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Time-dependent injection strategies are commonly employed to control thenumber of viscous fingers emerging at the interface separating two fluidsduring radial displacement in Hele-Shaw flows. Here we demonstratetheoretically that such a usual controlling method is significantly improved bytaking advantage of an electro-osmotic flow generated by applying an externalelectric field. More specifically, under the coupled action of time-varyingelectric currents and injection rates, we design a strategy capable ofcontrolling not only the number of fingers emerging at the interface but alsowhen (and if) the self-similar evolution occurs. In addition, the level ofinstability of the $n$-fold fingered patterns can also be tuned. This improvedcontrol over the interfacial features cannot be realized by the soleconsideration of a time-varying injection rate. Perturbative second-ordermode-coupling analysis and boundary integral simulations confirm that thevalidity and effectiveness of the controlling protocol go beyond the linearregime.

Electrically-controlled self-similar evolution of viscous fingering patterns in radial Hele-Shaw flows