Question

If the Mach number tends to infinity, what would be the measured density?

• A. Infinity
• B. Zero
• C. \(\sqrt{6}\) times higher than initial density
• D. 0.378 times higher than initial density

Hint:

As Mach number increases, shock strength increases and static properties decrease.

Answer 1

The Correct Option is B

In aerospace engineering, the Mach number (M) is a crucial parameter that represents the ratio of an object's speed compared to the speed of sound in the surrounding medium. The density of an object moving at high speeds can be analyzed using the formula derived from the isentropic flow relations. The relation connecting Mach number and density ratio is given as:

\( \frac{\rho}{\rho_0} = \left(1 + \frac{\gamma - 1}{2}M^2\right)^{-\frac{1}{\gamma-1}} \)

Where:

• \(\rho\) is the local density.
• \(\rho_0\) is the initial density.
• \(\gamma\) is the specific heat ratio (typically 1.4 for air).
• M is the Mach number.

When the Mach number tends to infinity, \(M \to \infty\), the term \(\frac{\gamma - 1}{2}M^2\) tends to infinity, leading to \(1 + \frac{\gamma - 1}{2}M^2 \to \infty\).

Further simplifying, the expression for density ratio becomes:

\( \frac{\rho}{\rho_0} = \infty^{-\frac{1}{\gamma-1}} \)

As any positive number raised to a negative power that increases without bound tends to zero, this results in:

\( \frac{\rho}{\rho_0} \to 0 \)

Thus, the density \(\rho\) approaches zero.

Therefore, when the Mach number tends to infinity, the measured density is Zero.