Numerical simulation of nonlinear oscillations of a viscoelastic pipeline with fluid

In this paper, mathematical models of nonlinear dynamic problems with fluid and gas flows through pipelines have been developed based on the Boltzmann–Volterra integral models with weakly singular hereditary kernels. Using the Bubnov–Galerkin method for the boundary conditions, the resulting nonlinear integrodifferential equations with partial derivatives are reduced to solving systems of nonlinear ordinary integrodifferential equations with both constant and variable coefficients as functions of time. It is proposed to investigate oscillating processes occurring in a pipeline by a numerical algorithm for solving the nonlinear integrodifferential equations with weakly singular hereditary kernels, which is convenient for a computer implementation. On the basis of the developed computational algorithm, a complex of computer application programs allowing one to explore a completely new class of mathematic simulation problems, such as an oscillatory process of viscoelastic thin-walled pipelines with a large diameter, in terms of the shell theory is designed. The influence of a singularity in the hereditary kernels on oscillations of the construction with viscoelastic properties has been numerically investigated. When simulating the nonlinear problems, a number of new dynamic effects were explored. It was found that the determination of the effect of viscoelastic properties of the construction material on vibrations of the pipeline with a flowing liquid requires applying weakly singular hereditary kernels with an Abel type singularity.