BERNOULLI'S THEOREM 



75 



ART. 29. ELEMENTARY PROOF OF BERNOULLI'S THEOREM. 



Making the assumption that the fluid is non-viscoua and therefore exerts 

 only normal forces on any bounding surface, the proof of this theorem is 

 easily deduced. Consider the element length B s of a stream tube in 

 such a fluid, the only forces tending to produce motion of this element 

 being its own weight and the normal pres- 

 sures of the surrounding fluid (Fig. 34). 



Let its cross sectional area at the top be 

 " a + B a," and at the bottom " a," the pres- 

 sure intensities at the top and bottom being 

 p B p and p respectively. Let the average 

 normal pressure on the sides of the element 

 be q. Actually q will lie between p and 

 p Bp. 



This normal pressure will have an un- 

 balanced component q . B a along the axis, 

 and, since the fluid is non-viscous, the only 

 effect of this pressure in producing motion 

 in the direction of the axis will be due to 

 this component. 



Let the direction of the axis make an 

 angle 6 with the vertical. 



Then the magnitude of the resultant \ (, _ 



of forces on top and bottom faces of [ = -j 



element, in the direction of its motion] 



Resultant of normal pressures on sides \ 

 of element in same direction 



The only other force acting on the element is its weight, and the 

 resolved part of this in the same direction 



= I W (a + -~) B 



FIG. 34. 



> P) ( a + ^ a) 

 = p o a a B p. 



cos 



Resultant force in) 5. ^ * . J rr 



= p B a a o p q B a -f- W' 



direction of motion] 



= mass X acceleration 



s - cos 



x d v where v = velocity 

 <7 d t of element. 



Putting q = p k & p, and neglecting small quantities of the second 

 order, we get 



^ 8s - ^ v =WaBscose-aBp (1) 



Q ' B t 



