Magnetic droplets obtained by induced phase separation in a magnetic colloid show a large variety of shapes when exposed to an external field. However, the description of the shapes is often limited. Here, we formulate an algorithm based on three-dimensional boundary-integral equations for strongly magnetic droplets in a high-frequency rotating magnetic field, allowing us to find their figures of equilibrium in three dimensions. The algorithm is justified by a series of comparisons with known analytical results. We compare the calculated equilibrium shapes with experimental observations and find a good agreement. The main features of these observations are the oblate–prolate transition, the flattening of prolate shapes with the increase of magnetic field strength and the formation of starfish-like equilibrium shapes. We show both numerically and in experiments that the magnetic droplet behaviour may be described with a triaxial ellipsoid approximation. Directions for further research are mentioned, including the dipolar interaction contribution to the surface tension of the magnetic droplets, accounting for the large viscosity contrast between the magnetic droplet and the surrounding fluid.
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