Fluid Flow Parameters in Rotational and Irrotational Flows

Vorticity, Velocity potential, and Stream function are three essential parameters that are used to describe fluid flow. While vorticity is a measure of the local rotation of a fluid element, the velocity potential and stream function are scalar and vector quantities that help describe the velocity field of a fluid.

Rotational Flows

A rotational flow is one in which the fluid particles rotate about their own axis as they move in the flow. In such flows, the vorticity is non-zero and varies from point to point. Examples of rotational flows include turbulence, eddies, and whirlpools.

Irrotational Flows

An irrotational flow is one in which the fluid particles do not rotate about their own axis as they move in the flow. In such flows, the vorticity is zero everywhere. Examples of irrotational flows include laminar flows and potential flows.

Parameters in Rotational and Irrotational Flows

Of the three parameters, vorticity is the only one that exists in rotational flows but not in irrotational flows. On the other hand, both velocity potential and stream function exist in both rotational and irrotational flows.

- Vorticity: Vorticity is non-zero in rotational flows and is a measure of the local rotation of a fluid element. It is defined as the curl of the velocity vector and is given by the formula:

ω = ∇ x v

where ω is the vorticity, v is the velocity vector, and ∇ x v is the curl of the velocity vector.

- Velocity potential: Velocity potential is a scalar quantity that describes the velocity field of a fluid. In irrotational flows, the velocity potential exists and is defined as the negative gradient of a scalar function called the potential function. It is given by the formula:

v = -∇Φ

where v is the velocity vector, Φ is the potential function, and ∇Φ is the negative gradient of the potential function.

- Stream function: Stream function is a vector quantity that also describes the velocity field of a fluid. In irrotational flows, the stream function exists and is defined as the scalar function whose gradient is perpendicular to the velocity vector. It is given by the formula:

v = ∇ x Ψ

where v is the velocity vector, Ψ is the stream function, and ∇ x Ψ is the curl of the stream function.

Conclusion

In summary, vorticity exists only in rotational flows, while both velocity potential and stream function exist in both rotational and irrotational flows. Therefore, the correct answer to the question is option C (1 and 3).