### **Detailed Step-by-Step Solution (GATE AE 2025 Q.21)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to identify the physical definition and ratio represented by the **Mach number** ($M$) in fluid dynamics.
**2. Step-by-Step Fluid Dynamic Derivation**:
1. **Definition of Mach Number**:
- The Mach number is the ratio of flow velocity $V$ to the speed of sound $a$:
M = \frac{V}{a}
2. **Physical Forces Ratio Representation**:
- Dynamic pressure represents fluid **inertia force**:
F_{\text{inertia}} \sim \rho V^2 L^2
- The speed of sound in a fluid is directly related to the bulk elastic modulus $K$:
a = \sqrt{\frac{K}{\rho}} \implies K = \rho a^2
- The **elastic force** resisting compression is:
F_{\text{elastic}} \sim K L^2 \sim \rho a^2 L^2
- Taking the ratio of inertia force to elastic force:
\frac{F_{\text{inertia}}}{F_{\text{elastic}}} = \frac{\rho V^2 L^2}{\rho a^2 L^2} = \frac{V^2}{a^2} = M^2
- Taking the square root:
M = \sqrt{\frac{\text{inertia force}}{\text{elastic force}}}
**3. Comprehensive Options & Distractor Analysis**:
- **(A) √(inertia force / viscous force)**: ❌ **INCORRECT**. This is related to Reynolds number ($Re = \text{inertia force} / \text{viscous force}$).
- **(B) √(inertia force / elastic force)**: ✅ **CORRECT**. Mach number represents the square root of the ratio of inertia force to elastic (compressibility) force.
- **(C) √(elastic force / viscous force)**: ❌ **INCORRECT**.
- **(D) √(viscous force / inertia force)**: ❌ **INCORRECT**. Inverse of Reynolds number.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: Fundamental dimensionless numbers in fluid mechanics:
- Reynolds: Inertia / Viscous
- Mach: $\sqrt{\text{Inertia / Elastic}}$
- Froude: $\sqrt{\text{Inertia / Gravity}}$
- Weber: $\sqrt{\text{Inertia / Surface Tension}}$.