  We establish the existence and uniqueness of some smooth accelerating
transonic flows governed by the steady compressible Euler equations with an
external force in cylinders with arbitrary cross sections, which include both
irrotational flows and Beltrami flows with nonuniform proportionality factors.
One of the key ingredients in the analysis of smooth transonic irrotational
flows is the well-posedness theory for a linear elliptic-hyperbolic mixed
second order differential equation of Keldysh type in cylinders, which is
achieved by extending the problem to an auxiliary linear
elliptic-hyberbolic-elliptic mixed problem in a longer cylinder where the
governing equation becomes elliptic at the exit of the new cylinder, so that
one can use the multiplier method and the cut-off techniques to derive the
$H^2$ and higher order estimates in transonic regions. It is further shown that
the energy estimate can be closed in the $H^4$ framework. For smooth transonic
Beltrami flows, we solve a transport equation for the proportionality factor
and a type-changing enlarged deformation-curl system with mixed boundary
conditions. The compatibility conditions for the $H^4$ estimate to the enlarged
deformation-curl system near the intersection between the entrance and the
cylinder wall play a crucial role in the analysis.
