Problems of statics, stability, and small oscillations of an ideal incompressible fluid in a partially filled container with holes in the bottom.
The paper deals with the problems on statics, stability, eigenoscillations and small movements of an ideal incompressible fluid in a vessel with bottom holes.
A rectangular channel (plane problem) and cylindrical container (axisymmetric problem) are considered. It is assumed that hydrosystem is under low gravity conditions, and therefore the action of surface tension forces and weak gravitational forces are considered.
Under these assumptions, the free surface of a fluid is disconnected and consists of an upper part (located inside vessel) and surfaces hanging drops held by capillary forces. The cases of a horizontal upper part of the free surface, as well as an option when it is curved, are studied.
In the investigation the methods of linear operators, acting in Hilbert spaces, as well as variations and operator approaches are used.
In static problem the boundary value problems for the system of second order differential equations with an additional integral condition are obtained and the algorithm of the numerical solution is proposed.
In the problem on stability of the hydrosystem equilibrium state the statements of the static stability of the equilibrium state, based on the sign of the minimum eigenvalue of associated spectral problem are proved. By using of operator approach the problem is transformed to the spectral problem in a Hilbert space and the properties of its operator matrices are studied. It is proved that this problem has a discrete spectrum consisting of two branches of eigenvalues, with a limit point at $+\infty$ and the boundary of the stability of the hydrosystem is found.
In the eigenoscillations problem by using of auxiliary boundary value problems a corresponding spectral problem is studied; the theorem on the properties of the spectrum and statement on the dynamic stability and instability are proved.
In this paper an initial-boundary value problem for small movements of hydrosystem are formulated. It is proved that if the initial data of the problem satisfy certain smoothness conditions, then unique strong solution of this problem exists, as well as solution of associated Cauchy problem for differential operator equation in corresponding Hilbert space.
Keywords: ideal incompressible fluid, low gravity, equilibrium state, oscillations, operator approach, Hilbert space, initial boundary value problem, spectral problem, solvability, strong solution, instability.