An incompressible liquid flows through a uniform cross section tube with velocity 12 cm/s. the thickness of liquid layer is `0.8` cm then velocity of gradient of flow is

Options

A. `15s^(-1)`
B. `12s^(-1)`
C. `18s^(-1)`
D. `5s^(-1)`

Answer

To solve the problem of finding the velocity gradient of an incompressible liquid flowing through a uniform cross-section tube, we can follow these steps:

### Step-by-Step Solution

1. **Identify Given Values:**
- Velocity of the liquid, \( V = 12 \, \text{cm/s} \)
- Thickness of the liquid layer, \( y = 0.8 \, \text{cm} \)

2. **Understand the Concept of Velocity Gradient:**
- The velocity gradient (\( \frac{dV}{dy} \)) is defined as the change in velocity (\( dV \)) over the change in distance (\( dy \)). In this case, it represents how the velocity of the liquid changes with respect to the thickness of the liquid layer.

3. **Set Up the Formula for Velocity Gradient:**
- The formula for velocity gradient is given by:
\[ \text{Velocity Gradient} = \frac{dV}{dy} \]
- Here, \( dV \) is the change in velocity, which is equal to the maximum velocity \( V \) since the velocity at the wall of the tube is zero (the liquid is stationary at the wall).

4. **Calculate the Velocity Gradient:**
- Substitute the values into the formula:
\[ \frac{dV}{dy} = \frac{V}{y} = \frac{12 \, \text{cm/s}}{0.8 \, \text{cm}} \]
- Perform the division:
\[ \frac{dV}{dy} = \frac{12}{0.8} = 15 \, \text{cm/s/cm} \]

5. **Final Result:**
- The velocity gradient of the liquid flow is:
\[ \frac{dV}{dy} = 15 \, \text{cm/s/cm} \]