Wave occurrences mathematical modeling in two geometrically nonlinear elastic coaxial cylindrical shells, containing viscous incompressible liquid

Yu. A. Blinkov ; A. V. Mesyanzhin ; L. I. Mogilevich

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 2, pp. 184-197

Résumé

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 shell dynamics and viscous incompressible liquid equations in the form of generalized Korteweg–de Vries equations. In addition, mathematical models of 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 equation 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 between and inside them. The difference schemes of Crank–Nicholson type are obtained for the considered equation system by taking into account liquid and with the help of Gröbner basis construction. To generate these difference schemes, the basic integral difference correlations, approximating the initial equation system, were used. The usage of Gröbner basis technology provides generating the schemes, for which it becomes possible to obtain discrete analogs of the laws of preserving the initial equation system. To do that, equivalent transformations were made. Based on the computation algorithm the corresponding software, providing graphs generation and numerical solutions under exact solutions of coaxial shell dynamics equation system obtaining, was developed.