In this article, we derive a viscous Boussinesq system for surface water waves from Navier–Stokes equations for non-vanishing initial conditions. The relevance of such initial conditions is proved. We use neither the irrotationality assumption, nor the Zakharov–Craig–Sulem formulation. During the derivation, we find the bottom shear stress and also the decay rate for shallow water. In order to justify our derivation, we derive the viscous Korteweg–de Vries equation from our viscous Boussinesq system and compare it to the ones found in the bibliography. We also extend the system to the 3D geometry.
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