Library / Fluid Mechanics / FLU-07
Fluid Mechanics
Viscous shear stress
Shear stress in a fluid layer from the velocity gradient across it.
- Ref
- FLU-07
- Rev
- v1
- Field
- Fluid Mechanics
- Result
- τ [Pa]
For a Newtonian fluid in simple parallel shear. For a film of thickness h between a fixed plate and one moving at U, du/dy = U/h.
A thick liquid resists layers sliding over each other; the resistance rises with viscosity and with how fast neighbouring layers move relative to each other.
Calculator
- Velocity gradient2,000 1/s
- Viscosity in centipoise100 cP
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.
- Thinner films give much larger shear stresses and drag at the same speed.
- Viscosity of oils drops quickly with temperature, so the shear stress does too.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Pa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| h Film thickness | 0.01 to 20 mm | 10 | 20,000 | 19,990 | +10% | −9.09 % |
| μ Dynamic viscosity | 5e-4 to 5 Pa·s | 1 | 10,000 | 9,999 | +10% | +10 % |
| U Plate speed | 0.01 to 20 m/s | 1 | 2,000 | 1,999 | +10% | +10 % |
Sweep of Film thickness
| Film thickness (mm) | Shear stress (Pa) |
|---|---|
| 0.01 | 20,000 |
| 0.02586 | 7,734 |
| 0.06687 | 2,991 |
| 0.1729 | 1,156 |
| 0.4472 | 447.2 |
| 1.156 | 172.9 |
| 2.991 | 66.87 |
| 7.734 | 25.86 |
| 20 | 10 |
Sweep one variable
Variables and how each one contributes
Shear stress
What the equation gives you. Everything else feeds it.
Dynamic viscosity
Fluid viscosity. Directly proportional.
Plate speed
Speed of the moving plate. Directly proportional.
Film thickness
Gap between the plates. Inversely proportional.
Worked example with the values above
- Velocity gradient: U/h = 2 / 0.001 = 2,000 1/s.
- Multiply by μ = 0.1 Pa·s.
- τ = 200 Pa.
Assumptions and limits
- Newtonian fluid.
- Linear velocity profile across a thin film.
Common mistakes
- Mixing Pa·s and cP.
- Using the film thickness in mm without converting.
Statistics and uncertainty
Viscosity varies strongly with temperature (about 5–10% per K for oils), so quote the temperature.
Where it is used
- Bearing and lubrication calculations.
- Drag in thin fluid layers.
Maths you need
- Viscosity: Pa s
- Velocity gradient: du/dy