Question

In 2D potential flow, the doublet is the limit of the superposition of which two singularities?

• A. A uniform stream and a source
• B. A source and a sink of equal strength
• C. A uniform stream and a sink
• D. A source and a vortex

Hint:

Remember: Doublet = limit of source–sink pair, vortex = rotating singularity, Rankine body = uniform flow + source/sink.

Answer 1

The Correct Option is B

Step 1: Definition of a doublet.
A doublet (or dipole) is formed by placing a source and a sink of equal strength \(Q\), separated by a small distance \(\ell\), and then letting \(\ell \to 0\) while keeping the product \(Q\ell\) finite (the dipole strength).

Step 2: Mathematical potential.
- Source potential at distance \(r_1\): \(\phi_s = \frac{Q}{2\pi}\ln r_1\).
- Sink potential at distance \(r_2\): \(\phi_k = -\frac{Q}{2\pi}\ln r_2\).
Superposition: \(\phi = \phi_s + \phi_k\). Taking the limit as \(\ell \to 0\) with \(Q\ell\) finite gives the dipole (doublet) potential: \[ \phi = \frac{\mu \cos \theta}{2\pi r}, \mu = Q \ell (\text{dipole strength}). \]

Step 3: Eliminate wrong options.
- (A) Uniform stream + source = Rankine half–body, not doublet.
- (C) Uniform stream + sink = Rankine body (with stagnation point).
- (D) Source + vortex = spiral flow.

Final Answer:
\[ \boxed{\text{Source + Sink of equal strength}} \]