ON SONIC CURVES OF SMOOTH SUBSONIC-SONIC AND TRANSONIC FLOWS

This paper concerns properties of sonic curves for two-dimensional smooth subsonic sonic and transonic steady potential flows, which are governed by quasi-linear degenerate elliptic equations and elliptic-hyperbolic mixed-type equations with degenerate free boundaries, respectively. It is shown that a sonic point satisfying the interior subsonic circle condition is exceptional if and only if the governing equation is characteristic degenerate at this point. For a C-2 subsonic-sonic flow whose sonic curve S is a nonisolated C-2 simple curve from one solid wall to another one, it is proved that S is a disjoint union of three connected parts (possibly empty): S-e, S-, S+, where S-e, is the set of exceptional points, which is empty or a point or a closed segment where the velocity potential equals identically to a constant; S- and S+ denote the other two connected parts before and after S-e if S-e not equal empty set , respectively. Moreover, if each interior point of S is not exceptional, then the velocity potential is strictly monotone along S; otherwise, on S-e the velocity is along the normal direction of S, and the velocity potential is strictly increasing (decreasing) along S- and strictly decreasing (increasing) along S+ if on S-e, the velocity is along the outer (inner) normal direction of S from the subsonic region. For a C-2 transonic flow, it is shown that there exist uniquely two distinct characteristics from each nonexceptional point (except the ones on the walls) pointing into the supersonic region, while two characteristics from an interior point of an exceptional segment coincide with this segment and never approach the supersonic region locally. Furthermore, it is proved that the curvature of the streamline in the physical plane near a nonexceptional point must be nonzero.