Labyrinthine and secondary wave instabilities of a miscible magnetic fluid drop in a Hele-Shaw cell

The experimental conditions include different magnetic fields (by varying the maximum pre-set magnetic field strengths, (Formula presented.), and sweep rates, (Formula presented.), where (Formula presented.) is the instant magnetic field strength), gap spans, (Formula presented.), and magnetic fluid samples, and are further coupled into a modified Péclect number (Formula presented.) to evaluate the instabilities. Two distinct instabilities are induced by the external magnetic fields with different sweep rates: (i) a labyrinthine fingering instability, where small fingerings emerge around the initial circular interface in the early period, and (ii) secondary waves in the later period. Based on 81 sets of experimental conditions, the initial growth rate of the interfacial length, (Formula presented.), of the magnetic drop is found to increase linearly with (Formula presented.), indicating that (Formula presented.) is proportional to the square root of the (Formula presented.) and (Formula presented.) at the onset of the labyrinthine instability. In addition, secondary waves, which are characterised by the dimensionless wavelength (Formula presented.), can only be triggered when the three-dimensional magnetic microconvection is strong enough to make (Formula presented.) exceed a critical value, i.e. (Formula presented.), where (Formula presented.) is the wavelength of the secondary wave. In this flow regime of high (Formula presented.), the length scale of the secondary wave instability is found to be (Formula presented.), corresponding to the Stokes regime; meanwhile, in the flow regime of low (Formula presented.), the flow corresponds to the Hele-Shaw regime introduced by Fernandez et al. (J. Fluid Mech., vol. 451, 2002, pp. 239–260).