Evolution of solitary hydroelastic strain waves in two coaxial cylindrical shells with the Schamel physical nonlinearity

Abstract

The paper considers the formulation and solution of the hydroelasticity problem for studying wave processes in the system of two coaxial shells containing fluids in the annular gap between them and in the inner shell. We investigate the axisymmetric case for Kirchhoff–Lave type shells whose material obeys a physical law with a fractional exponent of the nonlinear term (Schamel nonlinearity). The dynamics of fluids in the shells is considered within the framework of the incompressible viscous Newtonian fluid model. The derivation of the Schamel nonlinear equations of shell dynamics makes it possible to develop a mathematical formulation of the problem, which includes the obtained equations, the dynamics equations of two shells, the fluid dynamics equations and the boundary conditions at the shell-fluid interfaces and at the flow symmetry axis. The asymptotic analysis of the problem is performed using perturbation techniques, and the system of two generalized Schamel equations is obtained. This system describes the evolution of nonlinear solitary hydroelastic strain waves in the coaxial shells filled with viscous fluids, taking into account the inertia of the fluid motion. In order to determine the fluid stress at the shell-fluid interfaces, we perform linearization of the fluid dynamics equations for fluids in the annular gap and in the inner shell. The linearized equations are solved by the iterative method. The inertial terms are excluded from the equations in the first iteration, while, in the second iteration, these are the values found in the first iteration. A numerical solution of the system of nonlinear evolution equations is obtained by applying a new difference scheme developed using the Gröbner basis technique. Computational experiments are performed to investigate the effect of fluid viscosity and the inertia of fluid motion in the shells on the wave process. In the absence of fluids in the inner shell, the results of calculations demonstrate that the strain waves in the shells during elastic interactions do not change their shape and amplitude, i.e., they are solitons. The presence of viscous fluid in the inner shell leads to attenuation of the wave process.