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Abstract

Two-dimensional steady supersonic ramp flows are important and well-studied flow patterns in aerodynamics. Vimercati, Kluwick and Guardone [J. Fluid Mech., 885 (2018) 445--468] constructed various self-similar composite wave solutions to the supersonic flow of Bethe-Zel'dovich-Thompson (BZT) fluids past compressible and rarefactive ramps. We study the stabilities of the self-similar fan-shock-fan and shock-fan-shock composite waves constructed by Vimercati et al. in that paper. %In order to study the stabilities of the composite waves, we solve some classes of shock free boundary problems. In contrast to ideal gases, the flow downstream (or upstream) of a shock of a BZT fluid may possibly be sonic in the sense of the flow velocity relative to the shock front. In order to study the stabilities of the composite waves, we establish some a priori estimates about the type of the shocks and solve some classes of sonic shock free boundary problems. We find that the sonic shocks are envelopes of one out of the two families of wave characteristics, and not characteristics. This results in a fact that the flow downstream (or upstream) a sonic shock is not $C^1$ smooth up to the shock boundary. We use a characteristic decomposition method and a hodograph transformation method to overcome the difficulty cased by the singularity on sonic shocks, and derive several groups of structural conditions to establish the existence of curved sonic shocks.