Question

The shape of Rankine oval of equal axes can be found out by substituting

• A. \(\psi = 0\)
• B. \(\psi = 1\)
• C. \(U = 0\)
• D. \(U = 1\)

Hint:

\(\psi = 0\) corresponds to the dividing streamline in Rankine bodies.

Answer 1

The Correct Option is A

In fluid dynamics, a Rankine oval is a potential flow pattern that involves the superimposition of a source and a sink at equal strength. The streamline representing the Rankine oval can be described using a stream function \(\psi\). The oval shape occurs at a specific value of \(\psi\).

The stream function for the Rankine oval is given by:

\(\psi = \frac{\Gamma}{2\pi}\left(\tan^{-1}\frac{y}{x-a}-\tan^{-1}\frac{y}{x+a}\right)\)

Where:

• \(\Gamma\) is the circulation strength of the source/sink pair.
• \(a\) is the distance from the origin to each source or sink along the x-axis.

Setting \(\psi = 0\) solves for the shape of the Rankine oval with equal axes, because when \(\psi = 0\), it corresponds to the closed streamline that represents the boundary of the resulting oval.

Therefore, the correct substitution to find the shape of the Rankine oval of equal axes is:

\(\psi = 0\).