The Method of Singular Integral Equation in Problems of Liquid Oscillations in Coaxial Shells

Abstract

The paper deals with problems of free vibrations of an ideal incompressible fluid in coaxial shells of revolution. It is assumed that the motion of the fluid is irrotational that allows us to introduce a velocity potential. In these suppositions the potential is satisfied to the Laplace equation. Boundary conditions are formulated on wetted surfaces of the shells and on a free liquid surface. The non-penetration conditions are applied to the wetted surfaces. On the free surface we consider dynamical and kinematical boundary conditions. The dynamical condition consists in equality of the liquid pressure on the free surface to the atmospheric one. The kinematic condition requires that total time derivative of the free surface elevation will be equal to zero at any instant. Regarding the potential of velocities, a boundary value problem is formulated that is further reduced to an eigenvalue problem. To solve the boundary value problem for the Laplace equation, boundary element methods are used in a direct formulation.