Under a specific set of assumptions, the velocity of a viscous fluid in a straight circular pipe has a parabolic profile. The result comes from balancing the pressure gradient with viscous resistance.
Assumptions first
- Steady, incompressible, laminar flow of a Newtonian fluid.
- A straight circular pipe with constant radius and stationary walls.
- A fully developed axial velocity profile and a positive inlet-to-outlet pressure drop.
- Constant dynamic viscosity; no entrance, exit, or additional body-force effects in the model.
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.
Understand the result
The maximum velocity is twice the mean velocity for this profile. At fixed pressure drop, viscosity, and pipe length, the volumetric flow rate is proportional to the fourth power of radius. Small radius changes can therefore have a large effect.
Checkpoint
At fixed pressure drop, viscosity, and length, what happens to flow rate when the radius doubles?
The analytical flow rate increases by a factor of sixteen. Check whether the resulting Reynolds number still supports the laminar assumption before treating that prediction as physically applicable.
What this example can and cannot establish
This is an analytical teaching calculation, not a numerical flow solver or an experimental validation. Agreement with it can help verify a numerical model under matching conditions; it cannot establish accuracy for a different physical regime.