Mach Number
An introduction to the Mach Number.
The Mach Number is a dimensionless value useful for analyzing fluid flow dynamics problems where compressibility is a significant factor.
The Mach Number can be expressed as
M = v / c (1)
where
M = Mach number
v = fluid flow speed (m/s, ft/s)
c = speed of sound (m/s, ft/s)
Alternatively the Mach Number can be expressed with the density and the bulk modulus for elasticity as
M = v (ρ / E)^{1/2} (2)
where
ρ = density of fluid (kg/m^{3}, lb/ft^{3})
E = bulk modulus elasticity (N/m^{2 }(Pa), lb_{f}/in^{2} (psi))
The bulk modulus elasticity has the dimension pressure and is commonly used to characterize the fluid compressibility.
The square of the Mach number is the Cauchy Number.
M^{2} = C (3)
where
C = Cauchy Number
Subsonic and Supersonic speed
 If the mach number is < 1, the flow speed is lower than the speed of sound  and the speed is subsonic.
 If the mach number is ~ 1, the flow speed is approximately like the speed of sound  and the speed is transonic.
 If the mach number is > 1, the flow speed is higher than the speed of sound  and the speed is supersonic.
 If the mach number is >> 1, the flow speed is much higher than the speed of sound  and the speed is hypersonic.
Example  Calculating an Aircraft Mach Number
An aircraft flies at speed 500 mph at an altitude of 35000 ft. The surrounding temperature is 60 ^{o}F.
The speed of sound at this altitude and temperature can be calculated
c = [k R T]^{1/2}
= [ 1.4 (1716 ft lb/slug ^{o}R) ((60 ^{o}F) + (460 ^{o}R)) ]^{1/2}
= 980 ft/s
where
k = 1.4
R = 1716 (ft lb/slug ^{o}R)
The speed of the aircraft can be calculated as
v = (500 miles/hr) (5280 ft/miles) / (3600 sec/hr)
= 733 ft/sec
The Mach Number can be calculated as
M = (733 ft/s) / (980 ft/s)
= 0.75  the aircraft is flying at subsonic speed
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Fluid Mechanics
The study of fluids  liquids and gases. Involving velocity, pressure, density and temperature as functions of space and time.
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