Formulation of the problem of dynamics of interaction of a compressed layer of a viscous compressible fluid with elastic coaxial shells

Аuthors

Grigoriev S. A.^1*, Kondratov D. V.^{1, 2**}

1. Yuri Gagarin State Technical University of Saratov, 77, Politechnicheskaya str., Saratov, 410054, Russia
2. Institute of Precision Mechanics and Control, Russian Academy of Sciences, 24, Rabochaya str., Saratov, 410028, Russia

*e-mail: sagrigorev0703@yandex.ru
**e-mail: kondratovdv@yandex.ru

Abstract

The article presents the development of a mathematical model for a mechanical system consisting of two elastic coaxial shells with a layer of viscous compressible fluid in between. The shells are rigidly fixed at the ends, with no rotation or movement along their axis of symmetry. The thickness of the fluid layer is significantly smaller than the outer radius of the inner shell. A model of a viscous compressible barotropic fluid is used. Viscosity is taken into account because it provides damping properties, which prevent excessively large deflections during resonance. Compressibility must be taken into account, since in systems with varying pressure, as a barotropic fluid can affect the behavior of the system. The system is affected by an external vibration source and time-harmonic pressure at the ends. The system is considered thermally stable. The mathematical model of the mechanical system considered in this work is a system of coupled equations. To describe the dynamics of a fluid, we use the nonlinear partial differential Navier-Stokes equations and the continuity equation. The conditions for matching pressure and flow rate of liquid during the transition from cylindrical slit to end chambers are used as boundary conditions. To describe the dynamics of inner and outer cylindrical shells, we write partial differential equations based on Kirchhoff-Love hypotheses. Equations of shell displacement at ends and along Oy axis are used as boundary conditions. Further study of this mathematical model will allow us to better understand the processes occurring in such systems and develop more efficient methods for their analysis. These methods can be applied in various fields of science and technology, such as mechanical engineering, aviation and space industries.

Keywords:

viscous compressible fluid, elastic coaxial shells, Navier-Stokes equation, continuity equation

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