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Hydraulics, open channels and fluid power
Gravity-driven laminar film discharge per unit width
Discharge per unit width of a uniform laminar liquid film of thickness δ running down a plane inclined at θ above horizontal.
- Ref
- HYD-10
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- q [m²/s]
A uniform liquid film running down a plane inclined at θ balances gravity along the plane against viscous shear, giving a parabolic velocity profile. The discharge per unit width is q = ρ g sinθ δ³/(3μ). It rises with the cube of the film thickness δ, so a slightly thicker film carries much more liquid. This assumes laminar flow and no shear at the free surface.
Discharge per unit width of a uniform laminar liquid film of thickness δ running down a plane inclined at θ above horizontal. The wall has no slip, and the free surface has no shear.
Calculator
- Mean film velocity, q/δ0.1635 m/s
- Free-surface velocity, ρ g sinθ δ²/(2μ)0.2452 m/s
- Film Reynolds number, 4ρq/μ65.4
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.
- The 1 mm film flows at a mean 0.1635 m/s, carrying 1.635×10−4 m²/s, or 0.1635 litres per second per metre of width.
- The film Reynolds number is 65.4; the laminar formula holds only while it is low (roughly below 20 before waves develop).
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m²/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| δ Film thickness | 1e-5 to 0.02 m | 1.635e-10 | 1.308 | 1.308 | +10% | +33.1 % |
| μ Dynamic viscosity | 5e-4 to 10 Pa·s | 1.635e-7 | 0.00327 | 0.00327 | +10% | −9.09 % |
| g Gravitational acceleration | 1 to 30 m/s² | 1.667e-5 | 5e-4 | 4.833e-4 | +10% | +10 % |
| θ Plane angle above horizontal | 0.5 to 90 ° | 2.854e-6 | 3.27e-4 | 3.241e-4 | +10% | +8.93 % |
| ρ Liquid density | 500 to 2,000 kg/m³ | 8.175e-5 | 3.27e-4 | 2.452e-4 | +10% | +10 % |
Sweep of Film thickness
| Film thickness (m) | Discharge per unit width (m²/s) |
|---|---|
| 1e-5 | 1.635e-10 |
| 2.586e-5 | 2.828e-9 |
| 6.687e-5 | 4.89e-8 |
| 1.729e-4 | 8.456e-7 |
| 4.472e-4 | 1.462e-5 |
| 0.001156 | 2.529e-4 |
| 0.002991 | 0.004374 |
| 0.007734 | 0.07563 |
| 0.02 | 1.308 |
Sweep one variable
Variables and how each one contributes
Discharge per unit width
What the equation gives you. Everything else feeds it.
Liquid density
The liquid density, in kg/m³. Raises the discharge in direct proportion. Raising it by 10 % (to 1,100) moves the result by +10 %, from 1.635×10−4 to 1.798×10−4 m²/s.
Gravitational acceleration
The gravitational acceleration, in m/s². Raises the discharge in direct proportion. Raising it by 10 % (to 10.79) moves the result by +10 %, from 1.635×10−4 to 1.799×10−4 m²/s.
Plane angle above horizontal
The plane’s angle above horizontal, in degrees. A steeper plane raises the discharge in proportion to sin θ. Raising it by 10 % (to 33) moves the result by +8.93 %, from 1.635×10−4 to 1.781×10−4 m²/s.
Film thickness
The film thickness, in m. Raises the discharge with the cube of the thickness. Raising it by 10 % (to 0.0011) moves the result by +33.1 %, from 1.635×10−4 to 2.176×10−4 m²/s.
Dynamic viscosity
The dynamic viscosity, in Pa·s. A thicker liquid reduces the discharge in inverse proportion. Raising it by 10 % (to 0.011) moves the result by −9.09 %, from 1.635×10−4 to 1.486×10−4 m²/s.
Worked example with the values above
- Driving term: ρ g sinθ = 1,000 × 9.81 × 0.5 = 4,905 N/m³.
- Thickness term: δ³/(3μ) = 1×10−9 / (3 × 0.01) = 3.3333×10−8.
- q = 1.635×10−4 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