Variational method for the solution of nonlinear boundary-value problems of the dynamics of bounded volumes of liquid with variable boundaries

Abstract

We study nonlinear boundary-value problems of the hydrodynamic type formulated for the domains whose boundaries are unknown in advance. The investigations are based on the variational formulations of these problems with the help of specially introduced integral functionals with variable domains of integration. It is shown that the required solutions of the boundary-value problems are, in a certain sense, equivalent to finding stationary points of the analyzed functionals. These are pairs formed by the families of domains and the functions defined in these domains. On an example of the problem of space motion of a vessel with elastic walls partially filled with an ideal incompressible liquid, we propose a method for the construction in the analytic form both of the required solutions and of the boundaries of the domains (deformed walls and the shape of the perturbed free surface of the liquid).