The dynamics of a confined polymer melt between weakly adsorbing surfa
ces is considered theoretically. The finite chain extensibility is tak
en into account explicitly, and the tangential stress and the first an
d the second normal-stress differences are calculated as functions of
shear rate gamma. For small shear velocities (u < u*) the surface sli
p is large, and the apparent viscosity, eta(app), is proportional to t
he layer thickness h and is independent of the shear rate. For very hi
gh shear rates, the surface slip is small and the tangential stress in
creases with velocity to the power 1/3. Alternatively, the apparent vi
scosity eta decreases as a function of gamma with a characteristic -2/
3 power law.