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Hydraulics, open channels and fluid power
Flow per unit width between plates with a moving wall
Discharge per unit width between a stationary lower plate and an upper plate moving at speed U, with gap h and a pressure gradient G = −dp/dx.
- Ref
- HYD-09
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- q [m²/s]
Between a fixed plate and a plate moving at speed U, the flow per unit width is the sum of a shear-driven part Uh/2 (the Couette flow) and a pressure-driven part G h³/(12μ) (the Poiseuille flow). G = −dp/dx is positive when pressure falls in the direction of the moving wall. A suitable negative G can make the net flow zero or reverse it.
Discharge per unit width between a stationary lower plate and an upper plate moving at speed U, with gap h and a pressure gradient G = −dp/dx. It is the sum of the Couette and Poiseuille parts, for steady laminar flow.
Calculator
- Couette part, U h/21×10−4 m²/s
- Poiseuille part, G h³/(12 μ)4×10−5 m²/s
- Mean velocity across the gap, q/h0.07 m/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.
- Of the 1.4×10−4 m²/s, 1×10−4 m²/s is dragged along by the wall and 4×10−5 m²/s is pushed by the pressure gradient.
- Halving the gap cuts the pressure-driven part by a factor of 8 but the wall-driven part by only 2, so narrow gaps are dominated by the moving wall.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m²/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| h Gap between the plates | 1e-5 to 0.1 m | 5e-7 | 5.005 | 5.005 | +10% | +16.6 % |
| U Speed of the moving wall | −10 to 10 m/s | −0.00996 | 0.01004 | 0.02 | +0.4 m/s | +286 % |
| G Pressure gradient, −dp/dx (signed) | −100,000 to 100,000 Pa/m | −0.006567 | 0.006767 | 0.01333 | +4000 Pa/m | +190 % |
| μ Dynamic viscosity | 5e-4 to 10 Pa·s | 1e-4 | 9e-4 | 8e-4 | +10% | −2.6 % |
Sweep of Pressure gradient, −dp/dx (signed)
| Pressure gradient, −dp/dx (signed) (Pa/m) | Flow per unit width (m²/s) |
|---|---|
| −100,000 | −0.006567 |
| −75,000 | −0.0049 |
| −50,000 | −0.003233 |
| −25,000 | −0.001567 |
| 0 | 1e-4 |
| 25,000 | 0.001767 |
| 50,000 | 0.003433 |
| 75,000 | 0.0051 |
| 100,000 | 0.006767 |
Sweep one variable
Variables and how each one contributes
Flow per unit width
What the equation gives you. Everything else feeds it.
Speed of the moving wall
The speed of the moving wall, in m/s. Raises the Couette part in direct proportion. Raising it by 10 % (to 0.11) moves the result by +7.14 %, from 1.4×10−4 to 1.5×10−4 m²/s.
Gap between the plates
The gap between the plates, in m. Raises the flow, with a cubic effect on the pressure-driven part. Raising it by 10 % (to 0.0022) moves the result by +16.6 %, from 1.4×10−4 to 1.632×10−4 m²/s.
Pressure gradient, −dp/dx (signed)
The pressure gradient −dp/dx, in Pa/m, signed. Raises the pressure-driven part in direct proportion. Raising it by 10 % (to 660) moves the result by +2.86 %, from 1.4×10−4 to 1.44×10−4 m²/s.
Dynamic viscosity
The dynamic viscosity, in Pa·s. A thicker liquid reduces the pressure-driven part in inverse proportion. Raising it by 10 % (to 0.011) moves the result by −2.6 %, from 1.4×10−4 to 1.364×10−4 m²/s.
Worked example with the values above
- Couette part: U h/2 = 0.1 × 0.002 / 2 = 1×10−4 m²/s.
- Poiseuille part: G h³/(12μ) = 600 × 8×10−9 / (12 × 0.01) = 4×10−5 m²/s.
- q = 1.4×10−4 m²/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