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