  This paper concerns the structural stability of subsonic flows with a contact
discontinuity in a finitely long axisymmetric cylinder. We establish the
existence and uniqueness of axisymmetric subsonic flows with a contact
discontinuity by prescribing the horizontal mass flux distribution, the swirl
velocity, the entropy and the Bernoulli's quantity at the entrance and the
radial velocity at the exit. It can be formulated as a free boundary problem
with the contact discontinuity to be determined simultaneously with the flows.
Compared with the two-dimensional case, a new difficulty arises due to the
singularity near the axis. One of the key points in the analysis is the
introduction of an invertible modified Lagrangian transformation which can
overcome this difficulty and straighten the contact discontinuity. Another one
is to utilize the deformation-curl decomposition for the steady Euler system
introduced in \cite{WX19} to effectively decouple the hyperbolic and elliptic
modes. Finally, the contact discontinuity will be located by using the implicit
function theorem.
