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In this paper, the interfacial motion between two immiscible viscous fluidsin the confined geometry of a Hele-Shaw cell is studied. We consider theinfluence of a thin wetting film trailing behind the displaced fluid, whichdynamically affects the pressure drop at the fluid-fluid interface byintroducing a nonlinear dependence on the interfacial velocity. In thisframework, two cases of interest are analyzed: The injection-driven flow(expanding evolution), and the lifting plate flow (shrinking evolution). Inparticular, we investigate the possibility of controlling the development offingering instabilities in these two different Hele-Shaw setups when wettingeffects are taken into account. By employing linear stability theory, we findthe proper time-dependent injection rate $Q(t)$ and the time-dependent liftingspeed ${\dot b}(t)$ required to control the number of emerging fingers duringthe expanding and shrinking evolution, respectively. Our results indicate thatthe 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, aspectrally accurate boundary integral approach is utilized to examine thevalidity and effectiveness of the controlling protocols at the fully nonlinearregime of the dynamics and confirms that the proposed injection and liftingschemes are feasible strategies to prescribe the morphologies of the resultingpatterns in the presence of the wetting film.

Controlling fingering instabilities in Hele-Shaw flows in the presence of wetting film effects