Solitary deformation waves in a Kirchhoff-Love cylindrical shell made of a material with a combined nonlinearity containing a viscous fluid


Аuthors

Mogilevich L. I.1*, Popova E. V.2**, Evdokimova E. V.1***, Popov V. S.1****

1. Yuri Gagarin State Technical University of Saratov, 77, Politechnicheskaya str., Saratov, 410054, Russia
2. Saratov State University named after N. G. Chernyshevsky, 83, Astrakhanskaya str., Saratov, 410012, Russia

*e-mail: mogilevichli@gmail.com
**e-mail: elizaveta.popova.97@bk.ru
***e-mail: eev2106@mail.ru
****e-mail: vic_p@bk.ru

Abstract

This paper presents the hydroelasticity problem formulation for a Kirchhoff-Love cylindrical shell whose material obeys a generalized Hooke's law, accounting for its nonlinearity in the form of a combination of a quadratic function and a power function with exponent 3/2. The case of an infinitely extended shell filled with a viscous Newtonian fluid of constant density is studied. An asymptotic analysis of the boundary value problem of mathematical physics is carried out using the multiscale perturbation method. Considering the first (linear) approximation with respect to a small parameter of the problem, it is established that the fluid filling the shell does not affect the wave process. The profile of the longitudinal strain wave front in the shell is an arbitrary function, and the strain waves in the shell propagate at the speed of sound. Considering the problem in the second approximation, a nonlinear evolution equation is obtained, generalized Schamel–Korteweg– de Vries equation, which allows one to study nonlinear solitary hydroelastic waves of longitudinal strain in the shell. It is shown that, in the particular case of an incompressible shell material and a creeping fluid flow in the shell, this equation has an exact soliton solution. The viscous fluid filling the shell has no effect on the wave process in its walls, and the soliton velocity is higher than the speed of sound. To study the general case, a new difference scheme is proposed for converting to a discrete analog of the generalized Schamel–Korteweg–de Vries equation. This scheme is obtained using an integro-interpolation approach based on the Gröbner base construction technique. A numerical study of this equation is performed with initial conditions specified as an exact particular soliton solution. Computational experiments revealed that the velocity of solitary longitudinal deformation waves in the shell is lower than the speed of sound, and the initially excited soliton decays over time if the shell material is compressible and the inertia of the fluid motion is taken into account. 

Keywords:

cylindrical shell, combined nonlinearity, viscous fluid, hydroelasticity, deformation solitons, computational experiment

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