Hydrodynamics of fingering instabilities in dipolar fluids

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Abstract
Domains of magnetic and electric dipolar fluids are known to undergo fingering instabilities in Hele-Shaw flow, forming complex labyrinthine patterns. The hydrodynamics of this process are studied theoretically with a generalization of Darcy’s law. The boundary condition on the pressure at the interface between the dipolar fluid and that surrounding it is shown to include competing Young-Laplace and Biot-Savart terms. The spectrum of growth rates in the linear stability analysis of a circular domain has a complicated wave vector dependence as a consequence of the long-range forces, and reveals that the fingering arises from a negative effective surface tension. The free boundary problem for the interface motion is solved numerically using conformal mapping methods, and is compared with experiment. A simple model is proposed for mode competition and pattern selection under time-dependent magnetic fields. Aspects of this analysis may be relevant to the description of the intermediate state of type-I superconductors.

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