Examples of steady subsonic flows in a convergent-divergent approximate nozzle

We construct special solutions of the full Euler system for steady compressible flows in a convergent-divergent approximate nozzle and study the stability of the purely subsonic flows. For a given pressure p₀ prescribed at the entry of the nozzle, as the pressure p₁ at the exit decreases, the flow patterns in the nozzle change continuously: there appear subsonic flow, subsonic-sonic flow, transonic flow and transonic shocks. Our results indicate that, to determine a subsonic flow in a two-dimensional nozzle, if the Bernoulli constant is uniform in the flow field, then this constant should not be prescribed if the pressure, density at the entry and the pressure at the exit of the nozzle are given; if the Bernoulli constant and both the pressures at the entrance and the exit are given, the average of the density at the entrance is then totally determined.