Explore the analytical velocity profile of fully developed laminar flow. Change the inputs, inspect the profile table, and watch how the computed flow rate responds.
Change the conditions. See the flow.
Steady, incompressible, fully developed Newtonian flow in a straight circular pipe. Water-like density is fixed at 998 kg/m³.
u(r) = Δp (R² − r²) / (4 μ L)
Maximum velocity 0.0125 m/s
Mean velocity 0.00625 m/s
Flow rate 4.909 × 10⁻⁷ m³/s
Reynolds number 62.4
Laminar-flow estimate. No entrance effects or experimental validation are included.
Read the profile as a table
| r/R | u (m/s) |
|---|---|
| 0 | 0.0125 |
| 0.5 | 0.009375 |
| 1 | 0 |
The normalized parabola keeps its shape; calculated speeds and the profile table change with the inputs. Δp = p in − p out > 0. Reynolds number uses diameter 2R and mean velocity.
Use it with a clear model
The calculation assumes a straight circular pipe, stationary walls, steady incompressible Newtonian flow, and a fully developed laminar profile. Density is fixed at 998 kilograms per cubic metre. The normalized diagram is intentionally unchanged as the dimensional velocities change.
Check the solution
At r = R the velocity is zero. At r = 0 it is Δp R² / (4 μ L). Integrating over the circular section gives Q = π Δp R⁴ / (8 μ L), so the mean speed is half the maximum. A warning appears when the calculated Reynolds number reaches 2,300; this threshold is a guide, not a guarantee of the flow regime.