Abstract:

This paper studies the existence and uniqueness of transonic shock solutions to the steady compressible Euler equations with gravity in a three-dimensional spherically symmetric divergent nozzle. Assuming that the influence of gravity on the fluid is sufficiently small and the supersonic initial conditions are given at the entrance, it can be proved that when the pressure $p$ at the exit falls in certain range, there exists a unique transonic shock solution within the nozzle by demonstrating that the pressure at the outlet is a strictly monotone function of the shock location.

Key words: transonic solution, Bernoulli's function, Euler equations, spherically symmetrical flow, gravity