Velocity Distribution
The velocity distribution for flow over a flat plain is given by:
u = (3/2)(y - (y^3/2))
This equation represents the velocity profile where \( u \) is the velocity, and \( y \) is the vertical distance from the flat surface.
Shear Stress Calculation
To determine the shear stress at \( y = 9 \, \text{cm} \), we first need to calculate the velocity gradient (du/dy) at that point.
Step 1: Calculate the Derivative
- Differentiate the velocity equation:
\[ \frac{du}{dy} = \frac{d}{dy} \left( \frac{3}{2} \left( y - \frac{y^{3/2}}{2} \right) \right) \]
- Calculating the derivative:
\[ \frac{du}{dy} = \frac{3}{2} \left( 1 - \frac{3}{4}y^{1/2} \right) \]
Step 2: Evaluate at \( y = 9 \, \text{cm} \)
- Convert \( y \) into meters: \( y = 0.09 \, \text{m} \)
- Substitute \( y \) in the derivative:
\[ \frac{du}{dy} = \frac{3}{2} \left( 1 - \frac{3}{4} \times (0.09)^{1/2} \right) \]
- Calculate \( (0.09)^{1/2} = 0.3 \):
\[ \frac{du}{dy} = \frac{3}{2} \left( 1 - \frac{3}{4} \times 0.3 \right) \]
\[ \frac{du}{dy} = \frac{3}{2} \left( 1 - 0.225 \right) = \frac{3}{2} \times 0.775 = 1.1625 \, \text{m/s/m} \]
Step 3: Calculate Shear Stress
The shear stress (\( \tau \)) can be calculated using:
\[ \tau = \mu \frac{du}{dy} \]
Assuming the dynamic viscosity \( \mu = 0.001 \, \text{Pa.s} \):
- Substitute values:
\[ \tau = 0.001 \times 1.1625 = 0.0011625 \, \text{Pa} \]
Conclusion
The shear stress at \( y = 9 \, \text{cm} \) is approximately:
0.0011625 Pa