CHAPTER IV. 



Bernoulli's Theorem Elementary Proof Experimental Verification Applications Con- 

 verging Flow Loss due to sudden change in Section of a Stream Special Cases 

 Gradual Enlargement of Section Change of Pressure across Stream Lines Vortex 

 Motion Forced Vortex Free Vortex Compound Vortex. 



ART. 28. 

 Bernoulli's Theorem. The theorem expressed in the equation ^ + 



F 2 

 1- z = constant, is commonly known, from its discoverer, as Bernoulli's 



^ fl 



theorem. 



It expresses the fact that the total energy of the fluid per Ib. in any 

 stream tube is constant. The three terms denote (1) the pressure energy 



n V 2 



per Ib. -^ ; (2) the kinetic energy per Ib. ^- ; (3) the potential energy 



per Ib. z y where z is the height above some datum to be fixed for any 

 particular problem. The significance of the second and third of these 

 terms is obvious, but some difficulty is often experienced in grasping the 

 precise significance of the first, or pressure energy term. If p be the 

 pressure intensity in pounds per square foot, and W the weight per cubic 



foot, the expression J^gives the height in feet of a column of water which 



would produce the statical pressure "p" 



Now, if water is compressed in a cylinder fitted with a movable piston 

 its pressure is enormously increased by an extremely small movement of 

 the piston. Exactly the same thing would occur if some elastic solid, 

 such as indiarubber, were compressed in the cylinder, and, just as with 

 rubber, so with the water, the work done on the substance during com- 

 pression would be returned during a slow retrograde motion of the piston. 



Since, due to an increase in pressure of p Ibs. per square foot, the pro- 

 portional decrease in the volume of water is given by 



B V _ p ^ 



T ~ 31 X 144 X 10 4 K 

 .'. Work expended per cubic foot in compressing water to this pressure 



r=^ . |? = ^, and this amount of work is stored in the water, in virtue 



