144 HANDBOOK OF PHYSIOLOGY ^ CIRCULATION I 



FIG. I. The pressure -flow relation of a sample of dog de- 

 fibrinated blood: hematocrit 49%; tube radius 480 m; tube 

 length 155 cm; mean shearing stress (dyne. cm~^j = 1.38 X 

 pressure (cm Hg). The "asymptotic" line, on extrapolation, 

 intercepts the line of zero flow at a pressure of 6.3 cm Hg; from 

 its slope the asymptotic relative viscosity of the blood is deduced 

 to be 3.05. The "calculated" line is that derived from the 

 Buckingham-Reiner equation (equation 15), the "friction" 

 pressure being J^ of the "intercept" pressure. The "slippage" 

 line is the pressure-flow relation for "plug" flow within a 

 marginal sheath 2 y. wide. 



Other properties of suspensions. Whetlier it is suffi- 

 cient, by itself, to account for the departure of the 

 pressure-flow Hne of blood from that to be expected 

 from the Buckingham-Reiner equation cannot be 

 decided until some other aspects of the flow of blood 

 have been discussed. 



FLOW OF BLOOD IN VERY SM.'^LL TtrBES 



If the apparent viscosity of blood depends only on 

 the shearing stress applied to it, its value (as measured 

 in terms of the pressure-to-flow ratio), at any given 

 value of the average shearing stress in the tube, should 

 be independent of the radius of the tube. So long as 

 this radius is more than about 100 times the radius 

 of the red cell, this, on the whole, is found to be true. 

 But if the radius of the tube is made smaller than this, 

 the apparent viscosity of the blood is found to be less 

 than the value observed in larger tubes; and the 

 smaller the tube, the smaller the viscosity. The effect 

 is of considerable magnitude, as may be seen in figure 

 2. In a tube of radius 20 //, for example (approximately 

 that of the arterioles), the asymptotic apparent 

 viscosity of blood is about two-thirds of the value 

 obtained in large tubes, such as are ordinarily used 

 in viscometers. 



300 100 50 40 30 25 2 



FIG. 2. The ratio of the viscosity observed in a small tube to 

 the viscosity of the same (or a similar) sample of blood ob- 

 served in a tube of radius greater than 200 ^, plotted against 

 the reciprocal of the radius of the tube. •: Data of FShraeus & 

 Lindqvist (ig): human blood, 3 samples. O: Data of Kiimin 

 (31) : ox blood, 5 samples. C: Data of Bayliss (3) : dog's blood, 

 7 samples. The lines are calculated from equations 18 and 22. 



Attention was first called to this phenomenon in 

 blood by Fahraeus & Lindqvist (19), and it has since 

 been observed by many others (e.g., 3, 23, 31). But it 

 is a phenomenon which has been observed also in 

 many kinds of suspension, such as paint (7), clay, 

 glass beads, etc. Dix & Scott Blair (13) have termed 

 it the "sigma Phenomenon." It has its origin in two 

 features which are peculiar to the flow of suspensions 

 in which the particles are large enough to be com- 

 parable in size with the radius of the tube. 



The Finite Summation Correction 



In these circumstances, we cannot regard the 

 fluid within the tube as a continuous system with 

 uniform viscous properties, and we cannot properly 

 use the method of the infinitesimal calculus in order 

 to deduce the relation between rate of flow and 

 applied pressure from the relation between rate of 

 shear and shearing stress, as is done when deriving 

 the Poiseuille equation. Dix & Scott Blair (13), 

 assuming for simplicity that the shear occurs in 

 layers of suspended fluid, separated by unsheared 

 layers of thickness 5, and applying a method of 

 summation over finite intervals, arri\ed at an equation 

 which may he put in the form: 



P 



7, |_ a fl-J 



(18) 



The apparent viscosity of the blood will thus become 

 smaller as a becomes smaller, and the ratio b/a 

 becomes larger. 



