Longitudinal–Transverse Natural Waves in a Cylindrical Shell in Contact with a Viscous Fluid

Natural waves are widely used in seismology and seismic exploration as tools for nondestructive testing of the surface layer. The study examines longitudinal and transverse vibrations of a polymer pipeline transporting petroleum products, which is modelled as a viscoelastic cylindrical shell filled with a viscous fluid. This work examines the longitudinal–transverse vibrations of a viscoelastic cylindrical shell filled with a viscous fluid, considering the viscous properties of both the fluid and the cylindrical shell during longitudinal–transverse oscillations. The differential equations governing the longitudinal–transverse vibrations of a cylindrical shell in contact with a viscous fluid are derived based on thin-shell equations satisfying the Kirchhoff–Love hypotheses, while the motion of the viscous fluid obeys the Navier–Stokes equations. The viscoelastic properties of the shell are described using the Boltzmann–Volterra hereditary integral. After applying the “freezing method” to the system of integro-differential equations, we obtain ordinary differential equations with complex coefficients, which are subsequently solved by the method of separation of variables and Godunov’s orthogonal sweep combined with Müller’s and Gauss’s methods in complex arithmetic. It is established that for small viscosity, the frequencies of both modes are close to each other in the low-frequency region, while at high frequencies, the phase velocity of the first mode tends toward the velocity of the dry shell.