Construction of solutions of the problem of free oscillations of viscous fluid in a half-filled spherical tank

Abstract

We study free oscillations of a viscous incompressible fluid with free boundary in a hemispherical tank. We determine four systems of coordinate functions that satisfy the equations of the problem inside the domain and possess the properties of the required solution. Then, using these systems, we construct a coordinate system that satisfies all but one boundary condition of the problem. This system is used for the solution of the problem by a projection method. We compute the eigenfrequencies and logarithmic decrements of oscillations of the fluid. The results obtained are compared with asymptotic and empirical formulas.