Question

The velocity potential \(\phi\) for a vortex flow is given by

• A. \(-\dfrac{\Gamma}{2\pi} \ln r\)
• B. \(-\dfrac{\Gamma}{2\pi r}\)
• C. \(\dfrac{\Gamma}{2\pi} \theta\)
• D. \(\dfrac{\Gamma}{2\pi} r\)

Hint:

\(\phi = \dfrac{\Gamma}{2\pi} \theta\) describes the potential for irrotational vortex flow.

Answer 1

The Correct Option is C

In vortex flow, the velocity potential \(\phi\) is a scalar function used to describe the flow field, particularly for irrotational flows. In cylindrical polar coordinates \((r, \theta)\), where the flow properties depend only on the angular coordinate \(\theta\) and not on the radial coordinate \(r\), the velocity potential for a vortex is expressed through its relationship with circulation and angular position.

The correct expression for the velocity potential \(\phi\) in a vortex flow is given by:

\(\phi = \dfrac{\Gamma}{2\pi} \theta\)

where \(\Gamma\) is the circulation and \(\theta\) is the angular position in radians. This expression indicates that the velocity potential is directly proportional to the angular coordinate, which is characteristic of vortex flows.

None of the other options—such as those involving natural logarithms or inverses of \(r\)—correctly represent the potential for this specific flow type. The proportionality to \(\theta\) is essential for maintaining the characteristics of a vortex flow in an irrotational context.