Under sluice is the correct option for maintaining a deep channel in front of the head regulator and disposing of heavy silt and a part of flood discharge on the downstream side of the barrage.

Explanation:
An under sluice is a hydraulic structure that is used to regulate the flow of water and maintain the required depth in a channel. It is typically located at the bottom of the head regulator or barrage. Here is a detailed explanation of how under sluice helps in maintaining a deep channel and disposing of silt and flood discharge:

1. Definition of Under Sluice:
- An under sluice is a type of gate or opening at the base of a dam or head regulator.
- It is designed to release water from the reservoir or barrage to maintain a desired water level and prevent excessive water buildup.

2. Maintaining a Deep Channel:
- The under sluice gate can be operated to release water from the reservoir or barrage.
- By controlling the opening of the under sluice gate, the flow of water can be regulated to maintain a deep channel in front of the head regulator.
- The flow of water through the under sluice helps to prevent the accumulation of silt and sediment in the channel, ensuring its proper depth.

3. Disposing of Heavy Silt:
- As water is released through the under sluice gate, it carries sediment and silt along with it.
- The force of the flowing water helps to transport the heavy silt downstream, preventing its accumulation in front of the head regulator.
- This ensures that the channel remains clear and deep, allowing for the smooth flow of water.

4. Disposing of Flood Discharge:
- During periods of high rainfall or flooding, the under sluice gate can be fully opened to release a significant amount of water from the reservoir or barrage.
- This helps in managing the flood discharge by diverting a portion of it downstream through the under sluice, preventing excessive water buildup and potential damage to the structure.

In conclusion, an under sluice is an effective hydraulic structure for maintaining a deep channel in front of the head regulator and disposing of heavy silt and a part of flood discharge on the downstream side of the barrage. Its operation allows for the regulated flow of water, preventing silt accumulation and facilitating the smooth flow of water through the channel.