Mechanics of Solids and Fluids


   
 
Bernoulli's Theorem
This theorem is a consequence of the principle of conservation of energy, applied to ideal liquids in motion. The theorem states that:
 
For the streamline flow of an ideal liquid, the total energy (sum of pressure energy, potential energy and kinetic energy) per unit mass remains constant at every cross-section, throughout the flow.
 
 
Consider a tube AB of varying cross-section and at different heights. Let an ideal liquid (an ideal liquid is incompressible and non-viscous) flow through it in a streamline. Since the liquid is flowing from A to B, p1 > p2. Now A1V1r = A2V2r = m (according to the equation of continuity)

Here A1 > A2 so V1 < V2

 
Now, the work done per second on the liquid at section A = r1A1v1 (v1 is velocity and V1 is volume of liquid per sec)
 
 
 
Now, the work done per second on the liquid at
 
 
Net work done per second on the liquid by the pressure energy in moving from A to B = p1V - p2V
 
The net work done per second, in turn, increases the P.E. per second and also increases the K.E. per sec, from A to B. This is in accordance with the law of conservation of energy.
 
 
 
 
 
 
+ potential energy per unit mass (gh)
 
+ kinetic energy per unit mass is constant for
 
Streamline flow of an ideal liquid
 
Other forms of Bernoulli's theorem
 
 
 
If the liquid flows trough a horizontal tube, the two ends of the tube at the same level, h=0.
 
 
i.e., If p increases, then v decreases so that the their sum is a constant.
 
Limitations of the theorem
 
Since a velocity gradient exists across the tube, the mean velocity of the liquid is to be considered.
 
The viscous drag which comes into play when the liquid is in motion, is not taken into account.
 
In above conservation principle, part of K.E. is converted into heat.
 
  
     
   
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