continuity equation formulas and interpretation
For steady incompressible flow, the same volume flow rate passes every section of a closed conduit.
The calculator preserves section geometry helpers and unknown-variable solving in a lighter layout.
How to use the continuity equation calculator
- Choose a model: Select the relationship matching the problem.
- Choose the unknown: Select the quantity to calculate.
- Enter values: Enter all known values with matching units and signs.
- Calculate: Review the result, formula, units, and direction.
Formula and variables
Cross-sectional area multiplied by average velocity remains constant for steady incompressible flow.
A₁v₁ = A₂v₂- A₁,A₂ — Areas
- Flow cross-sections (m²)
- v₁,v₂ — Velocities
- Average section velocities (m/s)
- Q — Flow rate
- Constant volume flow (m³/s)
Pipe contraction example
Area decreases from 0.02 m² to 0.01 m² while inlet speed is 2 m/s.
- Area 1
- 0.02 m²
- Velocity 1
- 2 m/s
- Area 2
- 0.01 m²
- v₂ = A₁v₁/A₂
- v₂ = 4 m/s
Result: Outlet velocity is 4 m/s.
Halving area doubles velocity for incompressible steady flow.
Understanding your results
Interpreting the result
Use mass continuity instead when density changes materially between sections.
Assumptions
- The selected equation represents the physical system.
- Inputs use a consistent reference direction.
- Values are converted through coherent SI units.
Limitations
- Vector components must be resolved along a common axis.
- External forces or energy losses are not added automatically.
- Results depend on the accuracy of entered measurements.
Common mistakes
- Mixing incompatible units.
- Dropping negative signs that represent direction.
- Using weight where mass is required.
- Entering a zero divisor.
Practical use cases
Physics problems
Check classroom, laboratory, and mechanics calculations.
Practical estimates
Estimate motion, forces, and energy for real systems.
Frequently asked questions
Can a result be negative?
Yes. For directional quantities, the sign indicates direction relative to the chosen positive axis.
Should I use SI units?
The interface can convert supported units, while the formulas are evaluated through coherent SI units.
Sources and review
- SI Brochure, 9th edition — BIPM. Accessed 2026-07-11.
- Special Publication 811 — NIST. Accessed 2026-07-11.
Reviewed 2026-07-11.