  We propose a two-dimensional flow model of a viscous fluid between two close
moving surfaces. We show that its asymptotic behavior, when the distance
between the two surfaces tends to zero, is the same as that of the the
Navier-Stokes equations. The solutions of the new model and Navier-Stokes
equations converge to the same limit problem, that depends on the boundary
conditions. If slip velocity boundary conditions are imposed on the upper and
lower bound surfaces, the limit is solution of a lubrication model, but if the
tractions and friction forces are known on both bound surfaces, the limit is
solution of a shallow water model. The model proposed has been obtained to be a
valuable tool for computing viscous fluid flow between two close moving
surfaces, without the need to decide a priori whether the flow is typical of a
lubrication problem or it is of shallow water type, and without the enormous
computational effort that would be required to solve the Navier-Stokes
equations in such a thin domain.
