ON FLUID FLOW; BLOOD 213 



If P, and P 2 are the pressures measured at the points 1 and 2 in the tube, 

 and R is the distance from the center bore of the tube, the driving force is 

 given by 



ttR 2 (P x - P 2 ) 



The frictional force on the layer at distance R from the center is propor- 

 tional to the area of the layer (2ttRI), and to the velocity difference between 

 the layer we are considering and its nearest neighbors; in the limit this is 

 dv/dR. 



After the two forces have been equated, integration (or summing all veloc- 

 ities from that at the center of the tube to zero at the wall) gives 



P — P 



v = £j £2( r 2 _ R 2) 



4/ 



where r is the radius of the tube. This expression gives the linear speed of 

 the layer which is R cm from the center. is the proportionality constant, 

 and is called the fluidity (the higher its value the higher the velocity). 



The total volume of fluid flowing per second through the tube is calculated 

 by summing all the elemental volumes, 2irRdR, for which v is expressed. The 

 result is the celebrated Poiseuille equation which' expresses rate of flow 

 (cc/sec) of liquid through a tube of radius r and length / under an applied 

 pressure difference of AP = P ] — P 2 : 



irr 4 

 dVldt = AP cc/sec 



8/ 



If A P is given in dynes per cm 2 , r and / in cm, and the speed of flow in cc per 

 sec, the fluidity, 0, must be cm per sec for a force gradient of 1 dyne per cm; 



i.e., has the dimensions: — /— — . It is the velocity of flow of a fluid 



sec/ cm 



under a unit force gradient. 



The case for gases is slightly more complicated because of the added fact 



that the volume depends strongly upon the pressure and the temperature. 



With the proper modifications the expression for rate of flow of gases 



approximates: 



Trr 4 P. 2 - P 2 



dV/dt = ! 



16/ - P n 



if P is the pressure at which the volume is measured. 



Fluidity, 0, and Viscosity, rj 



Table 8-8 gives values of the fluidities of various substances at different 

 temperatures. Of the liquids, ether is the most fluid one listed; glycerine at 

 0°C is the least fluid — indeed at 0°C it is almost a glass! The fluidity of 



