For almost flat nozzles, the paper proves that a scalar solvability condition determines admissible transonic shock locations, with multiple locations possible when the nozzle wall has both expanding and contracting parts.
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On Admissible Locations of Transonic Shock Fronts for Steady Euler Flows in an Almost Flat Finite Nozzle with Prescribed Receiver Pressure
For almost flat nozzles, the paper proves that a scalar solvability condition determines admissible transonic shock locations, with multiple locations possible when the nozzle wall has both expanding and contracting parts.