Effects of viscosity, surtace tension and wind on hydrodynamic waves for shallow water

Abstract

This thesis is devoted to the theoretical study of the effects of viscosity, surface tension and wind on hydrodynamic waves for shallow water. Various theories have been formulated for the study of weakly damped free-surface flows. These theories have essentially focused on forces relatively perpendicular to the fluid volume such as gravity forces, while neglecting forces relatively parallel to the fluid volume such as pressure forces due to wind and shear forces due to viscosity. In this work, some corrections due to viscosity are applied to the kinematics boundary condition at the surface and the dynamics condition modeled by Bernoulli’s equation. By using a linear approximation applied to the Navier-Stokes equation, we obtain a system of equations for the potential flow that includes the dissipative effect due to visosity for rightand left-moving waves. The pertubation theory applied to the Boussinesq system leads to some new higher-order generalized Korteweg De Vries (KdV) equations with nonlinear, dissipative and wind forcing terms. The wind effects are integrated into the model equations through the expression of atmospheric pressure proposed by the Miles model in which only the terms participating in the energy transfer (terms in phase quadrature with the surface elevation) are considered. In the absence of wind effects, these model equations describe the propagation of solitons in the viscous medium. We find the soliton solutions of each corresponding equation and then investigate on the effects of surface tension, viscosity and wind on the waves dynamics. The results show that such effects can strongly impact the group and phase velocities and the soliton dynamics. We can conclude that, these new equations obtained in this thesis, can be considered as improved versions of the KdV equation and can better describe the shallow water soliton dynamics. In addition, these equations can lead to several applications in various fields of nonlinear science.

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