A new method in nonlinear problems of waves in a heavy stratified liquid excited by a vertically moving body
Solving nonlinear problems of waves in a heavy stratified liquid excited by a vertically moving solid body is a very complex task, since the boundary conditions on the surface of the body and on the interfaces between the uniform fractions of the ideal incompressible fluid are to be taken into account. In the case of a so-called free surface, the pressure on it is constant, hence a so-called boundary integral equation method can be applied. When, however, the fluid boundary separates heavy uniform fractions of different density, this method is no longer applicable. Therefore, in the paper under consideration a system of integrodifferential equations is formulated with respect to the jump in the velocity component tangential to the interface. The equations of motion for axisymmetric inviscid incompressible flows are formulated using spherical coordinates and dimensionless variables. Especially a stream function and a vector potential, which is defined as a sum of integrals of the single-layer potential type for certain electrical charges, are introduced. The velocity of the solid body, which is not prescribed, in combination with the non-penetration condition of the fluid at the body surface form the main boundary condition. An initial condition is added. The resulting problem admits a formal solution in the form of series in integral powers with respect to the time. The example of a slowly rising sphere initially being at rest as well as the fluid itself is discussed in detail. The solution of the problem is sought in series form. A sequence of systems of linear integrodifferential equations is obtained. Finally, the problem is found to consist of a linear system of differential equations with slowly varying coefficients, the solution of which is a set of harmonics with slowly varying frequencies and amplitudes. The dependency of the occuring waves on the governing parameters is discussed in detail.
scientific article; zbMATH DE number 515023