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

Denys Dutykh (1) , Olivier Goubet (2)

1 LAMA - Laboratoire de Mathématiques
2 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée

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 (2004) and D. Dutykh & F. Dias (2007)). 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 asymptotic expansions of the Dirichlet-to-Neumann (D2N) operator. Both employed methods yield the same system by different ways.

Mots clés: Boussinesq equations; viscosity; dissipation; dispersive waves; boundary layer

Domaines: Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]; Dynamique des Fluides [physics.flu-dyn]; Equations aux dérivées partielles [math.AP]

Denys Dutykh, Olivier Goubet. Derivation of dissipative Boussinesq equations using the Dirichlet-to-Neumann operator approach. 2011. ⟨hal-00596804v1⟩