METHODS OF MEASURING BLOOD FLOW 



I2 95 



diameter must also be considered : 



fig I. Parabolic velocity profile according to Poiseuille's 

 law in steady laminar flow. For explanation see text. 



dition, the fluid velocity at particular points within 

 the cross section may be of interest, especially in hy- 

 drodynamic studies. In these cases, the flowmeter 

 is calibrated in terms of fluid velocity. 



Since different flow types occur in the circulation, 

 and even in the same blood vessel, any calibration in 

 terms of flow rate presupposes an examination of the 

 dependence of flowmeter response on the velocity 

 distribution over the cross section. 



In case of steady laminar flow, the velocity distri- 

 bution is in the form of a paraboloid, the profile of 

 which is represented in figure i . If v is the velocity 

 at the distance r from the axis and R is the radius of 

 the tube, then we have, according to Poiseuille's law: 



v = K(R z -r z ) 



(I) 



where A' = (AP/Ax)-(i 4m); AP Ay = pressure 

 gradient in axial direction; /x = viscosity of the fluid. 

 The maximum velocitv is at the axis where r = o: 



v *KR' 

 ax 



(2) 



while the lamina adhering to the wall (r = R) is at 

 rest. When equation I is integrated over the cross- 

 sectional area, the flow rate Q, is obtained : 



Q = 2irfvrdr= -jKirR 



(3) 



The average velocity taken over the cross-sectional 

 area is v A : 



v. ' —*— * —KR 



(4) 



It follows from equations i and 4 that the fluid lamina 

 moving at the velocity f., is at a distance of R/y/2 

 from the axis. 



With respect to the performance of some flow- 

 meters, the velocity v K averaged over the radius or 



; VD ; 



o 



From equations 2, 4 and 5 we obtain the ratios: 



(5) 



ax 



= 2:1 



v K = 4:3 



A 



(6) 



(7) 



If the critical Reynolds number is exceeded, the 

 flow becomes turbulent; the profile of the net forward 

 velocities is then flattened and approaches, with 

 increasing turbulence, complete flatness. Other condi- 

 tions, which will be mentioned below, may also give 

 rise to a flattening of this profile. In the extreme case 

 of complete flatness, all fluid particles are moving 

 uniformly at the net forward velocity v A . 



Now we may consider how different flowmeter 

 types will behave when the velocity profile changes 

 from the parabolic shape to complete flatness. A 

 flowmeter which responds to the axial flow only has 

 a relatively high sensitivity when the flow profile is 

 parabolic, since, according to equation 6, Vox'-Va = 

 2:1. When the profile is completely flat, the axial 

 velocity will be as high as the velocity at any other 

 point so that the sensitivity in terms of flow rate is now 

 reduced by 50 per cent from the case of a parabolic 

 profile. In theory, this loss in sensitivity could be 

 avoided by placing the flow-sensing element at a dis- 

 tance of R/y/o. from the axis where, in the parabolic 

 profile, the local velocity equals v A as discussed above. 



If the response of a flowmeter is determined by the 

 sum of the velocities at all points covering the diameter 

 or the radius, this response is proportional to v R . The 

 sensitivity in terms of flow rate will then decrease by 

 25 per cent when the profile changes from the para- 

 bolic shape to complete flatness since vr'.v a = 4:3. 

 It is obvious that the sensitivity of those flowmeters 

 which respond primarily to the velocity v A averaged 

 over the cross-sectional area is independent of the 

 velocity profile. The conditions are more complicated 

 if the flowmeter's response to the fluid velocity is not 

 linear, as is the case with most devices based on hydro- 

 dynamic principles. 



Particular conditions are given in the inlet section 

 of a tube into which fluid is driven from a larger 

 reservoir as is the case in the trunks of the aorta and 

 pulmonary artery. At the entrance of such a tube the 

 velocity profile is flat except in a small marginal zone 

 where a thin boundary layer showing a steep radial 

 velocity gradient exists. When the site of observation 



