148 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



the motion of a spherical particle in an infinite sheared 

 liquid. If the motion of the liquid is along the x-axis 

 (e.g., along a tube) and its velocity, u, at any point 

 along the .j-axis (e.g., at any particular radius of the 

 tube) varies with .c in a parabolic manner (as in 

 Newtonian flow along a tube), then the particle will 

 have a component of velocity in the ^-axis, in addition 

 to its main velocity along the x-axis (e.g., it will move 

 towards the center of the tube). The magnitude of 

 this component is determined by the fourth power of 

 the radius of the particle, by the parameters defining 

 the relation of u to z, and by the kinematic viscosity 

 of the suspending fluid. We cannot necessarily assume 

 that this conclusion can be applied to the flow of a 

 relatively concentrated suspension in a tube, where 

 account must be taken not only of the interactions 

 between the particles, but also of the boundary 

 conditions introduced by the wall of the tube and the 

 reversal of the velocity gradient at the axis. But it 

 makes it somewhat more probable that some axial 

 movement of the red cells may occur. That it occurs, 

 if at all, to an extent which is quantitatively insuffi- 

 cient to account for the curvature of the pressure-flow 

 line, is suggested by two lines of evidence. 



DIRECT OBSERVATION OF BLOOD FLOWING IN A TUBE. 



Study of the optical transmittance of the marginal 

 layers of the blood in a tube, viewed tangentially, 

 has shown that when the rate of flow is increased, the 

 transmittance is also increased. This has been ob- 

 served in suspensions of red cells in saline solutions 

 (38), and in dog defibrinated blood (4). In the 

 defibrinated blood, however, the magnitude of the 

 increase was very variable as between one sample and 

 another, and had no relation to the magnitude of the 

 reduction in apparent viscosity with increase in rate 

 of flow. The changes in optical transmittance may 

 be due, in part at least, to an orientation of the cells; 

 but the observations suggest that some axial move- 

 ment of the cells probably occurs. But even if it were 

 due entirely to an axial movement, the consequent 

 increase in the effective width of the marginal slippage 

 layer in the defibrinated blood was insufficient to 

 account for the observed reduction in the apparent 

 viscosity of the blood (using equation 23). Neverthe- 

 less, both in the suspension of red cells and in the 

 defibrinated blood, the component of velocity of the 

 red cells towards the axis of the tube, necessary to 

 produce the observed apparent increase in the width 

 of the marginal layer, was an order of magnitude 

 greater than that to be expected from the equation 

 developed by Saffman (37). The evidence is imsatis- 



factory and somewhat confused, but it does not 

 suggest that axial accumulation of the red cells occurs 

 to any considerable extent. 



DEDUCTION FROM THE VARI.\TION OF APPARENT 



VISCOSITY WITH RADIUS OF THE TUBE. The estimates 

 of the width of the marginal sheath, given in the 

 previous section, were all derived from measurements 

 made at large rates of flow. Similar calculations 

 applied to pressure-to-flow ratios observed at very 

 small rates of flow should indicate whether there is 

 any change in the width with change in the rate of 

 flow. Very few adequate measurements are at present 

 available; but it has been concluded (5) that the 

 width of the marginal sheath increases at most by a 

 factor of 2 when the average shearing stress increases 

 from 4 dynes per cm- (or, by extrapolation, from 

 zero) to infinity (i.e., in the asymptotic conditions of 

 flow). These conclusions must, of necessity, be derived 

 from observations in tubes of small radius. It is con- 

 ceivable that when the radius of the tube exceeds 100 

 times the radius of the red cell, the width of the 

 marginal sheath ceases to be independent of the size 

 of the tube, and when the shearing stress is large, 

 becomes more or less proportional to it: the width of 

 the sheath would then be underestimated by the 

 method used. There is no evidence that this occurs, 

 and it does not seem very probable. 



GENERAL CONCLUSIONS 



The reduction in the observed apparent viscosity 

 of blood with reduction in the radius of the tube 

 through which it flows may be ascribed, mainly, to 

 the presence of a marginal sheath of low viscosity, 

 with a width of some i/x to 3/1. The effect does not 

 depend on the existence of non-Newtonian properties 

 of the blood itself, although its magnitude may be 

 affected by them. But the disproportionate increase 

 in rate of flow through a tube of given radius, when 

 the applied pressure is increased, must result from 

 these non-Newtonian properties. In tubes of relatively 

 large radius (300/x or more), the presence of the 

 marginal sheath has little effect on the pressure-to- 

 flow ratio, and this will be sensibly proportional to the 

 apparent viscosity of the blood in the axial core 

 (equation 22a, if z/a = o). From the available 

 evidence, which is rather scanty, it seems that the 

 change in apparent viscosity with change in shearing 

 stress varies very greatly from one sample of blood to 

 another. At a mean shearing stress of i dyne per 



