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Hydraulics, open channels and fluid power
Laminar pipe velocity at a given radius
Local velocity in steady, fully developed Newtonian laminar flow in a straight circular pipe, from the mean velocity U and the pipe radius R.
- Ref
- HYD-06
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- u(r) [m/s]
In steady, fully developed laminar flow in a circular pipe, the velocity is a paraboloid, u(r) = 2U[1 − (r/R)²], where U is the mean velocity and R the pipe radius. It is zero at the wall (r = R, no slip) and twice the mean on the axis. The mean is half the centreline value.
Local velocity in steady, fully developed Newtonian laminar flow in a straight circular pipe, from the mean velocity U and the pipe radius R. It is twice the mean on the axis and zero at the wall.
Calculator
- Centreline velocity, 2U0.4 m/s
- Velocity as a share of the mean1.5
- Wall velocity gradient magnitude, 4U/R80 1/s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At r = 0.005 m the velocity is 0.3 m/s, against a mean of 0.2 m/s and 0.4 m/s on the axis.
- The wall velocity gradient of 80 1/s sets the wall shear stress, μ times this.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| U Mean velocity | 0.001 to 20 m/s | 0.0015 | 30 | 30 | +10% | +10 % |
| r Radial position | 0 to 0.01 m | 0 | 0.4 | 0.4 | +10% | −7 % |
| R Pipe radius | 0.01 to 1 m | 0.3 | 0.4 | 0.09999 | +10% | +5.79 % |
Sweep of Radial position
| Radial position (m) | Local velocity (m/s) |
|---|---|
| 0 | 0.4 |
| 0.00125 | 0.3938 |
| 0.0025 | 0.375 |
| 0.00375 | 0.3438 |
| 0.005 | 0.3 |
| 0.00625 | 0.2438 |
| 0.0075 | 0.175 |
| 0.00875 | 0.09375 |
| 0.01 | 0 |
Sweep one variable
Variables and how each one contributes
Local velocity
What the equation gives you. Everything else feeds it.
Mean velocity
The cross-sectional mean velocity, in m/s. Raises the local velocity in direct proportion. Raising it by 10 % (to 0.22) moves the result by +10 %, from 0.3 to 0.33 m/s.
Radial position
The radial position from the pipe axis, in m (0 ≤ r ≤ R). Moving outward lowers the velocity, with the fall getting steeper towards the wall. Raising it by 10 % (to 0.0055) moves the result by −7 %, from 0.3 to 0.279 m/s.
Pipe radius
The pipe radius, in m. A larger radius moves the point closer to the axis in relative terms, so the velocity rises. Raising it by 10 % (to 0.011) moves the result by +5.79 %, from 0.3 to 0.3174 m/s.
Worked example with the values above
- Position ratio: r/R = 0.005 / 0.01 = 0.5; squared 0.25.
- Bracket: 1 − 0.25 = 0.75.
- u = 2 × 0.2 × 0.75 = 0.3 m/s.
Assumptions and limits
- Steady, fully developed laminar flow of a Newtonian liquid with no slip at the walls.
- Smooth, straight geometry, and no body force other than gravity where stated.
Common mistakes
- Using these results for turbulent flow, which has a flatter profile.
- Entering the diameter where the radius is needed, which changes the answer by a factor of two or four.
- Mixing the sign of the pressure gradient: it is −dp/dx, positive when pressure falls along the flow.
Statistics and uncertainty
These laminar formulas are exact for their model, so the uncertainty comes from the inputs. Gap and film thickness enter with the third power, so a 1 % error in thickness gives about a 3 % error in flow.
Where it is used
- Lubrication films, coating, sealing gaps and thin-film runoff.
- Checking viscous pipe flow and wall shear from measured pressure drops.
Maths you need
- Viscosity: μ, the shear stress per unit velocity gradient
- No-slip condition: the liquid moves with the wall at the wall
- Reynolds number: the ratio of inertial to viscous forces