Solitary Waves on a Cylinder Shell with Liquid

An elastic cylindrical shell of infinite length is considered in this work. The geometrically nonlinear membrane equations are used to describe the shell. The ideal incompressible liquid fills the shell entirely. It is assumed that the velocity of the unperturbed motion of the liquid is constant. The problem is discussed in the axisymmetric formulation. The case of linear dispersion is studied, and the solutions in the form of nonlinear solitary waves are constructed. The solutions are expressed in the form of expansions in powers of a small parameter, the amplitude. For the nonlinear shell without liquid there are solutions in the form of a pair of waves with different phase velocities which may propagate in both directions along the axis of symmetry. For the shell filled with the liquid at rest, the same pattern is observed; however, the solutions themselves demonstrate a qualitatively different nature. In the case of the liquid flowing along the shell axis, the four different in the phase velocity solutions are constructed. We investigated how the obtained solutions depend on the physical parameters which characterize the system and a numerical example is given.