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Control of instability by injection rate oscillations in a radial Hele-Shaw cell

Small spatial perturbations grow into ﬁngers along the unstable interface of a ﬂuid displacing a more viscous ﬂuid in a porous medium or a Hele-Shaw cell. Mitigating this Saffman-Taylor instability increases the efﬁciency of ﬂuid displacement applications (e.g., oil recovery), whereas amplifying these perturbations is desirable in, e.g., mixing applications. In this work, we investigate the Saffman-Taylor instability through analysis and experiments in which air injected with an oscillatory ﬂow rate outwardly displaces silicone oil in a radial Hele-Shaw cell. A solution for linear instability growth that shows the competing effects of radial growth and surface tension, including wetting effects, is deﬁned given an arbitrary reference condition. We use this solution to deﬁne a condition for stability relative to the constant ﬂow rate case and make initial numerical predictions of instability growth by wave number for a variety of oscillations. These solutions are then modiﬁed by incorporating reference conditions from experimental data. The morphological evolution of the interface is tracked as the air bubble expands and displaces oil between the plates. Using the resulting images, we analyze and compare the linear growth of perturbations about the mean interfacial radius for constant injection rates with and without superimposed oscillations. Three distinct types of ﬂow rate oscillations are found to modulate experimental linear growth over a constant phase-averaged rate of ﬂuid displacement. In particular, instability growth at the interface is mildly mitigated by adding to the base ﬂow rate provided by a peristaltic pump a second ﬂow with low-frequency oscillations of small magnitude and, to a lesser extent, high-frequency oscillations of large amplitude. In both cases, the increased stability results from the selective suppression of the growth of large wave numbers in the linear regime. Contrarily, intermediate oscillations consistently destabilize the interface and signiﬁcantly amplify the growth of the most unstable wave numbers of the constant ﬂow rate case. Numerical predictions of low-frequency oscillations of opposite sign (initially decreasing) show promise of even greater mitigation of linear instability growth than that observed in this investigation. Looking forward, proper characterization of the unsteady, wetting, and nonlinear dynamics of instability growth will give further insight into the efﬁcacy of oscillatory injection rates.