# 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**