Lacey's regime theory is used to predict the velocity of flow in open channels. It is based on the principle that the velocity of flow is determined by the balance between gravitational forces and frictional forces. The theory proposes that the flow regime can be divided into three types, namely, laminar, transitional, and turbulent.

The velocity of flow in each regime is proportional to different factors. In this case, the correct answer is option C, which states that the velocity of flow is proportional to (Qf2)1/6. Let's break down the components of this formula:

Q: This represents the discharge or flow rate of the channel. It is the volume of water that passes through a cross-section of the channel per unit time.

f: This represents the friction factor, which is a measure of the resistance of the channel to flow. It depends on the roughness of the channel, the Reynolds number, and other factors.

The (Qf2)1/6 factor in the formula represents the hydraulic radius of the channel, which is a measure of the cross-sectional area of the channel relative to its wetted perimeter. The hydraulic radius is important because it determines the resistance of the channel to flow.

The (Qf2)1/6 factor is raised to the power of 1/6 because it is proportional to the sixth root of the hydraulic radius. This means that as the hydraulic radius increases, the velocity of flow increases at a slower rate.

In summary, Lacey's regime theory proposes that the velocity of flow in open channels is proportional to the (Qf2)1/6 factor, which represents the hydraulic radius of the channel. This formula is based on the principle that the velocity of flow is determined by the balance between gravitational forces and frictional forces.