Question

Derive equation of continuity.

Answer 1

Equations of continuity. Consider a non-viscous liquid in stream line flow through a tube AB of varying cross-section.

Let a₁, a₂ = area of cross-section of the tube at A and B respectively,

v₁, v₂ = velocity of flow of liquid at A and B respectively.

ρ₁, ρ₂ = densities of liquid at A and B respectively.

∴ Volume of liquid entering, per second at A

= a₁v₁

Mass of liquid entering per second at A

= a₁v₁ρ₁

Similarly, mass of liquid leaving per second at B

= a₂v₂ρ₂

If there is no loss of liquid in the tube and the flow is steady, then

Mass of liquid entering per second at A

= Mass of liquid leaving per second at B

Or a₁v₁ρ₁ = a₂v₂ρ₂ …(1)

If the liquid is incompressible,

then ρ₁ = ρ₂

From (1) a₁v₁ = a₂v₂

or av = a constant …(2)

This is knows as equation of continuity.