For classifying the speed performances of bodies in a fluid (e.g. aircrafts), as well as the flow regimes of fluids, the Swiss aeronautical engineer Jakob Ackeret (1898 – 1981) – one of the leading authorities in the 20th century aeronautics – proposed in 1929 that the ratio of the body or of the fluid's speed v and the local sound speed c in that fluid to be called Mach's number M=cv after the name of the great Czech (then in the Austrian empire) physicist and philosopher Ernst Mach (1838 – 1916). Primarily, the value of this non-dimensional quantity delimitates the incompressible from the compressible behavior of a flowing fluid, in aeronautics this limit being settled to M=0.3.
B3.1 Find the relative variation of the gas density ρΔρ as a function of Mach's number, when its motion is slowed down to a stop, its initial velocity being vc, and calculate its maximum value for a flow to be considered incompressible.
B3.2 The pressure at the nose of an aircraft in flight was found to be p=105 kPa and the speed of air relative to the aircraft was zero at this point. The pressure and temperature of the undisturbed air were p0=95 kPa and T0=280 K respectively. The adiabatic exponent for this temperature is γ=1.4. Find the speed and the Mach number of the aircraft.
B3.3 When a gas is flowing through a pipe, it exerts a friction force on the fluid, which is not always negligible. If at the entrance of such a pipe the static pressure in the flowing fluid is p1=300 kPa and the Mach number is M1=0.2, while at the exit M2=0.25, find the expression and the numerical value of the force with which the fluid is acting on the pipe. The adiabatic exponent is γ=1.4, the constant cross section of the pipe is A=0.01 m2, and the relative increase of the gas temperature through the pipe is T1T2−T1=0.06.