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 plates and the liquid between them. The key points are:

1. **Understanding the Setup**: We have two large horizontal plates that are 6 cm apart, filled with a liquid of viscosity 0.8 N/m². A thin plate of surface area 0.01 m² is moved parallel to the plates at a constant speed of 1 m/s. The thin plate is positioned 2 m from one plate and 4 cm from the other.

2. **Velocity Profile**: The velocity of the fluid layers will vary between the two plates. The layer of fluid in contact with the moving plate will have the same velocity as the plate, which is 1 m/s. The fluid layer in contact with the stationary plate will have a velocity of 0 m/s.

3. **Maximum Velocity Layer**: The maximum velocity of the fluid occurs at the moving plate, which is 1 m/s. As we move away from the moving plate towards the stationary plate, the velocity of the fluid decreases linearly to 0 m/s at the stationary plate.

4. **Fluid Layer Velocities**: The velocity of the fluid layers between the plates can be described by a linear velocity profile due to the viscous nature of the fluid. The velocity decreases from 1 m/s at the moving plate to 0 m/s at the stationary plate.

5. **Conclusion**: Since the plate is moving at a constant speed of 1 m/s, the layers of fluid in contact with the moving plate will also have a maximum velocity of 1 m/s. The fluid layer closer to the stationary plate will have a lower velocity compared to the fluid layer closer to the moving plate.

### Step-by-Step Solution:
1. **Identify the distance between the plates**: The distance between the two plates is given as 6 cm (0.06 m).

2. **Identify the viscosity of the fluid**: The viscosity of the fluid is given as 0.8 N/m².

3. **Determine the speed of the moving plate**: The speed of the moving plate is given as 1 m/s.

4. **Understand the velocity profile**: The velocity of the fluid layers varies linearly from the moving plate to the stationary plate. The layer in contact with the moving plate has a velocity of 1 m/s, while the layer in contact with the stationary plate has a velocity of 0 m/s.

5. **Identify the maximum velocity**: The maximum velocity of the fluid occurs at the moving plate and is equal to the speed of the plate, which is 1 m/s.

### Final Answer:
The maximum velocity of the fluid layer in contact with the moving plate is **1 m/s**.