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Introduction to Mach Number

It can be defined as the square root of the ratio of the inertia force of flowing liquid to the elastic force.

Mach Number=M

$\therefore M=\sqrt{\dfrac{\text{Inertia Force}}{\text{Elastic Force}}}$

$M=\sqrt{\dfrac{\rho AV^2}{KA}}$

$M=\sqrt{\dfrac{V^2}{\dfrac K\rho}}$

$M=\dfrac{V}{\sqrt{\dfrac K\rho}}$

Hence, $\left[M=\dfrac Vc\right]$

Significance of Mach number:-

The flow is defined according to the mach number, it is an important non-dimensional parameter

1. Sub-sonic flow:- When the mach number is less than '1.0' (M<1), i.e., velocity of flow is less than the velocity of sound wave (V<c). Such floor is known as Sub-sonic flow.</p>

2. Sonic flow:- When the mach number is equal to 1.0, i.e., the velocity of flow is equal to the velocity of sound (V = c). Such flow is known as Sonic flow.

3. Supersonic flow:- When the Mach number is greater than 1.0, i.e., the velocity of flow is greater than velocity of sound (V>C), such flow is known as Supersonic flow.

### Numericals

Q1. Find the Sonic velocity for the following fluids:-

I. Crude oil sp.gr. 0.8 and bulk modulus $153036N/cm^2$

II. Mercury having a bulk modulus of $2648700N/cm^2$

Solution:

I. For crude oil,

sp.gr=0.8

$K=153036N/cm^2=153036\times10^4N/m^2$

$\rho=0.8\times1000=800kg/m^3$

$\therefore c=\sqrt{\dfrac K\rho}$

$c=\sqrt{\dfrac {153036\times10^4}{800}}=1383.09m/s$

II. For mercury,

sp.gr=13.6

$K=2648700N/cm^2=2648700\times10^4N/m^2$

$\rho=13.6\times1000kg/m^3$

$\therefore c=\sqrt{\dfrac K\rho}$

$c=\sqrt{\dfrac {2648700\times10^4}{13.6\times1000}}=1395.55m/s$

Q2. Calculate the Mach number at a point on a jet-propelled aircraft, which is flying at 1100 km per hour at sea level where temperature is $20^\circ C$. Take K = 1.4, $R=287J/kg-K$

Solution: Given:

Speed, V=1100km/hr

$V=\dfrac {1100\times1000}{60\times60}$

V=305.55m/s

Temperature, $t=20^\circ C$

$T=273+20=293K$

$K=1.4, R=287J/kg-K$

$\therefore c=\sqrt{KRT}$

$c=\sqrt{1.4\times287\times 293}$

$c=343.11 \ m/s$

Mach Number, $M=\dfrac Vc=\dfrac {305.55}{343.11}$

$M=0.89$

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