The elemental force consists of surface forces due to pressure and gravitational forces and its expression is given by,

(2.4.13)

Substitute Eq. (2.4.13) in the linear momentum equation (Eq. 2.4.12).

(2.4.14)

Recalling continuity equation, the first and last term of RHS of Eq. (2.4.14) cancels out. Divide both sides by ρA and rearrange it to obtain the final desired relation;

(2.4.15)

This is the Bernoulli's equation for unsteady, frictionless flow along a streamline. It can be integrated between any two points ‘1' and ‘2' as given below;

(2.4.16)

When the flow is unsteady and incompressible (constant-density), Eq. (2.1.16) reduces to,

(2.4.17)

Eq. (2.4.17) is the Bernoulli equation for steady frictionless incompressible flow along a streamline. Many a times, the Bernoulli constant is known as energy grade line and the height corresponding to pressure and elevation is known as hydraulic grade line.