Study of Trapped Modes in Two and Three-Layer Fluids

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Trapped waves are of considerable interest in providing examples of discrete wave frequencies in the presence of a continuous spectrum. In this thesis, we ऒrst investigate the existence of trapped modes in a two-layer ओuid of ऒnite depth subject to diऍerent conditions: (i) upper surface bounded above by a rigid lid; (ii) upper surface bounded above by a thin ice-cover; (iii) ओuid ओowing over an elastic bottom at a ऒnite depth. In all these problems, a submerged horizontal circular cylinder is placed in either of the layers. The eऍect of surface tension at the surface of separation is neglected and each ओuid layer is considered to be immiscible. Furthermore, the assumptions of linear and time harmonic motions are followed. To solve the ice-cover problem, the standard idealization of ice as a thin elastic plate, which responds to only ओexural changes, is followed. In the elastic bottom problem, the ओexural bottom is considered as a thin elastic plate and is based on the Euler-Bernoulli beam equation. Later on, trapped mode frequencies are computed for a submerged horizontal circular cylinder with the hydrodynamic set-up involving an inऒnite depth three-layer incompressible ओuid with layer-wise diऍerent densities. The impermeable horizontal cylinder is fully immersed in either the bottom layer or the uppermost layer. In this problem we restrict the uppermost layer to be covered by a free surface only. In all these problems mentioned above, trapped mode frequencies are computed below a cut-oऍ value. In one of the later parts of the thesis, we compute trapped modes which are embedded in a continuous spectrum due to the presence of a pair of identical cylinders submerged in either layer of a two-layer ओuid which is covered by a thin ice-cover. In this case we assume the lower layer to be of inऒnite depth. Though numerical computation is carried out for a pair of cylinders only, we additionally provide the theoretical development for the case of a speciऒc arrangement of multiple cylinders. We have excluded the presence of surface tension at the free surface and interfaces for all the problems considered here. Its exclusion is justiऒed by presenting some numerical results in the last problem of the thesis. In this study of trapped waves, mixed boundary value problems are set up for the determi- nation of velocity potentials corresponding to each layer where the governing partial diऍerential equation happens to be modiऒed Helmholtz equation in two-dimensions for oblique incidence within the ओuid. The governing equation is accompanied by boundary conditions near the upper rigid boundary or the ice-cover surface or the free surface, at the interface between two ओuids and at the bottom boundary, if any, depending on the problem considered. The trapped mode condition arises which ensures that wave propagation to inऒnity does not take place at the interface(s) or at the upper surface. In order to examine the existence of trapped modes, multipole expansion method, along with the properties of an inऒnite system of linear equations, is used. A number of observations are made on the trapped modes with regard to diऍerent submergence depths and depths of all the layers. For the frequencies below a cut-oऍ value, there exist two modes (except for the rigid lid problem) for which trapped wave exists.
Supervisor: Swaroop Nandan Bora