arbor, MI.Google Scholar
  72. Zarda, P. R., Chien, S. and Skalak, R. (1977). Interaction of a viscous incompressible fluid with an elastic body. In Computational Methodsfor Fluid-Structure Interaction Problems (Belytschko, T. and Geers, T. L. (eds.)). American Society of Mechanical Engineers, New York, pp. 65–82.Google Scholar
  73. Zweifach, B. W. (1974). Quantitative studies of microcirculatory structure and function. I. Analysis of pressure distribution in the terminal vascular bed Circulation Res. 34: 843–857.CrossRefGoogle Scholar
  74. Zweifach, B. W. (1974). II. Direct measurement of capillary pressure in splanchmic mesenteries Circulation Res. 34: 858–868.CrossRefGoogle 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 Circulation Res. 41(3) : 380–390.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1984

Authors and Affiliations

  • Y. C. Fung
    • 1
  1. 1.University of CaliforniaSan Diego, La JollaUSA

Personalised recommendations