Comprehensive Perturbation Approach to Nonlinear Viscous Gravity–Capillary Surface Waves at Arbitrary Wavelengths in Finite Depth

This study presents a comprehensive analysis of the second-order perturbation theory applied to Navier–Stokes equations governing free surface flows. We focus on gravity–capillary waves in incompressible viscous fluids finite depth over flat bottom. The amplitude these is regarded as parameter. A systematic derivation nonlinear-surface-wave equation presented that fully takes into account dispersion, while nonlinearity included leading order. However, presence infinitely many over-damped modes has been neglected and only two least-damped are considered. new surface-wave formulated wave-number space rather than real nonlinear terms contain convolutions making an integro-differential equation. Some preliminary numerical results compared with computational-modelling data obtained via open source CFD software OpenFOAM.