  In this paper, the interfacial motion between two immiscible viscous fluids
in the confined geometry of a Hele-Shaw cell is studied. We consider the
influence of a thin wetting film trailing behind the displaced fluid, which
dynamically affects the pressure drop at the fluid-fluid interface by
introducing a nonlinear dependence on the interfacial velocity. In this
framework, two cases of interest are analyzed: The injection-driven flow
(expanding evolution), and the lifting plate flow (shrinking evolution). In
particular, we investigate the possibility of controlling the development of
fingering instabilities in these two different Hele-Shaw setups when wetting
effects are taken into account. By employing linear stability theory, we find
the proper time-dependent injection rate $Q(t)$ and the time-dependent lifting
speed ${\dot b}(t)$ required to control the number of emerging fingers during
the expanding and shrinking evolution, respectively. Our results indicate that
the consideration of wetting leads to an increase in the magnitude of $Q(t)$
[and ${\dot b}(t)$] in comparison to the non-wetting strategy. Moreover, a
spectrally accurate boundary integral approach is utilized to examine the
validity and effectiveness of the controlling protocols at the fully nonlinear
regime of the dynamics and confirms that the proposed injection and lifting
schemes are feasible strategies to prescribe the morphologies of the resulting
patterns in the presence of the wetting film.
