Non-linear waves in coaxial cylinder shells containing viscous liquid inside with consideration for energy dispersion

The equations which describe the strain waves by means of asymptotic methods of solving the hydro-elastic problem that includes the dynamic equations of two coaxial geometrically and physically nonlinear elastic shells are obtained. Energy dissipation and equations for an incompressible viscous fluid between cylindrical shells with appropriate boundary conditions are taken into account. Two cases are considered: one with structural damping in the material of shells, and the other with the viscoelastic material of the shell. Both cases lead to the same equations, which generalize the well-known modified Korteweg-de Vries–Burgers equations by introducing the term describing the liquid impact between the shells. The radius of the medial surface of the shell is significantly smaller than the wavelength of deformation, and therefore the asymptotic transition to the classical equation of hydrodynamic lubrication theory is made in the equations of viscous incompressible fluid. The presence of fluid between the co-axial shells gives rise to deformation waves not only in the outer shell but also in the inner one, where the initial deformation moment is equal to zero. Hence, the deformation wave of stable amplitude and velocity takes place. This fact is in accordance with the solitary wave solution, which cannot be described analytically. The construction under consideration can be characterized as a three layered packet, with liquid as a filler.