Active microrheology in the continuum limit: can the macrorheology be recovered?

Active microrheology differs from its passive counterpart in that the probe is actively forced through the material, rather than allowed to diffuse. Unlike in passive microrheology, active forcing allows the material to be driven out of equilibrium, and its nonlinear response to be probed. However, this also renders inoperable the fluctuation-dissipation theorem used to justify passive microrheology. Here we explore a question at the heart of active microrheology: are its results consistent with macrorheology? We study a simple model material -- a generalized Newtonian fluid, with a small but arbitrary shear-rate-dependent component -- and derive a general expression for dissipation due to probe motion, which remarkably does not require the non-Newtonian flow to be solved. We demonstrate that the straightforward application of active microrheology gives results that are inconsistent with macrorheology, even when the probe is large enough for material to behave as a continuum, unless the forcing is gentle enough to probe only the linear response. Regardless, each technique encodes information about the material; if suitably interpreted, the (macro-) constitutive relation can indeed be recovered from the microrheological data. We emphasize that more, rather than less, information would be obtained if the two methods disagree.