Method of singular integral equations in liquid vibration problems for coaxial shells

Abstract

The paper deals with the problem 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 the velocity potential. In these suppositions the potential is satisfied to Laplace equation. The boundary conditions are formulated on the wetted surfaces of the shells and on the 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 the eigenvalue problem. To solve the boundary value problem for the Laplace equation, the boundary element method is used in a direct formulation. The axial symmetric form of the shells allows us to reduce the obtained system of singular equations to one-dimensional equations. The kernels in singular operators of obtained integral equations are expressed on terms of elliptical integrals of the first and second kinds, and have the logarithmic singularities. The special numerical technique is elaborated to treat with such kind integral equations. The resulting one-dimensional singular equation is solved by the method of discrete singularities. The integration region contains the free surface of the fluid that in the case of coaxial shells is a ring. So, the possibility of using the boundary integral equation approach coupled with application of the discrete singularities method is established to solution of the singular integral equation with incoherent boundaries. A numerical study has been carried out that made it possible to determine the frequencies and modes of the liquid sloshing in the shells for different ratios of the inner and outer radii of cylindrical coaxial shells. The obtained modes of natural vibrations will be used for numerical simulation of forced liquid vibrations in the tanks and reservoirs.