Discussion
In a fluid flow, Mach number is an estimate of _____.
Answer: B
The Mach number estimates \(\sqrt{\text{inertia force}/\text{elastic force}}\).
Inertia force scales as \(\rho V^{2}L^{2}\), and the elastic (compressibility) force scales as the bulk modulus times area, \(K L^{2}\). Since \(a^{2} = K/\rho\):
\[ \begin{aligned} \frac{\rho V^{2}L^{2}}{K L^{2}} &= \frac{V^{2}}{a^{2}} \\ &= M^{2} \end{aligned} \]
The ratio of inertia to elastic force (the Cauchy number) is \(M^{2}\), so M is its square root. Inertia over viscous force is the Reynolds number, which is not under a square root, and the other two ratios do not define a standard flow number.
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