Rotational Component in Potential Function

The potential function is a mathematical representation of fluid flow that is used to describe the velocity field of a fluid. It is often used in the study of fluid mechanics and is based on the assumption that the fluid is irrotational, meaning that there is no rotation or angular momentum present in the fluid flow. However, in some cases, the fluid may have a certain degree of rotational motion, which needs to be accounted for in the potential function.

The rotational component in the potential function is directly related to the vorticity of the fluid flow. Vorticity is a measure of the local rotation of fluid particles within the flow. It is defined as the curl of the velocity vector field and is given by the equation:

Ω = ∇ × v

Where Ω is the vorticity vector and v is the velocity vector. The vorticity vector has three components, Ωx, Ωy, and Ωz, which represent the rotation of the fluid in the x, y, and z directions respectively.

Relation between Rotational Component and Vorticity

The rotational component in the potential function can be expressed as a function of the vorticity vector. Let's denote the rotational component as φrot and the vorticity vector as Ω. Based on mathematical derivations and analysis, it can be shown that:

φrot = 1/2 * Ω

This means that the rotational component in the potential function is equal to half of the vorticity vector. In other words, the rotational component is directly proportional to the vorticity, with a proportionality factor of 1/2.

Correct Answer and Explanation

According to the given options, the correct answer is option 'B', which states that the rotational component is 1/2 times the vorticity. This aligns with the derived relation between the rotational component and vorticity discussed above.

Therefore, the correct answer is option 'B' - 1/2 times vorticity.