Question

Consider the following momentum equation. Let \( A \), \( B \), and \( C \) denote the first, second, and third terms on the left-hand side respectively, and \( D \) and \( E \) denote the first and second terms on the right-hand side respectively. Which of the following statement(s) is/are correct? \[ \rho \left[ \frac{\partial \mathbf{V}}{\partial t} + \text{grad} \left( \frac{\mathbf{V}^2}{2} \right) + (\text{curl } \mathbf{V}) \times \mathbf{V} \right] = -\text{grad}(P + \rho gz) + \mu \nabla^2 \mathbf{V} \]

• A. If terms \( A \), \( C \), and \( E \) vanish, then the flow is irrotational.
• B. If term \( A \) vanishes, then the flow is steady.
• C. If term \( D \) vanishes, then it leads to the Euler's equation.
• D. If terms \( A \), \( B \), \( C \), and \( E \) vanish, then it leads to the hydrostatic equation.

Hint:

To interpret the Navier-Stokes equation: 1. Time-dependent terms indicate unsteady flow (A).
2. Convective terms represent flow inertia (B).
3. Vorticity terms (C) indicate rotational effects.
4. Pressure-gradient terms (D) lead to the hydrostatic condition when other terms vanish.
5. Viscous terms (E) model internal friction.

Answer 1

The Correct Option is A

Step 1: Understand the terms in the momentum equation.
The given equation is the Navier-Stokes equation. Each term in the equation represents a physical phenomenon: 1. Term A (\( \frac{\partial \mathbf{v}}{\partial t} \)): Represents the unsteady (time-dependent) term.
2. Term B (\( \text{grad} \, \frac{|\mathbf{v}|^2}{2} \)): Represents the convective acceleration term.
3. Term C (\( \text{curl} \, \mathbf{v} \times \mathbf{v} \)): Represents the rotational effects (vorticity).
4. Term D (\( -\text{grad}(P + \rho gz) \)): Represents the pressure and body force term.
5. Term E (\( \mu \nabla^2 \mathbf{v} \)): Represents the viscous dissipation term.
Step 2: Analyze each statement.
Option (A): If terms A, C, and E vanish:
- Term A vanishes, indicating the flow is steady.
- Term C vanishes, indicating no rotational effects (irrotational flow).
- Term E vanishes, indicating the flow is inviscid. Thus, the flow is irrotational.
Option (A) is correct. Option (B): If term A vanishes:
- The absence of \( \frac{\partial \mathbf{v}}{\partial t} \) indicates that the flow is steady. Option (B) is correct. Option (C): If term D vanishes:
- The term \( -\text{grad}(P + \rho gz) \) vanishes, but this does not directly lead to Euler's equation, as convective and rotational terms are still present.
Option (C) is incorrect.
Option (D): If terms A, B, C, and E vanish:
- Terms A (unsteady), B (convective), C (rotational), and E (viscous) vanish, leaving only term D, representing the pressure gradient and gravity. This reduces the equation to the hydrostatic condition:
\[ \text{grad}(P + \rho gz) = 0. \] Option (D) is correct.
Step 3: Verify the options.
Option (A): Correct. Option (B): Correct. Option (C): Incorrect. Option (D): Correct. Conclusion: The correct statements are (A), (B), and (D).