Labyrinthine instability of miscible magnetic fluids

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Abstract
The paper treats theoretically an inhomogeneous magnetic fluid (MF), modeling a miscible MF pair, in a Hele-Shaw cell subjected to a perpendicular magnetic field. As the existing experimental evidence indicates, a miscible form of the labyrinthine instability may occur in this system, with diffusion of magnetic particles playing the key role. Linear stability analysis is performed in the present paper: Analytically for a sharp interface and numerically for a diffused concentration distribution. For the sharp interface, assuming the Darcy law governs the flow, the neutral curves and the stability diagram are found along with the critical wavelength and the critical field intensity. Oscillatory and stationary instabilities are shown to substitute each other under certain conditions. For the diffused interface the viscous effects due to the flow nonuniformity in the plane of the cell are allowed for and found significant. Therefore, the conventional Darcy law that takes into account only the near-wall friction must be replaced by the Brinkman (Darcy–Stokes) equation. With the latter, the most unstable wavelength in strong fields tends to the limit of a few gap widths that quite weakly depends on the basic concentration gradient. A mechanism of the oscillatory instability is explained physically. Self-oscillations occur through the interplay between diffusion and advection driven via a magnetic body force by concentration inhomogeneity.

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