The evolution and stability of interfaces which arise between two fluids of finite vertical extent, caused by the interaction of two different harmonics of the fundamental, are very important for a number of applications, for example, stabilizing stratified oil and water in pipe flow, the creation of rogue waves and the understanding of the role of dissipation. This however, requires an in-depth study of the waves which may be formed at the horizontal interface of two ideal fluids each of finite vertical extent. The forces acting on the system are those of gravity and surface tension. It is assumed that when the system is unexcited the interface is horizontal and the fluids have different velocities parallel to the interface. It is well-known that for certain distinguished values of the parameters resonance can occur in the sense that the linearised problem has a two dimensional solution space spanned by the Mth and Nth harmonics of the motion and most of the work will be concerned with this situation. The case when M=2, N=1 has already been solved by Jones (2010). However, this situation is entirely different to the other resonances because the interactions occur at the quadratic rather than at the cubic level. The first step in the analysis would be to solve the linear problem. Having accomplished this, we would then proceed to the nonlinear problem. This problem would be studied by employing the method of multiple scales to derive three or four coupled nonlinear partial differential equations which model, up to cubic order, the evolution of the interface. Solutions of this system would then be sought. Owing to the large number of parameters in the system, a great variety of wave profiles would be likely to be detected. If there is time, the final stage of the project would be to examine the stability of these interfaces against small plane wave perturbations.
There is no funding for this project
Jones, M.C.W. On 1:2 resonant waves which arise between two stratified fluids of finite vertical extent in the presence of mean flow effects; European Journal of Mechanics B/Fluid, 77 (2010) 395-402.
Jones M.C.W. Resonant interfacial capillary–gravity waves in the presence of damping effects; European Journal of Mechanics /B Fluids 68 (2018) 20-29.
Jones, M.C.W. & Hunter, G.J.A. Conservation laws for a nonlinear system describing the propagation of triadic waves; Proc. Inst. Acoustics 41 (2019) 137-145.
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