The Circulation, 



h, 



61. The Flow of Fluids through Tubes. 



Toricelli's Theorem (1043) states that the velocity of efflux (v) of a fluid- 

 through an opening at the bottom of a cylindrical 

 vessel is exactly the same as the velocity which a body 

 falling freely would acquire, were it to fall from the 

 surface of the fluid to the base of the orifice of the out- 

 flow. If h be the height of the propelling force, the 

 velocity of efflux is given by the formula 



v = V 2 g h (where g 9 '8 metre). 



The rapidit} 7 of outflow increases (as shown experi- 

 mentally) with increase in the height of the propelling 

 force, h. The former occurs in the ratio, 1, 2, 3, when h 

 increases in the ratio, 1, 4, 9 i. e., the velocity of efflux 

 is as the square root of the height of the propelling 

 force. Hence, it follows that the velocity of efflux 

 depends upon the height of the liquid above the orifice 

 of outflow, and not upon the nature of the fluid. 



Resistance. Toricelli's theorem, however, is only 

 valid when all resistance to the outflow is absent ; but, 

 in fact, in every physical experiment such resistance 

 exists. Hence, the propelling force, h, has not only to 

 cause the efflux of the fluid, but has also to overcome 

 resistance. These two forces may be expressed by the 

 heights of two columns of water placed over each other 

 viz., by the height of the column of water causing the 

 outflow, F, and the height of the column, D, which 



Fig. 38. 



Cylindrical vessel filled 

 with water h, height 

 of the column of fluid ; 

 D, height of column of 

 fluid required to over- 

 come the resistance ; 

 and F, height of 

 column of fluid caus- 

 ing the efflux. 



overcomes the resistance opposed to the outflow of the fluid. So that 



h = F + D. 



62. Propelling Force Velocity of the Current, 

 and Lateral Pressure. 



In the case of a fluid flowing through a tube, which it fills completely, we have 

 to consider the propelling force, h, causing the fluid to flow through the various 

 sections of the tube. The amount of the propelling force depends upon two 

 factors : 



(1.) On the velocity of the current, v ; 



(2.) On the pressure (amount of resistance) to which the fluid is subjected at the 

 various parts of the tube, D. 



(1.) The velocity of the current, v, is estimated (a, ) from the lumen, I, of the 



