Consider a pair of point vortices with clockwise circulation Γ each. The distance
between their centers is a, as shown in the figure. Assume two-dimensional,
incompressible, inviscid flow. Which one of the following options is correct?
Options:
(A) The vortices translate downwards together with a velocity
Γ
2𝜋a .
(B) The vortices translate upwards together with a velocity
Γ
2𝜋a .
(C) The vortices rotate clockwise around each other about their centroid O.
(D) The vortices rotate counter-clockwise around each other about their centroid O.
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.20)**
**1. Intuitive Physical Setup & Given Parameters**:
A pair of 2D point vortices in an inviscid, incompressible fluid:
- Both have **identical clockwise circulation** $\Gamma$ (same sign and magnitude).
- Distance between their centers: $a$.
- Centroid of the two vortices is at point $O$ (midpoint, at distance $a/2$ from each).
We need to determine the resulting kinematic motion of the vortex pair.
**2. Step-by-Step Potential Flow Derivation**:
1. **Induced Velocity on Each Vortex**:
- By the Biot-Savart law in 2D potential flow, a point vortex does not move due to its own velocity field, but is advected by the velocity induced by all other vortices.
- Let Vortex 1 be located at $(0, a/2)$ and Vortex 2 at $(0, -a/2)$.
- Both have clockwise circulation $\Gamma$.
- The velocity induced by Vortex 2 on Vortex 1 is directed perpendicular to the line connecting them:
$V_{\text{ind}} = \frac{\Gamma}{2\pi a}$
- Because the circulation is clockwise, the velocity induced at Vortex 1 points to the right.
- By symmetry, the velocity induced by Vortex 1 on Vortex 2 has the same magnitude:
$V_{\text{ind}} = \frac{\Gamma}{2\pi a}$
but points to the left!
2. **Resulting Motion**:
- Because the induced velocity vectors at the two vortex locations are equal in magnitude and point in opposite directions perpendicular to the joining line, the pair undergoes a steady **pure rotation around their midpoint (centroid $O$)**.
- Angular velocity:
$\Omega = \frac{V_{\text{ind}}}{a/2} = \frac{\frac{\Gamma}{2\pi a}}{a/2} = \frac{\Gamma}{\pi a^2}$
- Tracing the direction of the velocity vectors confirms that the rotation of the pair about $O$ is in the **clockwise** direction.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) The vortices translate downwards together with a velocity Γ / (2𝜋a)**: ❌ **INCORRECT**. Co-translation occurs for a counter-rotating pair (opposite circulations $\pm \Gamma$), not an identical pair.
- **(B) The vortices translate upwards together with a velocity Γ / (2𝜋a)**: ❌ **INCORRECT**. Requires opposite signs of circulation.
- **(C) The vortices rotate clockwise around each other about their centroid O**: ✅ **CORRECT**. Identical clockwise circulations induce opposite tangential velocities, rotating the pair clockwise around their midpoint.
- **(D) The vortices rotate counter-clockwise around each other about their centroid O**: ❌ **INCORRECT**. Opposite direction of rotation.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: Like-sign vortices rotate around their centroid ($\Omega = \Gamma / (\pi a^2)$); opposite-sign vortices translate in a straight line with uniform speed $V = \Gamma / (2\pi a)$.
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