Propagation of Nonlinear Waves in Coaxial Physically Nonlinear Cylindrical Shells Filled with a Viscous Fluid

Investigation of deformation waves behavior in elastic shells is one of the important trends in 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 generalized KdV equations. Also, mathematical models of the wave process in infinitely long geometrically non-linear coaxial cylindrical elastic shells are obtained by means of disturbances method. These models differ from the known ones by the consideration of incompressible liquid presence between the shells, based on the related hydroelasticity problems. These problems are described by shells 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 of two geometrically non-linear elastic coaxial cylindrical shells model of Kirchhoff-Love type, containing viscous incompressible liquid between them, as well as inside. 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 Gro¨bner bases construction. To generate these difference schemes, the basic integral difference correlations, approximating initial equations system, were used. The usage of Gro¨bner bases technology provides generating the schemes, for which it becomes possible to obtain discrete analogs of the laws of preserving initial equations system. To do this, equivalent transformations were made. On the basis of computation algorithm the complex of programs, permitting to construct graphs and obtain numerical solutions under exact solutions of coaxial shell dynamics equations system, was made.

nonlinear waves, viscous incompressible liquid, elastic cylinder shell