Last updated: 2026-07-01

The Continuity Equation

The continuity equation is the statement of mass conservation for a flowing fluid. In its most common form, for a steady, incompressible flow through a pipe or channel, it says that the volume of fluid crossing any section per second is the same everywhere along the flow. That single idea explains why water speeds up when you squeeze the end of a garden hose and why rivers run fast through narrow gorges.

The equation

For steady, incompressible flow between two cross-sections of a pipe, the volumetric flow rate is constant:

$A_1 v_1 = A_2 v_2 = Q$

Here $A$ is the cross-sectional area, $v$ is the average velocity across that section, and $Q$ is the volumetric flow rate. Solving for the downstream velocity gives the form this calculator uses:

$v_2 = \frac{A_1 v_1}{A_2}$

and the flow rate is simply

$Q = A_1 v_1$

Because the fluid cannot pile up or vanish, a smaller area forces a proportionally larger velocity.

Symbol table

Worked example

Why area and velocity trade off

Because the flow rate $Q = A v$ is fixed, area and velocity are inversely related along a pipe. Cut the area in half and the velocity doubles; open it up fourfold and the velocity drops to a quarter. For a round pipe this coupling is even stronger in terms of diameter: since area scales with the square of the diameter ($A = \pi d^2/4$), halving the diameter quarters the area and so quadruples the velocity, giving the shortcut $v_2 = v_1 (d_1/d_2)^2$.

From continuity to Bernoulli

Continuity tells you the velocity at each section from geometry alone, but it says nothing about pressure. That is where Bernoulli's equation takes over: once continuity has fixed the speed change, Bernoulli relates it to a pressure change. In a constriction, continuity forces the fluid to speed up, and Bernoulli then predicts the static pressure there falls. This pairing is the basis of the Venturi effect and of Venturi and orifice flow meters, which infer flow rate from a measured pressure drop.

Compressible flow

The simple form $A_1 v_1 = A_2 v_2$ holds only when density is constant. For a gas moving fast enough that its density changes appreciably — roughly above Mach 0.3 — the conserved quantity is the mass flow rate rather than the volume flow rate:

$\rho_1 A_1 v_1 = \rho_2 A_2 v_2$

In that regime the area–velocity relationship can even reverse, which is why a supersonic nozzle widens rather than narrows to accelerate the flow. For the everyday liquids and low-speed gases this calculator targets, the incompressible form is an excellent approximation.