Continuity Equation Calculator

The continuity equation says that mass cannot disappear or appear in a fluid flow. For an incompressible fluid (water, oil, most liquids at moderate pressure), this reduces to a clean algebraic relationship:

A₁ v₁ = A₂ v₂

A is the cross-sectional area of the pipe (or channel, or any region the fluid passes through), v is the average velocity, and the subscripts 1 and 2 mark two points along the flow. The product Av is the volumetric flow rate Q, and Q stays constant from one cross-section to the next as long as no fluid enters or leaves between them.

What this means in plain English. If a pipe narrows, the fluid speeds up by exactly the area ratio. Halve the diameter and area drops by a factor of 4, so velocity quadruples. This is why putting your thumb over a garden hose shoots water further: smaller exit area, much higher exit velocity, same total flow rate.

Worked example — pipe narrowing. Water enters a 100 mm diameter pipe at 2 m/s. The pipe necks down to 50 mm. Find v₂. A₁ = π(0.05)² = 7.854 × 10⁻³ m². A₂ = π(0.025)² = 1.963 × 10⁻³ m². Area ratio = 4. v₂ = v₁ × A₁/A₂ = 2 × 4 = 8 m/s. Q stays at A × v = 0.01571 m³/s throughout.

Worked example — river flooding. A river runs 20 m wide × 2 m deep at 0.5 m/s, so Q = 20 m³/s. It enters a 10 m wide gorge (same depth assumed). A₂ = 20 m². v₂ = 20/20 = 1.0 m/s — double the speed in the gorge. Doubling speed quadruples the kinetic energy per unit mass, which is why narrow river channels flood violently even when the upstream river looks calm.

Where the simple form breaks down.

• Compressible flow (gases at high speed, generally above Mach 0.3). Density ρ changes with pressure, and the right form becomes ρ₁A₁v₁ = ρ₂A₂v₂. Aircraft wings, jet engines, and rocket nozzles all need the compressible version.
• Branching or leaking pipes. If fluid splits into multiple paths or leaves the control volume, you need to sum: Q_in = Σ Q_out.
• Unsteady flow. A water column being filled or drained does not satisfy steady A v = constant. The full differential form ∂ρ/∂t + ∇·(ρv) = 0 is needed.

Pair with Bernoulli for the complete picture. Continuity tells you about velocity changes due to geometry. Bernoulli’s equation tells you about the pressure changes that drive (and result from) those velocity changes. Together they solve almost every introductory pipe-flow problem.

Real-world hooks.

• The narrowing throat of a carburettor or fuel injector. Air speeds up through the venturi, pressure drops, fuel gets sucked into the airstream.
• Blood vessels branch from the aorta into smaller and smaller arteries. The total cross-sectional area actually grows downstream (because there are many parallel capillaries), so blood slows down dramatically at the capillary bed — exactly where it needs time to exchange oxygen.
• The nozzle of a fire hose. Same Q from the pumper truck, much smaller A at the tip, much higher exit velocity to reach upper floors.

This calculator lets you enter any three of A₁, v₁, A₂, v₂ and solves for the fourth. Q is reported in m³/s and L/s for convenience.