  We report an experimental and theoretical study of the global stability and
nonlinear dynamics of vertical jets of viscous liquid confined in the axial
direction due to their impact on a bath of the same liquid. Previous works
demonstrated that in the absence of axial confinement the steady liquid thread
becomes unstable due to an axisymmetric global mode for values of the flow
rate, $Q$, below a certain critical value, $Q_c$, giving rise to oscillations
of increasing amplitude that finally lead to a dripping regime (Sauter &
Buggisch, J. Fluid Mech., 2005; Rubio-Rubio et al., J. Fluid Mech., 2013). Here
we focus on the effect of the jet length, $L$, on the transitions that take
place for decreasing values of $Q$. The linear stability analysis shows good
agreement with our experiments, revealing that $Q_c$ increases monotonically
with $L$, reaching the semi-infinite jet asymptote for large values of $L$.
Moreover, as $L$ decreases a quasi-static limit is reached, whereby $Q_c \to 0$
and the neutral conditions are given by a critical length determined by
hydrostatics. Our experiments have also revealed the existence of a new regime
intermediate between steady jetting and dripping, in which the thread reaches a
limit-cycle state without breakup. We thus show that there exist three possible
states depending on the values of the control parameters, namely steady
jetting, oscillatory jetting and dripping. For two different combinations of
liquid viscosity, and injector radius, $R$, the boundaries separating these
regimes have been determined in the $Q-L$ parameter plane, showing that steady
jetting exists for small enough values of $L$ or large enough values of $Q$,
dripping prevails for small enough values of $Q$ or sufficiently large values
of $L$, and oscillatory jetting takes place in an intermediate region whose
size increases with the liquid viscosity and decreases with $R$.
