THE RHEOLOGY OF BLOOD 



139 



the linear velocity, u, of the liquid at any radius r is 

 given by: 



u, = — (a- 



r') 



(3) 



where 7; is the coefficient of viscosity, as already 

 defined. This equation defines the "parabolic dis- 

 tribution of velocities" : if iir is plotted against r, the 

 curve obtained is a parabola, with its vertex at the 

 axis of the tube. Finally, the volume rate of flow 

 through an elementary annulus of radius r and width 

 di will be ■2iTr-dr-Ur. Thus the total rate of flow will 

 be given by : 



■dr 



Jo 

 On inserting ecjuation 3 and integrating, we get: 



Q. 



fir, I 



(4) 



This is the well-known "Poiseuille equation." 

 Poiseuille himself deduced it from his experimental 

 observations on the flow of various kinds of liquid 

 through tubes of different dimensions, and used an 

 empirical constant in place of the coefficient 7r/8?j. 

 The deduction of the eqviation from the Newtonian 

 assumption, of the proportionality between rate of 

 shear and shearing stress, was made by Wiedemann 

 in 1856, and more precisely by Hagenbach in 1870. 

 If we combine equation 4 with equation 3, we get: 



^Q. 



(l -rV<22) 



(3a) 



The mean linear \elocity, w, is gi\en by: u = d'lra-, 

 and is thus one-half the maximum linear velocity at 

 the axis of the tube, where r = o. We may deduce 

 further, that the rate of change of velocity with 

 radius — i.e., the rate of shear — is given by: 





(5) 



The rate of shear at any value of the radius is, of 

 course, 1/77 times the shearing stress at that radius, is 

 zero at the axis of the tube, and has a maximum value 

 at the wall of the tube. It is important to remember, 

 however, that in deriving equation 3 it is assumed 

 that 7) is independent of r. When the liquid is non- 

 Newtonian, for example, or in certain other conditions 

 to be di.scussed later, this will not be justified; the 

 distribution of velocities will not be parabolic, and 



the velocity and rate of shear at any value of the 

 radius cannot be correctly calculated by means of 

 equations 3a and 5. 



According to the Poiseuille equation 4, the rate of 

 flow, CI, is directly proportional to the applied pressure 

 head, P. This will be true only when the pressure 

 gradient is uniform and the flow is laminar. The 

 first of these requirements cannot be satisfied in 

 practice without special arrangements, although the 

 error introduced can be made negligible; the second 

 is satisfied only if the pressure gradient and rate of 

 flow are less than certain limiting values. 



T/ie Kinetic Eiiergy Correction 



The fluid in the reservoirs at each end of the tube 

 is sensibly at rest. On entering the tube, each element 

 must be accelerated to its steady velocity Ur; this 

 requires the expenditure of power, which can be 

 derived only from the pressure head applied. On 

 leaving the tube, the kinetic energy of the fluid will, 

 in general, be dissipated in the reservoir as heat. In 

 order to create a volume rate of flow Q., it is necessary, 

 therefore, to apply a pressure head greater than that 

 expected from the Poiseuille equation. The conditions 

 at the entrance to the tube, before the parabolic 

 distribution of \elocities has been taken up, are 

 complicated; but it is generally accepted that the 

 value of this additional pressure head is given by: 



pQ: 



p = m 



7ra< 



(6) 



where p is the density of the liquid and m is a constant 

 the value of which is close to i.io (its precise value 

 depends on whether the tube is cut off sharplv at the 

 ends, or is opened out into a bell-mouth, and may 

 vary with the rate of flow). It is always desirable to 

 ensure that the kinetic energy correction is small — i.e., 

 that /) is small compared with P. Equation 6 may be 

 written in the form: 



P ^ ^ . — . p 

 P 647,' ' f- ' 



(6a) 



The correction, therefore, will become increasingly 

 significant as the applied pressure becomes greater. 

 If the correction is significant, but is not applied, the 

 rate of flow corresponding to a gix-en pressure head 

 will be smaller than that expected from the Poiseuille 

 equation: the line relating rate of flow to applied 

 pressure will become curved, concave to the axis of 

 pressure. 



