Wave Processes Modeling in Two Coaxial Shells Filled with a Viscous Liquid and Surrounded by Elastic Medium

The investigation of deformation waves behavior in elastic shells is one of the important trends in the contemporary wave dynamics. There exist mathematical models of wave motions in infinitely long geometrically non-linear shells, containing viscous incompressible liquid based on the related hydroelasticity problems, which are derived by the shells dynamics and viscous incompressible liquid equations in the form of centralized Korteweg-de Vries (KdV) equations. In addition, mathematical models or the wave process in infinitely long geometrically non-linear coaxial cylindrical elastic shells are obtained by the perturbation method. These models differ from the known ones by the consideration of incompressible liquid between the shells, based on the related hydroelasticity problems. These problems are described by shell dynamics and viscous incompressible liquid equations with corresponding edge conditions in the form of generalized KdV equations system. The paper presents the investigation of wave occurrences in two geometrically non-linear elastic coaxial cylindrical shells of Kirchhoff-Love type, containing viscous incompressible liquid both in between and inside, and surrounded by an elastic medium, acting in both normal and longitudinal directions. The difference schemes of Crank-Nicholson type are obtained for the considered equations system by taking into account liquid impact and with the help of Grobner basis construction. To generate these difference schemes, the basic integral difference correlations, approximating initial equations system, were used.