  In view of recent microrheology experiments we re-examine the problem of a
rigid sphere oscillating inside a dilute polymer network. The network and its
solvent are treated using the two-fluid model. We show that the dynamics of the
medium can be decomposed into two independent incompressible flows. The first,
dominant at large distances and obeying the Stokes equation, corresponds to the
collective flow of the two components as a whole. The other, governing the
dynamics over an intermediate range of distances and following the Brinkman
equation, describes the flow of the network and solvent relative to one
another. The crossover between these two regions occurs at a dynamic length
scale which is much larger than the network's mesh size. The analysis focuses
on the spatial structure of the medium's response and the role played by the
dynamic crossover length. We examine different boundary conditions at the
sphere surface. The large-distance collective flow is shown to be independent
of boundary conditions and network compressibility, establishing the robustness
of two-point microrheology at large separations. The boundary conditions that
fit the experimental results for inert spheres in entangled F-actin networks
are those of a free network, which does not interact directly with the sphere.
Closed-form expressions and scaling relations are derived, allowing for the
extraction of material parameters from a combination of one- and two-point
microrheology. We discuss a basic deficiency of the two-fluid model and a way
to bypass it when analyzing microrheological data.
