THE RHEOLOGY OF BLOOD 



143 



sheared. If the long axis of a rod lies along the axis 

 of the tube through which the suspension is flowing, 

 or the flat surface of a disc is parallel to the wall, 

 there will be a minimum disturbance to the lines of 

 flow, and the presence of the particle will have the 

 least efTect on the viscosity of the system. Such an 

 orientation, however, is unstable, and any small 

 departure from it would result in a rotation of the 

 particle through 180°. Indeed, if the motion of any 

 one particle is not seriously interfered with by the 

 presence of neighboring particles, it may be observed 

 to undergo continuous rotation as it passes down the 

 tube. But its angular velocity is a minimum when it 

 is in the state of metastable orientation along the 

 lines of flow, and is a maximum when it is at right 

 angles to this; there will thus be a ''statistical" 

 orientation in the direction which leads to a minimum 

 value of the viscosity of the suspension. The difference 

 between the maximum and minimum angular 

 velocities will increase with increase in the rate of 

 shear, and so also will the fraction of the particles 

 which are in the optimum condition of orientation 

 at any moment : the viscosity will tlius be shear- 

 dependent. 



Direct microscopic observation of suspensions of 

 small rod-shaped particles, flowing through a tube, 

 has shown that with increase in the rate of flow the 

 particles become increasingly orientated along the 

 axis of the tube (16); the viscosity of the suspension 

 also decreases with increase in the rate of shear. A 

 similar effect has been observed in suspensions of 

 tobacco mosaic virus, the average orientation of the 

 particles being measured in terms of the birefringence 

 of the suspension (36). When blood is sheared, both 

 its optical transmittance (30, 41) and its electrical 

 conductivity, measured in the direction of flow (12, 

 41), increase. These changes would be expected if 

 the red blood cells became partially orientated along 

 the lines of flow. It is probable, therefore, that 

 orientation effects will contribute to the reduction in 

 the apparent viscosity of blood with increase in the 

 rate of shear. 



Coherence Resistance and Friction 



In 1912, Hess (24), having observed that the rate 

 of flow of blood through a tube decreased, as the 

 pressure head was reduced, more than in proportion 

 to the reduction in pressure, suggested that this 

 resulted from the existence of a "coherence resistance" 

 between the red cells, independent of the rate of 

 shear, in addition to the viscous resistance, propor- 



tional to the rate of shear. Bingham (6), indepen- 

 dently, made the analogous suggestion that the 

 anomalous flow properties of paints were due to 

 "friction" between the particles of the pigment in 

 suspension. 



The essential feature of both hypotheses is the 

 replacement of the simple Newtonian assumption, 

 that the shearing stress (t) is equal to the product of 

 the viscosity (57) and the rate of shear (D), by an 

 equation of the form: 



■n-D 4-/ 



(14) 



where / is the "friction" per unit area. Thus D is 

 zero unless r is equal to, or is greater than / (/ is of 

 the nature of a "static" friction or "stiction," and D 

 does not become negative when t is less than /). The 

 expected relation between the rate of flow of a 

 suspension in which there is friction and the applied 

 pressure head was worked out by Buckingham (10) 

 and by Reiner (35), independently. It may be written 

 in the form : 



'? \ 3 3^7 



(15) 



There will be a finite pressure head Pj ijelow which 

 the suspension will not flow at all; if the applied 

 pressure head is large compared with Pj, the line 

 relating the rate of flow, Q,6, to the pressure head will 

 approach an asymptotic straight line; this line, when 

 extrapolated back to Qj, = o, will cut the axis of 

 pressure at 4P//3. 



The Buckingham-Reiner equation is not, in fact, 

 obeyed either by blood or by paint. It has not been 

 possible to discover a value of the pressure head below 

 which the flow ceases. If there is such a critical 

 pressure, it is an order of magnitude smaller than the 

 intercept of the asymptotic straight line on the axis 

 of pressure. This is illustrated in figure i , in which 

 the thick line represents the observed pressure-flow 

 relation of a sample of dog defibrinated blood under 

 small values of the shearing stress, the "asymptotic" 

 line is the extrapolation of the straight line obtained 

 under very large values of the shearing stress, and the 

 "calculated" line is the pressure-flow relation deduced 

 from the Buckingham-Reiner equation. Bingham 

 accounted for this by supposing that when the applied 

 pressure was less than the "friction" pressure, a 

 "plug" of unsheared suspension moved down the 

 tube within a thin layer of suspending fluid between 

 it and the wall of the tube. The existence of such a 

 "slippage" layer is to be inferred also from several 



