In the context of steady, inviscid, incompressible flows, consider the superposition
of a uniform flow with speed U along the positive x-axis (from left to right), and a
source of strength $\Lambda$ located at the origin. Which one of the following statements is
NOT true regarding the location of the stagnation point of the resulting flow?
Options:
(A) It is located to the left of the origin
(B) It moves closer to the origin for increasing $\Lambda$, while U is held constant
(C) It moves closer to the origin for increasing U, while $\Lambda$ is held constant
(D) It is located along the x-axis
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2024 Q.21)**
**1. Intuitive Physical Setup & Given Parameters**:
In 2D potential flow, we consider the superposition of a uniform stream $U_\infty$, a source of strength $\Lambda$, and an equal-strength sink of strength $-\Lambda$ placed at a distance $2a$ apart along the flow axis.
**2. Step-by-Step Mathematical Derivation**:
1. **Superposition of Elementary Potential Flows**:
The stream function for the combined flow is:
\psi(x, y) = U_\infty y + \frac{\Lambda}{2\pi}\theta_1 - \frac{\Lambda}{2\pi}\theta_2
where $\theta_1$ and $\theta_2$ are the polar angles from the source at $(-a, 0)$ and sink at $(+a, 0)$.
2. **Dividing Streamline & Body Shape**:
The dividing streamline $\psi = 0$ forms an oval-shaped closed streamline that isolates the fluid emitted by the source and absorbed by the sink from the outer streamflow.
3. **Physical Identification**:
This closed streamline represents flow over a symmetric elongated body known as a **Rankine oval** (or Rankine body).
- If source and sink coalesce ($a \to 0$) keeping $\Lambda \cdot (2a) = \text{const}$, it becomes a **doublet**, yielding flow past a circular cylinder.
- A uniform flow plus a single source yields a semi-infinite open body known as a **Rankine half-body**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) circular cylinder**: ❌ **INCORRECT**. Flow over a circular cylinder requires uniform flow + doublet (coincident source-sink pair).
- **(B) Rankine oval**: ✅ **CORRECT**. A uniform flow combined with a separated source-sink pair forms a closed Rankine oval.
- **(C) Rankine half-body**: ❌ **INCORRECT**. A half-body requires uniform flow + single source only.
- **(D) vortex sheet**: ❌ **INCORRECT**. Generated by a distribution of point vortices.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: Remember the classic potential flow bodies: 1) Uniform + Source = Half-body (open), 2) Uniform + Source + Sink = Rankine oval (closed), 3) Uniform + Doublet = Circular Cylinder (closed).