  This paper concerns studies on smooth transonic flows with nonzero vorticity
in De Laval nozzles for a quasi two dimensional steady Euler flow model which
is a generalization of the classical quasi one dimensional model. First, the
existence and uniqueness of smooth transonic flows to the quasi one-dimensional
model, which start from a subsonic state at the entrance and accelerate to
reach a sonic state at the throat and then become supersonic are proved by a
reduction of degeneracy of the velocity near the sonic point and the implicit
function theorem. These flows can have positive or zero acceleration at their
sonic points and the degeneracy types near the sonic point are classified
precisely. We then establish the structural stability of the smooth one
dimensional transonic flow with positive acceleration at the sonic point for
the quasi two dimensional steady Euler flow model under small perturbations of
suitable boundary conditions, which yields the existence and uniqueness of a
class of smooth transonic flows with nonzero vorticity and positive
acceleration to the quasi two dimensional model. The positive acceleration of
the one dimensional transonic solutions plays an important role in searching
for an appropriate multiplier for the linearized second order mixed type
equations. A deformation-curl decomposition for the quasi two dimensional model
is utilized to deal with the transonic flows with nonzero vorticity.
