Stream function:
The stream function is a mathematical function used to describe the flow of a fluid in two dimensions. It is defined such that its partial derivatives with respect to the coordinates of a point give the velocities at that point. In other words, it can be used to determine the flow velocity components in a fluid flow field. The stream function is commonly used to describe irrotational flows, where the fluid particles move in smooth, organized paths with no rotation.

Velocity potential:
The velocity potential is another mathematical function used to describe fluid flow. It is defined such that its gradient gives the velocity vector at each point in the flow field. The velocity potential is also commonly used to describe irrotational flows, where the fluid particles move in smooth paths without any rotation.

Vorticity:
Vorticity is a measure of the local rotation of a fluid flow. It is defined as the curl of the velocity vector field. In other words, it represents the tendency of fluid elements to rotate about their own axes. Vorticity is non-zero in rotational flows, where the fluid particles have both translational and rotational motion.

Explanation:
In rotational flows, the fluid particles have both translational and rotational motion, so vorticity is non-zero. The presence of vorticity indicates the presence of circulation and rotation in the flow.

However, in irrotational flows, the fluid particles move in smooth, organized paths with no rotation. This means that vorticity is zero in irrotational flows.

Both the stream function and the velocity potential are mathematical functions commonly used to describe fluid flow. They are particularly useful in describing irrotational flows, where the fluid particles move in smooth paths without any rotation.

Therefore, the correct answer is option 'B': Both the stream function and the velocity potential exist in both rotational and irrotational flows.