Bernoulli’s Theorem:

In streamline motion of an incompressible liquid, the total energy of the liquid i.e., the sum of potential energy, kinetic energy and pressure energy remains constant at all points.

Proof:

Let is consider a tube of flow \( AB \) as shown in the fig.1 . Let, at the point \( A \), \( {\alpha}_1 \) be the cross sectional area, \( v_1 \) be the velocity of the liquid and \( P_1 \) be the pressure. On the other hand, let \( {\alpha}_2 \) be the pressure, \( v_2 \) be the velocity and \( P_2 \) be the pressure at the point \( B \). Let, \( h_1 \) and \( h_2 \) be the heights of the tube by which the tube is raised from the reference level \( xy \).

The force exerted by the liquid at A is \( P_1{\alpha}_1 \).

The work done by the mass entering the tube through the cross sectional area \( {\alpha}_1 \) at the point A per unit time is \( P_1{\alpha}_1=P_1{V} \).

where, \( V={\alpha}_1{v}_1 \) is the volume of the liquid entering \( A \).

The work done is stored in the liquid and is known as pressure energy.

So the pressure energy per unit volume at the end \( A \) is \( P_1 \).

If \( m \) be the mass of the liquid entering at the point \( A \) per unit time then the pressure energy of the liquid at \( A=m\frac{P_1}{\rho} \).

where, pressure energy per unit mass is \( \frac{P_1}{\rho} \) and \( \rho \) is the density of the liquid.

The kinetic energy of the liquid at the point \( A \) is \( \frac{1}{2}m{v_1}^2 \)

The potential energy of the liquid at the point \( A \) is \( mg{h}_1 \)

So the total energy at the point \( A \) is given by,

\( {m\frac{P_1}{\rho} + \frac{1}{2}m{v_1}^2+ mg{h}_1}\tag{1} \)

Similarly, The total energy at the point B is given by,

\( {m\frac{P_2}{\rho} + \frac{1}{2}m{v_2}^2+ mg{h}_2}\tag{2} \)

From the principle of coservation of energy,

\( m\frac{P_1}{\rho} + \frac{1}{2}m{v_1}^2+ mg{h}_1= m\frac{P_2}{\rho} + \frac{1}{2}m{v_2}^2+ mg{h}_2 \)

or, \( \frac{P_1}{\rho} + \frac{1}{2}{v_1}^2+ g{h}_1= \frac{P_2}{\rho} + \frac{1}{2}{v_2}^2+ g{h}_2 \)

or, \( \frac{P}{\rho}+\frac{1}{2}{v}^2+gh=constant \)

This is Bernoulli’s theorem.

\( \displaystyle{\frac{P}{{\rho}g}+\frac{v^2}{2g}+h=constant} \)

\( \frac{P}{{\rho}g} \) is called the pressure head,

\( \frac{v^2}{2g} \) is called the velocity head,

\( h \) is called elevation head.

So Bernoulli’s theorem may be stated in the following form :

In the streamline motion of an incompressible liquid the sum of the pressure head, the velocity head and the elevation head is constant at all point.