Drop in Ferrofluids Subjected to an Azimuthal Field

We perform direct numerical simulations of a non-magnetic drop immersed in immiscible ferrofluids in a confined Hele-Shaw cell under an azimuthal field by a diffuse-interface method. The interface is unstable in such a condition because of the inward attraction of the ferrofluids induced by the magnetic field gradient. We focus on the fingering onset and pattern influenced by the coupling viscous effect with different viscously stable conditions, which is achieved by varying the viscosity contrast of the ferrofluids and non-magnetic drop. In a viscously stable condition, in which the viscosity of the ferrlfluids is greater than the immerse drop. The fingering pattern is simpler with numerous straightly developed fingers. On the other hand, a viscously unstable interface of less viscous ferrofluids results in ramified fingering pattern associated with the secondary phenomena, e.g., competitions and tip-splits of fingers. However, the fingering onset is delayed because the drop is less mobile. The interfacial instability of a circular non-magnetized drop (drop2) with radius R0 surrounded by immiscible ferrofulids (fluid 1) in a Hele-Shaw cell, as the principle sketch shown in Fig. 1. The Hele-Shaw cell has gap spacing h. An azimuthal field H is generated by wire with electric I placed at the center of the drop. The viscosity of ferrofluids and the drop are η1 and η2, respectively. We focus on the magneto-induced motion, but allow the inner drop (drop 2) to be either more or less viscous than other ferrofluids (fluids 1).