Variational principles of nonlinear dynamical fluid–solid interaction systems

Abstract

Based on the fundamental equations of continuum mechanics, the concept of Hamilton's principle and the adoption of Eulerian and Lagrangian descriptions of fluid and solid, respectively,variational principles admitting variable boundary conditions are developed to model mathematically the nonlinear dynamical behaviour of the responses and interactions between fluid and solid. The nonlinearity of the fluid is introduced through nonlinear field equations and nonlinear boundary conditions on the free surface and fluid–solid interaction interface. The structure is treated as a nonlinear elastic body. This model assumes the fluid inviscid, incompressible or compressible and the fluid motion irrotational or rotational but isentropic along the flow path of each fluid particle. The stationary conditions of the variational principles include the governing equations of nonlinear elastic dynamics, fluid dynamics and those relating to the fluid-structure interaction interface as well as the imposed boundary conditions. A family of variational principles are obtained depending on the assumptions introduced into the mathematical model (i.e. fluid incompressible, motion irrotational, etc.) and these provide a foundation to construct numerical schemes of study to assess the dynamical behaviour of nonlinear fluid–solid interaction systems. Two simple illustrative examples are presented demonstrating the applicability of the proposed theoretical approach.