We study a ferrofluid in a horizontal Hele-Shaw geometry subjected to a vertical magnetic field. Specifically, we calculate the energy of a single ferrofluid finger using an idealized model for the finger. By minimizing this energy, we find the preferred finger width as a function of the applied field. Our model predicts a first order transition as the fluid abruptly transforms from a circular drop to a finite finger. This behavior arises because of a double energy minimum that yields two different stable configurations for the system. Interestingly, this system exhibits hysteresis as the circle-to-finger (increasing field) transition occurs at a different applied field than the finger-to-circle (decreasing field) transition. We carry out a simple experiment and observe good overall agreement with the theoretical predictions.

Hillier, Narelle J., and David P. Jackson. Width of a Ferrofluid Finger: Hysteresis and a Double Energy Minimum. Physical Review E 75, no. 3 (2007): e036314. https://journals.aps.org/pre/pdf/10.1103/PhysRevE.75.036314

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