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.

I guess both are true..................

Case 1

U use a machine that decreases the air pressure by sucking in some air..............To satisfy Bernoulli's equation, the air from the surroundings starts rushing in to fill the space(P decreases and v increases)...........

This proves statement 2...........

Case 2

We use two machines, one that can measure pressure and another, that pushes air out to increase the velocity..........As soon as we start the second machine, we'll notice that the reading in the first machine decreases, indicating a decrease in air pressure..............

This proves statement 1.................

CONCLUSION: Both the parameters, fluid pressure at a region and fluid velocity at that region, can be causes or results, as the need be.........All depends on what process is going on...............