Mathematical and computational modeling of nonlinear waves in inviscid fluids

This research proposal focuses on two thematic axes: (1) the study of aquatic waves and (2) waves in electrohydrodynamic (EHD) flows. Regarding axis (1), we investigate waves propagating over a spatially or temporally varying seabed, focusing on nonlinear interactions between the free surface and seabed dynamics. Specifically, we examine wave formation and propagation due to seabed displacements, with direct applications to tsunami generation. Additionally, we analyze fluid properties, including pressure distribution and particle trajectories beneath these waves, considering cases with and without underlying currents. For axis (2), we aim to explore the effect of electric fields on the velocity field of fluids in EHD flows, where the fluid movement is coupled with an electric field. This topic has significant industrial applications, as electric fields are used to control fluid motion. The investigation will focus on the manifestations of nonlinearities in particle trajectories and the propagation of waves in channels with variable topography. The general problem addressed in this proposal is the understanding and modeling of nonlinear phenomena in hydrodynamics and electrohydrodynamics. The modeling will be carried out using nonlinear Euler equations, formulated as free boundary problems for harmonic functions in planar domains, where the free boundary is determined as the solution to a system of nonlinear partial differential equations. Our hypothesis is that combining conformal mapping with spectral numerical methods will effectively resolve the research problems. Evidence supporting this hypothesis is found in existing literature. These methods will allow for the extraction of detailed information about surface waves and a comprehensive understanding of the velocity field in the fluid.