  Cold, non-self-gravitating clumps occur in various astrophysical systems,
ranging from the interstellar and circumgalactic medium (CGM), to AGN outflows
and solar coronal loops. Cold gas has diverse origins such as turbulent mixing
or precipitation from hotter phases. We obtain the analytic solution for a
steady pressure-driven 1-D cooling flow around cold, local over-densities,
irrespective of their origin. Our solutions describe the slow and steady
radiative cooling-driven gas inflow in the saturated regime of nonlinear
thermal instability in clouds, sheets and filaments. Such a cooling flow
develops when the gas around small clumps undergoes radiative cooling. These
small-scale, cold `seeds' are embedded in a large volume-filling hot CGM
maintained by feedback. We use a simple two-fluid treatment to include magnetic
fields as an additional polytropic fluid. To test the limits of applicability
of these analytic solutions, we compare with the gas structure found in and
around small-scale cold clouds in the CGM of massive halos in the TNG50
cosmological MHD simulation from the IllustrisTNG suite. Despite qualitative
resemblance of the gas structure, we find deviations from steady state profiles
generated by our model. Complex geometries and turbulence all add complexity
beyond our analytic solutions. We derive an exact relation between the mass
cooling rate ($\dot{\rm M}_{\rm cool}$) and the radiative cooling rate
($\dot{\rm E}_{\rm cool}$) for a steady cooling flow. A comparison with the
TNG50 clouds shows that this cooling flow relation only applies in a narrow
temperature range around $\rm \sim 10^{4.5}$ K where the isobaric cooling time
is the shortest. In general, turbulence and mixing, instead of radiative
cooling, may dominate the transition of gas between different temperature
phases.
