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A rotating particle in electrorheological (ER) fluid leads to a displacementof its polarization charges on the surface which relax towards the externalapplied field ${\bf E}_0$, resulting in a steady-state polarization at an anglewith respect to ${\bf E}_0$. This dynamic effect has shown to affect the ERfluids properties dramatically. In this paper, we develop a dynamic effectivemedium theory (EMT) for a system containing rotating particles of finite volumefraction. This is a generalization of established EMT to account for theinteractions between many rotating particles. While the theory is valid forthree dimensions, the results in a special two dimensional configuration showthat the system exhibits an off-diagonal polarization response, in addition toa diagonal polarization response, which resembles the classic Hall effect. Thediagonal response monotonically decreases with an increasing rotational speed,whereas the off-diagonal response exhibits a maximum at a reduced rotationalangular velocity $\omega_0$ comparing to the case of isolated rotatingparticles. This implies a way of measurement on the interacting relaxationtime. The dependencies of the diagonal and off-diagonal responses on variousfactors, such as $\omega_0$, the volume fraction, and the dielectric contrast,are discussed.

Dynamic polarizability of rotating particles in electrorheological fluids