Continuity Equation Calculator

Calculate flow velocity, cross-sectional area, or volume flow using the continuity equation A₁v₁=A₂v₂.

A₁v₁ = A₂v₂ = Q
Across-sectional aream²
vmean flow velocitym/s
Qvolumetric flow ratem³/s
10cm×10cm = 0.01m²
Please check your inputs and try again.

Continuity Is a Statement of Mass Conservation

The continuity equation says that steady flow cannot lose or create mass between two sections of a streamtube. The general one-dimensional steady relation is ρ1A1v1=ρ2A2v2. For an incompressible fluid with essentially constant density, this reduces to A1v1=A2v2=Q, the volume-flow rate.

A smaller cross-sectional area therefore requires a larger average speed when incompressible steady flow passes through the same streamtube. That speed increase does not come from continuity alone; pressure and energy changes are described by momentum or Bernoulli relations. For gases at high Mach number, density variation must be retained.

˙m=ρAv=constant,   incompressible: Q=Av=constant
SymbolMeaningWhy it appears / units
˙mMass-flow ratekg/s; conserved in steady flow with no mass addition.
ρDensitykg/m³; may change in compressible flow.
ACross-sectional aream²; area normal to the average flow.
vAverage flow speedm/s across the section.

If density is constant, halving area doubles average speed. If density changes, area and velocity alone are insufficient; the mass-flow relation with density is the correct conservation statement.

A narrowing incompressible streamtube must speed the flow. If area is cut in half, velocity should double for the same volumetric flow rate. When density changes appreciably, check mass flow ρAv instead of volume flow Av; otherwise a compressible-flow result may look internally consistent while violating mass conservation.

Worked Examples

Example 1: Pipe narrows: A₁=0.01m², v₁=2m/s, A₂=0.005m²
Q=0.02m³/s, v₂=0.02/0.005
Result: v₂=4 m/s — doubles when area halves
Conservation of mass in flow
Example 2: Nozzle: A₁=100cm², v₁=0.5m/s, A₂=1cm²
Q=0.005m³/s, v₂=0.005/0.0001
Result: v₂=50 m/s — 100× faster!
Ideal continuity result for a 100:1 area ratio
Example 3: Pipe contraction
A1=0.010m², v1=2m/s, A2=0.004m²
Result: v2=5m/s
The same 0.020m³/s volume flow passes both sections.
Example 4: Mass flow of water
ρ=1000kg/m³, A=0.002m², v=3m/s
Result: ˙m=6kg/s
Mass flow combines density with volume flow.

Common Mistakes

⚠️
Applying A₁v₁=A₂v₂ to strongly compressible flow

That simplified form assumes constant density. Use ρAv when density changes.

⚠️
Using diameter directly instead of area

For a circular pipe, A=πD²/4. Velocity therefore varies inversely with diameter squared, not diameter.

⚠️
Assuming continuity predicts pressure

Continuity enforces mass conservation. Pressure changes require additional physics such as Bernoulli, momentum, friction, or compressible-flow equations.

Connected Formulas

solved together with →Bernoulli Equationuses Q = Av →Flow Ratesupplies velocity to →Reynolds Number

Frequently Asked Questions

Why does water speed up in a narrow pipe?
Mass conservation: same mass must pass any cross-section per second. ṁ=ρAv=const. If A decreases and ρ is constant (incompressible): v must increase proportionally.
Compressible flow?
For gases at high speed (Mach>0.3), density changes significantly. The compressible form: ρ₁A₁v₁=ρ₂A₂v₂. At supersonic speeds, a diverging nozzle actually accelerates the flow (de Laval nozzle).
Why does fluid speed up in a narrower pipe?
For steady incompressible flow, the same volume must pass every cross-section each second. A smaller area therefore requires a proportionally larger average velocity.
Is continuity valid for unsteady flow?
Mass conservation is always valid, but the simple steady ρAv=constant form may not be. Unsteady control-volume analysis includes accumulation of mass with time.
What is the difference between mass flow and volume flow?
Volume flow Q=Av measures m³/s. Mass flow ṁ=ρQ measures kg/s. They are proportional only when density is known.
Can branches be handled with continuity?
Yes. At a junction in steady flow, total incoming mass flow equals total outgoing mass flow, provided there is no accumulation or source within the junction.
When is A₁v₁=A₂v₂ valid?
That form assumes steady incompressible flow, so density is effectively the same at both sections. For compressible flow the conserved mass-flow relation is ρ1A1v1=ρ2A2v2. Gases at large pressure or temperature changes may require the density terms explicitly.