THE RHEOLOGY OF BLOOD 



147 



SO that 



where an is the dynamic liematocrit, and «<„ is the 

 hematocrit value of the blood in the idealized axial 

 core. By analogy with equation 19, for the total flow 

 through the tube, we can write for the flow of red 

 cells : 



CtoOji = ashQ_,th + CIC0Q.CO 



Putting ash = o (since it is supposed to be free from 

 red cells) we get : 



Olto/ao = Qh/Q,co 



Inserting the values of Q.fo and Qj, from equations 21 

 and 22 we find: 



aco _ I - 7H1 — I A*) 

 ao 2->2 - y(2 - 1/7,6) 



1 - 7^1 - l/r,b) 



2 — 7=(2 — 1/17*) 



(25) 



(26) 



The corresponding expression for the relative transit 

 times may be derived by inserting the value of ao 

 from equation 26 in equation 24. Alternatively, we 

 may put ; 



and 



«e = Q.J'^l'^a'' = olcaQ^cohlV 



(IJ-Ka^ = [QsA + (i - a„)a„]/7ra2 



and then insert the values of Q^sh and Qco from equations 

 20 and 21. 



Thus from measured values of the dynamic hemato- 

 crit, or of the relative transit times of cells and plasma, 

 the value of 7 and hence the width of the marginal 

 sheath may be estimated from equations 24 and 26. A 

 method of successive approximation may have to be 

 used, since the value of a^o, and thus the appropriate 

 value of r;;,, depends on the value of 7. 



We may take as illustrative examples first some 

 measurements of Fahraeus (18) of the dynamic 

 hematocrit of human ijlood flowing in glass tubes. If 

 these are inserted in equation 26, it appears that in 

 the two smaller tubes used, 47. 5m and 25^ in radius, 

 the width of the equivalent cell-free marginal zone 

 was 12^ and lO/i, respectively. Such values are not 

 consistent with the measurements of optical trans- 

 mittance (4, 38) to be discussed below, and are very 

 much larger than those to be expected from the 



viscosity measurements, as discussed above. The origin 

 of the discrepancy is not known. No measurements of 

 transit times seem to have been made on blood flowing 

 in glass tubes. But as a second illustrative example, 

 we may take the measurements of Freis et al. (20) of 

 the transit times of plasma and of cells between the 

 brachial artery of a man and a large vein draining 

 chiefly the deeper tissues of the arm; these were found 

 to be about 10 sec and 7 sec, respectively, so that 

 'tp/'tc = I ■45- If oio is taken to be 45 per cent, ac must 

 have been 36.5 per cent. Applying equation 26, we 

 find that this value of an would be obtained if all but 

 I sec of the transit times was occupied in flow through 

 a tube lo/i in radius, with an equivalent cell-free 

 marginal sheath 2/u in width. Such a solution is a 

 reasonable one, but of course it is not unique, and 

 many others are possible. 



ORIGIN OF THE .^NOM.^LOUS FLOW 

 PROPERTIES OF BLOOD 



If the flow of blood along a tube, when the applied 

 pre.ssure is equal to or less than the "friction" pres- 

 sure, results from the movement of a solid plug within 

 a slippage layer of constant width, the pressure-flow 

 line should be straight, as may be seen from equation 

 22. This is by no means so, as is illustrated for example 

 in figure 1 . We must suppose, therefore, either that 

 the width of the slippage layer increases rather 

 rapidly, with increase in the applied pressure, to a 

 value some 10 times as great as that likely to result 

 from the wall effect, or that the axial core is not a 

 solid plug but has a finite viscosity which decreases 

 with increase in the shearing stress, even when this is 

 less than the value corresponding to the friction 

 pressure. 



Motion of Red Cells Toward the 

 Axis of the Tube 



Observations made on blood flowing in the small 

 vessels of an animal have led to the idea that when a 

 suspension is made to flow in a tube, the particles 

 leave the region near the walls and collect near the 

 axis. There is little evidence that this occurs generally 

 in all kinds of suspension, and the evidence that it 

 occurs in blood is by no means conclusive. Until 

 recently, moreover, there was no theoretical reason to 

 expect that such a movement should occur. SafTman 

 (37), however, as a rider to his studies on the motion 

 of spheroidal particles in a viscous liquid, has studied 



