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Nonlinear waves in viscoelastic physically nonlinear cylindrical shell with a viscous incompressible fluid and surrounded by an elastic medium

Blinkov Yu.A., Kuznetsova E.L., Mogilevich L.I., Rabinskiy L.N.

#### Abstract:

A well-known Gardner equation modeling deformation waves was generalized. A new equation was obtained by means of the asymptotic approach applied to the coupled hydroelasticity problem including the dynamic equations for a nonlinearly deformed viscoelastic shell surrounded by an elastic media and the dynamic equations for a viscous incompressible liquid in the shell with their boundary conditions. The radius of the shell midsurface is significantly less than the deformation wavelength, so that the asymptotic transformation of the dynamic equations for the viscous incompressible liquid to the ones

of the hydrodynamic lubrication theory becomes possible.

Here we use the new method of a finite-difference schema’s construction based

on the overdetermined finite-difference equations system that follows from the approximation of the integral conservation laws as well as the integral relations between the unknown functions and their derivatives. The interference of the liquid, the elastic media, and the shell is taken

into account. As a result, the finite difference schema is defined as a compatibility equation

for this system and secures the conservation laws for the domains compounded from the basic finite volumes.

The presence of the liquid in the shell surrounded by the elastic media results the growth or drop of the deformation wave amplitude depending on the Poisson ratio of the viscoelastic media. The elastic media surrounding the shell results the growth of the velocity

of the nonlinear wave of deformation.

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