  This article studies the solutions of a two-dimensional grade-two fluid model
with a fully non-homogeneous boundary condition for velocity u. Compared to
problems with a homogeneous or tangential boundary condition, studied by many
authors , we must add a boundary condition, otherwise the problem is no longer
well-posed. We propose two conditions on z = curl (u -- $\alpha$$\Delta$u),
which differ according to the regularity of z, on the portion of
$\partial$$\Omega$ where $\alpha$u. n \textless{} 0. Following the approach of
V. Girault and L.R. Scott in the tangential boundary case, we split the problem
into a system with a generalized Stokes problem and a transport problem. But,
compared to the study of these authors, we are now led to solve transport
problems with boundary conditions. In two previous articles, we studied these
transport problems. The results obtained in these articles allow us, by a
fixed-point argument, to establish existence of the solutions for the fully
non-homogeneous grade-two problem. Uniqueness requires the boundary condition
with z in H 1 .
