Derivation of dissipative Boussinesq equations using the Dirichlet-to-Neumann operator approach

The water wave theory traditionally assumes the fluid to be perfect, thus neglecting all effects of the viscosity. However, the explanation of several experimental data sets requires the explicit inclusion of dissipative effects. In order to meet these practical problems, the theory of visco-potential flows has been developed (see P.-F. Liu & A. Orfila [42] and D. Dutykh & F. Dias [25]). Then, usually this formulation is further simplified by developing the potential in an entire series in the vertical coordinate and by introducing thus, the long wave approximation. In the present study we propose a derivation of dissipative Boussinesq equations which is based on two various asymptotic expansions of the Dirichlet-to-Neumann (D2N) operator. Both employed methods yield the same system at the leading order by different ways.