The space between two large horizontal metal plates, `6 cm` apart, is filled with a liquid of viscosity `0.8 N//m^(2)` . A thin plate of surface area `0.01 m^(2)` is moved parallel to the length of the plate such that the plate is at a distance of `2 m` from one of the plates and `4 cm` from the other. If the plate moves with a constant speed of `1 ms^(-1)`, then

Options

A. the layer of the fluid, which is having the maximum velocity, is lying mid-way between the plates
B. the layers of the fluid, which is in contact with the moving plate, is having the maximum velocity
C. the layer of the fluid, which is in contact with the moving plate and is on the side of farther plate, is moving with the maximum velocity
D. the layer of the fluid, which is in contact with the moving plant and is on the side of nearer plate, is moving with the maximum velocity

Explanation

To solve the problem, we need to analyze the situation involving the two large horizontal metal plates and the thin plate moving between them. The key points to consider are the viscosity of the fluid, the distance between the plates, and the movement of the thin plate.

### Step-by-step Solution:

1. **Identify the Given Values:**
- Distance between the two plates, \( d = 6 \, \text{cm} = 0.06 \, \text{m} \)
- Viscosity of the liquid, \( \eta = 0.8 \, \text{N} \cdot \text{s/m}^2 \)
- Surface area of the thin plate, \( A = 0.01 \, \text{m}^2 \)
- Distance of the thin plate from one plate, \( y_1 = 2 \, \text{m} \) (this seems incorrect as it exceeds the distance between the plates; it should be \( 2 \, \text{cm} = 0.02 \, \text{m} \))
- Distance of the thin plate from the other plate, \( y_2 = 4 \, \text{cm} = 0.04 \, \text{m} \)
- Speed of the thin plate, \( v = 1 \, \text{m/s} \)

2. **Understand the Flow Profile:**
- The fluid in contact with the stationary plates has a velocity of \( 0 \, \text{m/s} \).
- As we move away from the stationary plates toward the moving plate, the velocity of the fluid increases linearly due to the viscous drag.

3. **Determine the Maximum Velocity Layer:**
- The maximum velocity of the fluid occurs at the moving plate. Since the plate is moving at \( 1 \, \text{m/s} \), the layer of fluid in contact with the moving plate will also have a velocity of \( 1 \, \text{m/s} \).

4. **Conclusion:**
- The layer of fluid that is in contact with the moving plate has the maximum velocity compared to the other layers of fluid, which have lower velocities due to the viscous drag effect.

### Answer:
The layer of fluid which is in contact with the moving plate has a maximum velocity.