Subtopic: Conservation Equations of Mass, Momentum & Energy
Question Statement:
A pair of infinitely long, counter-rotating line vortices of identical circulation strength $\Gamma$ are situated a distance $h$ apart in an inviscid fluid. The vortices will:
Options:
(A) rotate counter-clockwise about the midpoint with tangential velocity $\frac{\Gamma}{2\pi h}$
(B) rotate counter-clockwise about the midpoint with tangential velocity $\frac{\Gamma}{4\pi h}$
(C) translate together in the $+y$ direction with velocity $\frac{\Gamma}{2\pi h}$
(D) translate together in the $+y$ direction with velocity $\frac{\Gamma}{4\pi h}$
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2020 Q.16)**
**1. Intuitive Physical Setup & Given Parameters**:
Two infinitely long parallel line vortices of equal and opposite strength ($+\Gamma$ and $-\Gamma$) are situated a distance $h$ apart along the $x$-axis in an unbounded inviscid fluid.
**2. Step-by-Step Mathematical Derivation**:
1. **Induced Velocity by a Line Vortex**:
The tangential velocity induced by a line vortex of strength $\Gamma$ at distance $r$ is:
$V_\theta = \frac{\Gamma}{2\pi r}$
2. **Interaction Analysis**:
- Vortex 1 ($+\Gamma$, counter-clockwise) is at $(-h/2, 0)$. At the location of Vortex 2 (distance $h$), it induces an upward velocity in the $+y$ direction of magnitude:
$V_2 = \frac{\Gamma}{2\pi h} \hat{j}$
- Vortex 2 ($-\Gamma$, clockwise) is at $(+h/2, 0)$. At the location of Vortex 1 (distance $h$), it also induces an upward velocity in the $+y$ direction of magnitude:
$V_1 = \frac{\Gamma}{2\pi h} \hat{j}$
3. **Coupled Motion**:
Because both mutually induced velocity vectors point in the identical direction ($+y$) with equal magnitude, the vortex pair does not rotate. Instead, it self-propels (translates) steadily along the $+y$ axis at speed $V = \frac{\Gamma}{2\pi h}$.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) rotate counter-clockwise...**: ❌ **INCORRECT**. Co-rotating vortices (same sign) rotate; counter-rotating vortices translate.
- **(B) rotate counter-clockwise...**: ❌ **INCORRECT**. Non-translating rotation requires like-sign vortices.
- **(C) translate along +y direction with velocity Γ / (2πh)**: ✅ **CORRECT**. Induced velocity from separation distance $h$ is precisely $V = \frac{\Gamma}{2\pi h}$ in the $+y$ direction.
- **(D) translate along +y direction with velocity Γ / (4πh)**: ❌ **INCORRECT**. Erroneously uses radius $h/2$ in denominator instead of full distance $h$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: Two vortices of SAME sign rotate about their midpoint ($\\omega = \Gamma / (\pi d^2)$). Two vortices of OPPOSITE sign translate rectilinearly ($V = \Gamma / (2\pi d)$).
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