A theoretical algorithm by united Lagrangian-Eulerian method for the problem in dealing with viscous fluid
and a circular cylindrical shell is presented. In this approach, each material is described in its preferred reference frame.
Fluid flows are given in Eulerian coordinates whereas the elastic circular cylindrical shell is treated in a Lagrangian
framework. The fluid velocity in a two-dimensional uniform elastic circular cylindrical shell filled with viscous fluid is
studied under the assumption of low Reynolds number. The coupling between the viscous fluid and the elastic circular
cylindrical shell shows kinematic conditions at the shell surface. Also, the radial velocity and axial velocity of the fluid
are discussed with the help of graphs.