  Scattering through natural porous formations (by far the most ubiquitous
example of disordered media) represents a formidable tool to identify effective
flow and transport properties. In particular, we are interested here in the
scattering of a passive scalar as determined by a steady velocity field which
is generated by a line of singularity. The velocity undergoes to erratic
spatial variations, and concurrently the evolution of the scattering is
conveniently described within a stochastic framework that regards the
conductivity of the hosting medium as a stationary, Gaussian, random field.
Unlike the similar one for uniform (in the mean) flow-fields, the problem at
stake results much more complex. Central for the present study is the
fluctuation of the driving field, that is computed in closed (analytical) form
as large time limit of the same quantity in the unsteady state flow regime. The
structure of the second-order moment X_{rr}, quantifying the scattering along
the radial direction, is explained by the rapid change of the distance along
which the velocities of two fluid particles become uncorrelated. Moreover, two
approximate, analytical expressions are shown to be quite accurate into
reproducing the full simulations of X_{rr}. Finally, the same problem is
encountered in other fields, belonging both to the classical and to the quantum
physics. As such, our results lend themselves to be used within a context much
wider than that exploited in the present study.
