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In this paper We consider the Coupling Navier-Stokes/Darcy equations in a two or three dimensional domain provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We establish a coupled variational formulation of this problem with three independent unknowns: the vorticity, the velocity and the pressure. We discuss coupling conditions and we analyze the global coupled model in order to prove its well-posedness and to characterize effective algorithms to compute the solution of its numerical approximation.

This paper is devoted to study the Navier-Stokes equations by applying the curl and using a current function, we obtain a non-linear biharmonic problem where the pressure disappears and instead of the velocity, we are working with a scalar function. After a linearization, we obtain a sequence of linear problems. We study the existence and uniqueness of its solutions. Finally we show the convergence of the sequence of the linearized problems obtained to the non-linear one.