Oleg Motygin

On the two-dimensional sloshing problem

We study an eigenvalue problem with a spectral parameter in a boundary condition. This problem for the two–dimensional Laplace equation is relevant to sloshing frequencies that describe free oscillations of an inviscid, incompressible, heavy fluid in a canal having uniform cross–section and bounded from above by a horizontal free surface. It is demonstrated that there exist domains such that at least one of the eigenfunctions has a nodal line or lines with both ends on the free surface (earlier, Kuttler tried to prove that there are no such nodal lines for all domains but his proof is erroneous). It is also shown that the fundamental eigenvalue is simple, and for the corresponding eigenfunction the behaviour of the nodal line is characterized. For this purpose, a new variational principle is proposed for an equivalent statement of the sloshing problem in terms of the conjugate stream function.

The wave resistance of a two-dimensional body moving forward in a two-layer fluid

A two-dimensional body moves forward with constant velocity in an inviscid, incompressible fluid under gravity. The fluid consists of two layers having different densities, and the body is totally submerged in one of them. The resulting fluid motion is assumed to be steady state in a coordinate system attached to the body. The boundary-value problem for the velocity potential is considered in the framework of linearized water-wave theory. The asymptotics of the solution at infinity is obtained with the help of an integral representation, based on the explicitly known Green function. The theorem of unique solvability is formulated, and the method applied to prove it is briefly explained (the detailed proof is given in another work). An explicit formula for the wave resistance is derived and discussed. A numerical example for the wave resistance serves to illustrate the so-called “dead-water” phenomenon.

Sloshing problem in a half-plane covered by a dock with two gaps: monotonicity and asymptotics of eigenvalues

Abstract The two-dimensional sloshing problem is considered in a half-plane covered by a rigid dock with two symmetric gaps. It is proved that the antisymmetric (symmetric) sloshing eigenvalues are monotonically decreasing (increasing) functions of spacing between gaps and formulae for their derivatives are obtained.

On numerical evaluation of the Heun functions

In the paper we deal with the Heun functions - solutions of the Heun equation. Despite the increasing interest to the equation and numerous applications of the functions in a wide variety of physical problems, it is only Maple amidst known software packages which is able to evaluate the functions numerically. But the Maple routine is known to be imperfect. The purpose of the work is to develop alternative algorithms for numerical evaluation of the Heun functions. A procedure based on power series expansions and analytic continuation is suggested.

The three-dimensional problem of the coupled time-harmonic motion of a freely floating body and water covered by brash ice

A mechanical system consisting of water covered by brash ice and a body freely floating near equilibrium is considered. The water occupies a half-space into which a surface-piercing body is immersed, thus allowing us to study the coupled motion which is assumed to be of small amplitude. The corresponding linear setting for time-harmonic oscillations reduces to a spectral problem whose parameter is the frequency. A constant that characterises the brash ice divides the set of frequencies into two subsets and the results obtained for each of these subsets are essentially different. For every frequency from a finite interval adjacent to zero, a family of motionless axisymmetric bodies trapping waves is constructed by virtue of the semi-inverse procedure. For sufficiently large frequencies outside of this interval, all solutions of finite energy are trivial.

The Problem of Steady Flow over a Two-Dimensional Bottom Obstacle

The linear boundary value problem describing a steady flow over a two–dimensional obstacle (bottom protrusion) is considered. This is a mixed problem for a harmonic function in an indented strip of constant width at infinity, where asymmetric conditions are imposed on the gradient. Under rather general assumptions on the obstacle, the existence of a unique solution is proved for all values of the nonnegative parameter (the reciprocal of the Froude number squared) of the problem, except possibly for a sequence of values that tends from above to the critical value.

Eigenvalues of the Steklov Problem in an Infinite Cylinder

The Steklov problem is considered in cylindrical domains; the coefficient in the boundary condition has a compact support and is an even function of a coordinate varying along the generators. We study the dependence of eigenvalues on the spacing between two symmetric parts of the coefficient’s support. It is proved that the antisymmetric (symmetric) eigenvalues are monotonically decreasing (increasing) functions of the spacing and formulae for their derivatives are obtained. Application to the sloshing problem in a channel covered by a dock with two equal rectangular gaps is given.