Hydroelasticity problems for viscous compressible liquids in a spherical coordinate system

Abstract

UDC 539.3 There are few papers devoted to the interaction between shells and viscous liquids among the many publications concerned with problems in the dynamics of liquid-filled shells. For the most part, axisymmetric problems for spherical and cylindrical shells which interact with an incompressible viscous liquid are investigated in these papers. Problems in the hydroelasticity of shells for a compressible viscous liquid have been considered only in a few papers. Thus, the axisymmetric problem for a cylindrical shell was investigated in [6], while the general (nonaxisymmetric ) problem for a cylindrical shell was investigated in [2], where use was made of the general solutions of the linearized Navier--Stokes equations for a compressible viscous liquid at rest that had been obtained in [1] and were in this case expressed in terms of the solutions of three independent second-order scalar equations. The present article is concerned with hydroelasticity problems for elastic solids interacting with a compressible viscous liquid. The three-dimensiona l equations of dynamic elasticity theory are used for describing the motion of the elastic solid [3]; linearized Navier--Stokes equations, the general solutions of which have been obtained in [1], are used for describing small oscillation of the compressible viscous liquid at rest. According to [5J, the solutions for the elastic solid and the compressible viscous liquid in a spherical coordinate system are represented in terms of the solutions of three second-order scalar equations. The derived presentations of solutions are general, and they apply to various steady-state and transient-state dynamic problems for elastic solids and compressible viscous liquids with spherical interfaces. In order to save space in the article, a number of quantities pertaining to the elastic solid are marked by the superscript 1, while the analogous quantities for the liquid are denoted by the superscript 2. 1. Basic Relationships. The displacement vector-~(l) for the elastic solid is given by