Problem Statement
A Newtonian fluid is filled in the clearance between a shaft and a concentric sleeve. The sleeve attains a speed of 50 cm/s when a force of 40 N is applied to the sleeve parallel to the shaft. Determine the speed if a force of 200 N is applied.
Given Data
- Initial force, F₁ = 40 N
- Initial speed, u₁ = 50 cm/s
- New force, F₂ = 200 N
- Fluid type: Newtonian
Solution
1. Establish the Relationship
Shear stress: \( \tau = \frac{F}{A} \)
Velocity gradient: \( \frac{du}{dy} = \frac{u - 0}{y} = \frac{u}{y} \)
Therefore: \( \frac{F}{A} = \mu \frac{u}{y} \)
2. Derive Proportional Relationship
Since \( A, \mu, \) and \( y \) are constant: \( F \propto u \)
Therefore: \( \frac{F_1}{F_2} = \frac{u_1}{u_2} \)
3. Calculate New Speed
\( u_2 = \frac{50 \times 200}{40} = 250 \, \text{cm/s} \)
- Speed when 200 N force is applied: 250 cm/s
Explanation
1. Newtonian Fluid Behavior:
The analysis uses the fundamental property of Newtonian fluids where shear stress is directly proportional to the velocity gradient. The constant of proportionality is the dynamic viscosity (μ).
2. Linear Force-Velocity Relationship:
The derivation shows that force is proportional to speed (F ∝ u) for this configuration. This relationship holds because the geometric parameters (area A and clearance y) and fluid properties (viscosity μ) remain constant.
3. Calculation Method:
Using the proportional relationship, the new speed is calculated by scaling the initial speed according to the force ratio (200N/40N = 5). Thus, 50 cm/s × 5 = 250 cm/s.
Physical Meaning
1. Newtonian Fluid Characteristics:
The linear relationship between force and speed confirms Newtonian behavior, where viscosity remains constant regardless of shear rate. This distinguishes Newtonian fluids from non-Newtonian fluids that exhibit variable viscosity.
2. Engineering Applications:
This principle is fundamental in designing lubrication systems, hydraulic equipment, and viscometers. Understanding the force-speed relationship helps engineers predict system performance under different operating conditions.
3. Practical Significance:
The result shows that increasing force by 5 times (40N → 200N) increases speed proportionally (50 cm/s → 250 cm/s). This linear relationship simplifies calculations for similar systems with Newtonian fluids in concentric configurations.
