The problem of viscous-fluid outflow from an elastic spherical vessel

A hydrodynamic problem of the liquid or gas outflow through a small hole in a spherical vessel with an elastic shell is formulated and solved. The fluid flows out in a slow regime with a constant velocity. Main attention is paid to studying the motion of fluid inside the vessel. In the reference system connected with the center of symmetry of the vessel, the problem is reduced to the interior Neumann problem for the Poisson equation with complex boundary conditions for the pressure and the velocity potential. The problem is solved with the help of elementary functions and a harmonic function that satisfies the standard Neumann equation for the Laplace equation. It is proved that the latter problem has a solution. It is found that the potential of the nonlinear velocity field and the pressure inside the vessel are described by harmonic functions, but these quantities have a singularity on the surface near the hole.