Continuity Equation Calculator
Enter the upstream area and velocity and the downstream area to find how fast an incompressible fluid moves after the pipe changes width.
See where the fluid speeds up
Enter A₁, v₁, and A₂ and the calculator returns the downstream velocity v₂ in metres per second from A₁v₁ = A₂v₂.
Use SI units
Square metres for both areas and metres per second for the inlet velocity give the outlet velocity in m/s — keep the units consistent.
What is the continuity equation?
Conservation of mass for a flowing fluid
This continuity equation calculator finds how fast a fluid moves after a pipe changes width. The continuity equation, A₁v₁ = A₂v₂, expresses conservation of mass for an incompressible fluid: the same volume passes every cross-section of a pipe each second. Because the volumetric flow rate — the area multiplied by the velocity — stays constant along the pipe, narrowing the channel forces the fluid to speed up and widening it lets the fluid slow down. Enter the upstream area A₁ and velocity v₁ together with the downstream area A₂, and the calculator returns the downstream velocity v₂ in metres per second. It is the number behind a garden hose nozzle, the throat of a venturi meter, and the design of any duct that must carry a fixed flow.
Enter the upstream area and velocity and the downstream area to get the downstream velocity in metres per second instantly.
Rearranging A₁v₁ = A₂v₂ for the unknown downstream velocity gives the upstream area times its velocity, divided by the downstream area.
v₂ = (A₁ × v₁) ÷ A₂Picture a pipe whose cross-section shrinks from 0.1 m² to 0.025 m² — one quarter of the inlet area — while the fluid enters at 2 m/s. The volume entering each second is A₁v₁ = 0.1 × 2 = 0.2 m³/s, and that same volume must leave through the smaller outlet. Dividing by the downstream area, v₂ = 0.2 ÷ 0.025 = 8 m/s: cutting the area to a quarter makes the fluid four times faster. Because area and velocity always multiply to the same flow, the velocity ratio is simply the inverse of the area ratio.
The relationship is exact under its assumptions, but those assumptions matter.
Incompressible, steady flow and consistent units
This form of the continuity equation assumes an incompressible fluid of constant density — water or low-speed air — in steady flow with no leaks or branches, so that conservation of mass reduces to A₁v₁ = A₂v₂. For a compressible gas at high speed the density changes and the mass form ρ₁A₁v₁ = ρ₂A₂v₂ is needed instead. Keep your units consistent — square metres for both areas and metres per second for the velocity — or the result will be wrong, and remember it gives the average velocity across the section, not the faster flow at the centre of the pipe.