Problem Statement
A shaft 70mm in diameter is being pushed at a speed of 0.4m/s through a bearing sleeve 70.2mm in diameter and 250mm long. The clearance, assumed uniform, is filled with oil of kinematic viscosity 0.005 m2/s and specific gravity 0.9. Find the force exerted by the oil on the shaft.
Solution
Given:
- Diameter of shaft (D) = 70mm = 0.07m
- Length of shaft (L) = 250mm = 0.25m
- Change in velocity (du) = 0.4m/s
- Kinematic viscosity of oil (υ) = 0.005 m2/s
- Specific gravity of oil (S) = 0.9
- Density of oil (ρ) = 0.9 × 1000 = 900 kg/m3
- Dynamic viscosity (μ) = υρ = 0.005 × 900 = 4.5 Ns/m2
- Clearance (dr) = dy = (70.2 – 70)/2 mm = 0.1mm = 0.0001m
- Force exerted by the oil on the shaft (F) = ?
Calculations:
Force exerted by the oil is given by:
F = Shear stress (τ) × Surface area (A)
Shear stress (τ) = μ (du/dy)
Surface area of the shaft (A) = πDL
F = μ (du/dy) (πDL)
Substitute the values:
F = 4.5 × 0.4 / 0.0001 × (π × 0.07 × 0.25)
Simplify:
F = 990 N
Result:
The force exerted by the oil on the shaft is 990 N.
Explanation
This problem involves determining the force exerted by the oil on a shaft as it moves through a bearing sleeve. The shaft is surrounded by a thin oil film that resists motion due to its viscosity.
- The kinematic viscosity and specific gravity of the oil are given. Using these, the dynamic viscosity of the oil is calculated. Dynamic viscosity (μ) is a measure of the oil’s resistance to flow.
- The clearance between the shaft and the sleeve is treated as a uniform thin film. The force exerted by the oil depends on the shear stress caused by the motion of the shaft relative to the sleeve.
- Shear stress (τ) is proportional to the dynamic viscosity (μ), the velocity gradient (du/dy), and the surface area of the shaft. These values are substituted into the formula for the force.
- The surface area of the shaft is calculated using its diameter and length (πDL), and the velocity gradient is determined using the clearance (dy).
- By substituting all the values into the formula, the force exerted by the oil on the shaft is calculated to be 990 N. This force represents the resistance offered by the oil film due to its viscosity.
This solution demonstrates the application of fluid mechanics concepts, including viscosity and shear stress, in solving real-world engineering problems.
