Maximum values of gas-dynamic flux densities

It is well known that, in steady isentropic expansion of a gas from a reservoir, the maximum mass flux density, pu, occurs at Mach number M=u/a= 1 (p, u, and a are the local values of density, velocity, and speed of sound, respectively; po, 0, and a0 are corresponding reservoir values.) The proof is usually given for the special case of a perfect gas and it does not seem to be well known that the result can be obtained in generality for any compressible fluid, as shown by Landau and Lifshitz.’ Extending their approach, the conditions for maximum flux density of momentum, pz?, and of kinetic energy, f pu3, can also be generally and simply obtained, for steady as well as unsteady flow. Consider the flux density pun, where n = 1, 2, or 3, and its variation with increasing flow velocity U. The value of pzJ’ is zero at u=O, where the pressure and density have their maximum values p. and po. Asp (and p) decrease, u increases in either steady or unsteady flow, attaining a maximum value u,, at p=O, where the density p has its minimum value (zero for a perfect gas). Thus pu” may have a maximum at some velocity between 0 and u,. This can be found by setting (d/c%) (pu”) =0, i.e.,