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

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