The continuity equation in detail

We investigate a volume element of a pipe section with a variable cross-section area assuming a stationary, laminar flow. The control volume ΔV1 has the mass Δm1 and the cross-section area A1. The fluid moves at the velocity v1 and covers the distance Δs1 in the time interval Δt.

ΔV1 = A1 · Δs1 = A1 · v1 · Δt

with density rho1:

Δm1 = rho1 · A1 · v1 · Δt

expressed as mass flow:

Δm1 / Δt = rho1 · A1 · v1

For the pipe section with the cross-section area A2, the same results: Δm2 / Δt = rho2 · A2 · v2

The principle of conservation of mass implies that for a control volume with a single inlet and a single outlet, the mass flow into the volume must be equal to the mass flow out of the volume.

Δm1 / Δt = Δm2 / Δt

rho1 · A1 · v1 = rho2 · A2 · v2 or in general: rho · A · v = const

For incompressible fluids, rho1 is equal to rho2. The continuity equation results in:

A1 · v1 = A2 · v2 or in general: A · v = const