Derive equation of continuity for steady flow of incompressible liquid.

Consider that a liquid is flowing through a pipe of varying cross-section as shown in figure.

Let the liquid enter at A with velocity 'v₁' whose area of cross-section is 'a₁' and exit from end B with velocity 'v₂' whose area of cross-section is 'a₂'.

Let 'ρ₁' and 'ρ₂' be the densities of liquid at ends A and B respectively.

Now the volume of liquid that enters in one second at end A is given by,

                        V₁ = a₁ v₁

Mass of liquid entering per second at end A is, 

                       m₁ = a₁v₁ρ₁

Similarly the mass of liquid leaving per second at end B is, 
                       m₂ = a₂v₂ρ₂

If there is no source or sink of liquid, then mass of liquid that enters at end A in one second is equal to the mass of liquid that leave in one second.

i.e.               a₁ v₁ ρ₁ = a₂ v₂ ρ₂        ... (1)

If the liquid is incompressible, then
                           ρ₁ = ρ₂.

Therefore equation (1) reduces to

                      a₁v₁= a₂v₂                   ....(2)

Equation (2) is called the equation of continuity.