2 The continuity equation

• Take two cross-sections of the tube, area A₁ with speed v₁ and area A₂ with speed v₂.
• Because the liquid is incompressible and cannot accumulate or disappear in the tube, the mass passing each section per second is the same (conservation of mass):

ρ A₁ v₁ = ρ A₂ v₂ → A₁ v₁ = A₂ v₂ (the continuity equation)

or v₁/v₂ = A₂/A₁.

Derivation (by volumes). In a time Δt the liquid at section 1 moves Δx₁ = v₁Δt, so the volume entering is A₁v₁Δt. The same volume must leave through section 2 (incompressible liquid): A₂v₂Δt. Therefore A₁v₁Δt = A₂v₂Δt, i.e. A₁v₁ = A₂v₂.

Meaning. The speed of the liquid at any point is inversely proportional to the cross-sectional area of the tube at that point:

• Wide section (large A) → slow flow, streamlines far apart.
• Narrow section (small A) → fast flow, streamlines close together.

For circular pipes A = πr² = πd²/4, so v₁/v₂ = (r₂/r₁)² = (d₂/d₁)². Halving the diameter makes the speed four times larger.

Both Q_v (m³/s) and Q_m (kg/s) are constant along the tube.