# 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.