On non-uniqueness in the 2D linear problem of a two-layer flow about interface-piercing bodies

Abstract

The two-dimensional Neumann-Kelvin problem describing the steady-state forward motion of a totally submerged tandem is considered in the case when the fluid consists of two superposed layers of different densities and bodies intersect the interface between them. For the so-called least singular solution, examples of non-uniqueness (trapped modes) are constructed using the inverse procedure. This procedure was previously applied by McIver (1996) to the problem of time-harmonic water waves and by Motygin (1997) and Kuznetsov & Motygin (1999) to the least singular and resistanceless statements of the Neumann-Kelvin problem involving a surface-piercing tandem in a homogeneous fluid. In the situation under consideration the inverse method involves investigation of stream lines generated by two vortices placed in the interface. The spacing of vortices delivering trapped modes depends on the forward velocity.