In fluid dynamics, d'Alembert's paradox refers to which one of the following?
Options:
(A) Deviation of drag from D ∝ v2 at very low speeds
(B) Deviation of drag from D ∝ v2 at high subsonic speeds
(C) Prediction of zero drag by potential flow theory
(D) Presence of shocks in transonic flows
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.15)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to define d'Alembert's paradox within fluid dynamics.
**2. Step-by-Step Mathematical Derivation**:
1. **Potential Flow Theory Formulation**:
In 1752, French mathematician Jean le Rond d'Alembert used Euler's equations of motion for steady, incompressible, irrotational, and inviscid potential flow around a body of arbitrary closed shape.
2. **Symmetry of Pressure Distribution**:
Because the fluid is assumed frictionless (zero viscosity, $\mu = 0$) and the flow does not separate, streamline patterns and static pressure distributions exhibit fore-and-aft symmetry:
$\oint_S p \vec{n} \cdot \hat{i} \, dA = 0$
3. **The Paradox**:
This mathematical derivation proved that a body moving at constant velocity through an inviscid fluid experiences **strictly zero net drag force**. This directly contradicted universal empirical observation (where every body experiences drag), presenting the paradox that stymied fluid dynamicists until Ludwig Prandtl introduced boundary layer theory in 1904.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) Deviation of drag from D ∝ v² at very low speeds**: ❌ **INCORRECT**. This refers to the Stokes drag regime ($D \propto v$) for creeping flow ($Re \ll 1$).
- **(B) Deviation of drag from D ∝ v² at high subsonic speeds**: ❌ **INCORRECT**. This refers to compressibility wave drag rise near drag-divergence Mach number.
- **(C) Prediction of zero drag by potential flow theory**: ✅ **CORRECT**. Potential flow theory for an inviscid, unseparated fluid mathematically predicts exactly zero net drag on any 2D or 3D closed body.
- **(D) Presence of shocks in transonic flows**: ❌ **INCORRECT**. Shocks are non-isentropic gasdynamic wave phenomena.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: d'Alembert's paradox arises because inviscid theory neglects the no-slip condition at the wall, which creates thin viscous boundary layers and wake separation responsible for real-world form drag and skin friction.
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