THE RHEOLOGY OF BLOOD 



145 



The Kail Effect 



The particles of a suspension cannot penetrate into 

 the wall of the vessel in which they are contained. 

 Imagine a plane surface within a suspension of 

 uniformly distributed spherical particles; the quantity 

 of suspended material, per unit volume, will be the 

 same on each side of the plane, and the plane will 

 pass through particles the centers of which lie on 

 each side of it within a distance equal to their radii. 

 If, now, this plane is made the interface between the 

 wall of the vessel and the suspension within it, we 

 shall have to remove not only all the particles which 

 were on the wall side of the plane, but also all those 

 on the suspension side throttgh which the plane 

 passed. Thus there will be a relative deficit of sus- 

 pended material up to a distance from the wall equal 

 to the radius of the particles. According to Vand (39), 

 this layer in which the concentration of particles is 

 reduced behaves hydrodynamically, when the suspen- 

 sion is sheared, as if it were a layer completely free 

 from suspended material with a width 1.301 times 

 the radius of the particles. Experimental evidence 

 suggests, however, that the equivalent width of this 

 hypothetical slippage layer is about one-half of that 

 e-xpected theoretically (26, 33, 40). In blood, there- 

 fore, we may expect its width to be some i/i to 3 ;u. 



Effect of the Marginal Sheath on 

 the Pressure-to-Flow Ratio 



When blood (or any other suspension) flows through 

 a tube, therefore, we may expect that there will be a 

 marginal slippage layer, or sheath, of relatively low 

 viscosity, surrounding an axial core of greater viscos- 

 ity. When a volume Q.^ of blood enters the tube per 

 second, part of it will travel in the marginal sheath 

 and part of it in the axial core. Let the volume 

 emerging from the sheath be (),,ft and the volume 

 emerging from the core be ()c„. Then, since the same 

 total quantity of blood must leave the tube in unit 

 time as enters it, we must have: 



Q.C. 



(19) 



There will not be a sharp boundary between the core 

 and the sheath; but in order to simplify the analysis, 

 it is convenient to assume that there is such a bound- 

 ary, and that the hypothetical sheath contains no 

 red cells, and has a viscosity equal to that of the 

 plasma. The width of this boundary may be defined 

 in one of two ways: for the moment, we define it as 

 being such that the flow properties of the hypothetical 



system (i.e., the ratio (Ib/P in any given conditions of 

 flow) are the same as those actually observed in the 

 sample of blood used. Let the radius of the tube be a 

 and the radius of the axial core be 70. Then, as in the 

 derivation of the Poiseuille equation (equation 4), we 

 have : 



f" 



Qaft = 2jr / r-Ur-dr 



^1a 



where Ur is given by equation 3. This reduces to: 



d.k = Gp(i - y')^-P (20) 



where Gp is defined by equation 12. Similarly, for the 

 axial core, we have: 



= -/' 



Jo 



r-u,-dr ■\- ■Kf'^a'-Uya 



The value of Ur is now defined by equation 3, after 

 inserting the viscosity of the blood in the core; and 

 the additional term is due to the fact that the outer 

 margin of the core has a velocity equal to that of the 

 inner margin of the sheath (i.e., «.,„). After integration 

 and simplification we get: 



e.o = Gp[27^(i - 7=) -I- yyvb]P 



Thus 



Q.b = Q.SH + Q.CO 



= Gp[(i - 7") + y*/r,b]P 

 By using equation 13, we find: 



t/r," = (I - 7*) + y'hb 



(21) 



(22) 



(23) 



If the width of the marginal sheath is z, then 7 = i 

 — C a; and if z/a is small compared with i, we may 

 simplify equation 22 to: 



Qb = Gp[(i - 4z/a)/nb + 4z/a)]P (22a) 



The modification of the Poiseuille equation, in order 

 to allow for slippage, by multiplying a^ by the quantity 

 (i -|- 4z/a) was first suggested by Helmholtz, and 

 was applied to the flow of paints by Buckingham (10). 

 The viscosity of the blood in the axial core, rji,, will be 

 a function of the radius, a, as given by equation 18. 

 If, moreover, c is not small compared with a, r]b will 

 vary with 7 : the smaller the radius of the axial core, 

 the greater will be the concentration of red cells 

 witliin it, as will be discussed later. 



We see from this that unless z is porportional to 

 a — and there is no reason why it should be — the 

 ratio of P to Qj,, and thus the observed apparent 



