This work deals with the fluid-structure interaction problem of two coaxial cylinders separated by a viscous stagnant fluid. The cylinders are flexible and are imposed a displacement corresponding to the vibration mode of an Euler-Bernoulli beam. We carry out a new three-dimensional theoretical viscous approach to determine an analytical expression of the fluid added-mass and added-damping matrices. This formulation differs from earlier theories [1,2] by taking into account the viscous effects and not assuming a narrow gap. This new formulation applies to finite-size cylinders and encompasses all classical types of boundary conditions. We show that the coefficients of the matrix depend on the aspect ratio of the flexible cylinder, the confinement of the duct, and the characteristics of the imposed vibration mode. To assess the validity of the theoretical predictions, we perform numerical simulations with the open-source code TrioCFD [3]. The fluid-structure interaction problem involving moving boundaries is solved in this code using an Arbitrary Lagrange-Eulerian technique. We show that the numerical simulations successfully corroborate the theoretical predictions for all types of classical boundary conditions, different confinements, and different aspect ratios of the vibrating cylinder. [1] R. Lagrange and M. A. Puscas. Hydrodynamic interaction between two flexible finite length coaxial cylinders: new theoretical formulation and numerical validation. Journal of Applied Mechanics, 2022. [2] R. Lagrange, M. A. Puscas, P. Piteau, X. Delaune, and J. Antunes. Modal added-mass matrix of an elongated flexible cylinder immersed in a narrow annular fluid, considering various boundary conditions. New theoretical results and numerical validation. Journal of Fluids and Structures, 2022. [3] D.Panunzio, M. A. Puscas, and R. Lagrange. FSI–vibrations of immersed cylinders. Simulations with the engineering open-source code TrioCFD. Test cases and experimental comparisons. Comptes Rendus. Mécanique, 2022.