Area A₁
Cross-sectional flow area at section 1, > 0.
Fluid mechanics · Fluid dynamics
Fluid continuity equation calculator
Compare two sections of the same incompressible stream with A₁v₁ = A₂v₂. Solve one area or speed from the other three values and obtain the common volume flow.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
02
Results
Speed v₁
Mean speed at section 1, ≥ 0.
Area A₂
Cross-sectional flow area at section 2, > 0.
Speed v₂
Mean speed at section 2, ≥ 0.
Volume flow
Same volume flow at both sections.
Relation and numerical substitution
A₁v₁ = A₂v₂ = Q—
Steady incompressible flow, one inlet and one outlet, no leaks or accumulation; cross-sectional mean velocities.
01Formulas and symbols
Formulas used
A₁
A₁ = A₂ × v₂ / v₁Cross-sectional flow area at section 1, > 0.v₁
v₁ = A₂ × v₂ / A₁Mean speed at section 1, ≥ 0.A₂
A₂ = A₁ × v₁ / v₂Cross-sectional flow area at section 2, > 0.v₂
v₂ = A₁ × v₁ / A₂Mean speed at section 2, ≥ 0.Q
Q = A₁ × v₁Same volume flow at both sections.02Assumptions and limits
Scope of validity
- Steady incompressible flow, one inlet and one outlet, no leaks or accumulation; cross-sectional mean velocities.
03Validation example
Reference numerical case
- A₁ = 0.05 m², v₁ = 4 m/s, A₂ = 0.02 m²: Q = 0.2 m³/s and v₂ = 10 m/s.
04References
Technical references
- University Physics 1, §14.5 Fluid dynamics — OpenStax
FAQ
Why does a narrower section have faster flow?
Conservation of volume flow requires v to increase as A decreases when density is constant.
Is this the same as the pipe-flow calculator?
No. Pipe-flow converts diameter and speed at one circular section; this page relates two arbitrary cross-sectional areas.
05Model and conventions
Compare two sections of the same incompressible stream with A₁v₁ = A₂v₂. Solve one area or speed from the other three values and obtain the common volume flow.
06Limits of this model
For compressible flow use mass continuity with density at each section. Zero flow cannot determine an unknown area: the area modes require strictly positive speeds.
07Common mistakes
Areas are not diameters. For a circular pipe convert diameter to πD²/4, or use pipe-flow for the one-section diameter-and-speed calculation.