  We consider the motion of incompressible viscous non-homogeneous fluid
described by the Navier-Stokes equations in a bounded cylinder under boundary
slip conditions. Assume that the third co-ordinate axis is the axis of the
cylinder. Assuming that the derivatives of density, velocity, external force
with respect to the third co-ordinate are sufficiently small in some norms we
prove large time regular solutions without any restriction on the existence
time. The proof is divided into two parts. First an a priori estimate is shown.
Next the existence follows from the Leray-Schauder fixed point theorem.
