State and Prove Bernoulli’s Theorem

The Swiss mathematician Daniel Bernoulli established a law for the streamlined flow of an ideal fluid (Which is incompressible and non-viscous). This law is known as Bernoulli’s Theorem. It is an important theorem in Hydrodynamics.

Bernoulli’s Theorem:

For a streamlined flow of an ideal liquid, the sum of the potential energy, kinetic energy, and energy due to pressure per unit volume of the liquid always remains constant at every point on the streamline.

Prove Bernoulli’s Theorem:

If the kinetic energy per unit volume of the liquid = 1/2 ρv2
Potential energy = ρgh, and energy due to pressure = p, then

1/2 ρv2 + ρgh + p = constant
or, 1/2 v2 + gh + p/ρ = constant …equ(1)

This is the mathematical form of Bernoulli’s Theorem. Dividing …equ(1) by we get,

v2 / 2g + h + p / ρg = constant …equ(2)

Here, v2 / 2g is called the velocity head, h is the elevation head and p / ρg is the pressure head. Each of these heads has the dimension of length.

So, velocity head + elevation head + pressure head = constant …equ(3)

According to relation …equ(3), Bernoulli’s theorem can also be stated as follows:

For a streamlined flow of an ideal liquid, the sum of the velocity head, elevation head, and pressure head always remains constant at any point in the liquid.

Bernoulli’s theorem is based on the law of conservation of energy for the streamlined motion of an ideal fluid. The theorem states that the energy remains conserved along any streamline.

When the flow of liquid is horizontal, the height of each point in the liquid is assumed to be the same, h = constant. We can rewrite equation(2) as,

v2 / 2g + p / ρg = constant
or, p + 1/2 ρv2 = constant

This implies that where the velocity of flow is high, the pressure is low and vice-versa.