Interaction between an oscillating sphere and a thin elastic cylindrical shell filled with a compressible liquid. Internal axisymmetric problem

The problem on the interaction between a spherical body that oscillates in a prescribed manner and a thin elastic cylindrical shell filled with an ide al compressible liquid is formulated. It is assumed that the geometrical ce nter of the sphere is located on the cylinder axis. The problem is solved b ased on the possibility of representing a partial solution of the Helmholtz equation written in cylindrical coordinates in terms of partial solutions in spherical coordinates, and vice versa. By satisfying the boundary condit ions on the surfaces of the sphere and the shell, we obtain an infinite sys tem of linear algebraic equations to determine the coefficients of expansio n of the liquid-velocity potential into a Fourier series in terms of Legend re polynomials. The hydrodynamic characteristics of the liquid filling the cylindrical shell are determined and compared with the cases where a sphere oscillates in an infinite liquid and in a rigid cylindrical vessel.