  We provide a integration of Navier-Stokes equations concerning the
unsteady-state laminar flow of an incompressible, isothermal (newtonian) fluid
in a cylindrical vessel spinning about its symmetry axis, say $z$, and inside
which the liquid velocity starts with a non-zero axial component as well. Basic
physical assumptions are that the pressure axial gradient keeps itself on its
hydrostatic value and that no radial velocity exists. In such a way the PDEs
become uncoupled and can be faced separately from each other. We succeed in
computing both the unsteady velocities, i.e. the axial $v_z$ and the
circumferential $v_\theta$ as well, by means of infinite series expansions of
Fourier-Bessel type under time exponential damping. Following this, we also
find the unsteady surfaces of dynamical equilibrium, the wall shear stress and
the Stokesian streamlines
