Fluid dynamics: flow rate and continuity

Eulerian description of a flow

In fluid mechanics, motion is generally described by the velocity field v(x,y,z,t) at each point in space (Eulerian description), rather than by tracking each particle (Lagrangian description). A streamline is a curve that is tangent to the velocity vector at every point at a given instant; in steady-state conditions, it coincides with the particle trajectory.

Volume flow rate and mass flow rate

The volume flow rate Qv through a cross-sectional area S is defined as:

Qv = v * S (m³/s)

and the mass flow rate as Qm = ρ * Qv (kg/s). These quantities are essential for characterising flow in a pipe, a hydraulic network or a ventilation system.

The continuity equation

For an incompressible fluid flowing through a pipe of variable cross-sectional area, the conservation of mass requires that the volume flow rate remains constant:

v₁ * S₁ = v₂ * S₂

In other words, the velocity increases as the cross-sectional area decreases. This explains why water accelerates at the narrow outlet of a garden hose when the nozzle is pinched.

Example

A pipe with a cross-sectional area of S1 = 20 cm² through which water flows at v1 = 2 m/s narrows to S2 = 5 cm². The new velocity is v2 = v1 * S1/S2 = 2 * 20/5 = 8 m/s.

Common pitfall

The continuity equation in this simple form (v * S = constant) assumes an incompressible fluid; for a gas travelling at high speed (close to or exceeding the speed of sound), variations in ρ must be taken into account, as the density is no longer constant.