Bernoulli's Theorem

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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.