Compressible Euler Flows on a Convergent–Divergent Surface: Steady Subsonic Flows

In this paper, we construct various special solutions on a convergent-divergent surface for the steady compressible complete Euler system and established the stability of the purely subsonic flows. For a given pressure p0 prescribed at the “entry” of the surface, as the pressure p1 at the “exit” decreases, the flow patterns on the surface change continuously as those happen in a de Laval nozzle: there appear subsonic flow, subsonic–sonic flow, transonic flow and transonic shocks. This work may help us understand subsonic flows in de Laval nozzles. Our results indicate that, to determine a subsonic flow in a two-dimensional de Laval 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, then the average of the density at the entrance is totally determined.