Problem Statement
The velocity distribution for flow over a flat plate is given by \( u = \frac{3}{4}y - y^2 \), where \( u \) is the velocity in meters per second at a distance \( y \) meters above the plate. Determine the shear stress at \( y = 0.15 \, \text{m} \). Take dynamic viscosity of fluid as 8.6 poise.
Given Data
- Velocity distribution: \( u = \frac{3}{4}y - y^2 \)
- Position: \( y = 0.15 \, \text{m} \)
- Dynamic viscosity: \( \mu = 8.6 \, \text{poise} \)
Solution
1. Convert Viscosity to SI Units
Convert from poise to Ns/m²:
\( 1 \, \text{poise} = 0.1 \, \text{Ns/m}^2 \)\( \mu = 8.6 \, \text{poise} \times 0.1 = 0.86 \, \text{Ns/m}^2 \)
2. Calculate Velocity Gradient
Differentiate velocity equation with respect to y:
\( u = \frac{3}{4}y - y^2 \)\( \frac{du}{dy} = \frac{d}{dy} \left( \frac{3}{4}y - y^2 \right) = \frac{3}{4} - 2y \)
3. Evaluate Velocity Gradient at y = 0.15 m
4. Calculate Shear Stress
Using Newton's law of viscosity:
\( \tau = \mu \frac{du}{dy} \)\( \tau = 0.86 \, \text{Ns/m}^2 \times 0.45 \, \text{s}^{-1} = 0.387 \, \text{N/m}^2 \)
- Shear stress (τ) at y = 0.15 m = 0.387 N/m²
Explanation
1. Viscosity Conversion:
The dynamic viscosity was converted from poise to SI units (Ns/m²) because all other parameters are in SI units. This ensures dimensional consistency in calculations.
2. Velocity Gradient:
The velocity gradient (du/dy) represents the rate of change of velocity with respect to distance from the plate. At y = 0.15 m, this gradient is 0.45 s⁻¹, indicating how rapidly the fluid velocity changes at that specific location above the plate.
3. Shear Stress Calculation:
Shear stress is calculated using Newton's law of viscosity, which states that shear stress is proportional to the velocity gradient. The proportionality constant is the dynamic viscosity.
Physical Meaning
1. Interpretation of Shear Stress:
The calculated shear stress of 0.387 N/m² represents the internal frictional force per unit area acting between adjacent layers of fluid at y = 0.15 m. This value quantifies the resistance to flow at that specific location.
2. Velocity Distribution Significance:
The parabolic velocity profile (\( u = \frac{3}{4}y - y^2 \)) is characteristic of laminar flow over a flat plate. The maximum velocity occurs where du/dy = 0.
3. Engineering Applications:
Understanding shear stress distribution is crucial for designing aerodynamic surfaces, calculating drag forces, and predicting boundary layer behavior in various engineering applications.
