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