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Cold, non-self-gravitating clumps occur in various astrophysical systems,ranging from the interstellar and circumgalactic medium (CGM), to AGN outflowsand solar coronal loops. Cold gas has diverse origins such as turbulent mixingor precipitation from hotter phases. We obtain the analytic solution for asteady pressure-driven 1-D cooling flow around cold, local over-densities,irrespective of their origin. Our solutions describe the slow and steadyradiative cooling-driven gas inflow in the saturated regime of nonlinearthermal instability in clouds, sheets and filaments. Such a cooling flowdevelops when the gas around small clumps undergoes radiative cooling. Thesesmall-scale, cold `seeds' are embedded in a large volume-filling hot CGMmaintained by feedback. We use a simple two-fluid treatment to include magneticfields as an additional polytropic fluid. To test the limits of applicabilityof these analytic solutions, we compare with the gas structure found in andaround small-scale cold clouds in the CGM of massive halos in the TNG50cosmological MHD simulation from the IllustrisTNG suite. Despite qualitativeresemblance of the gas structure, we find deviations from steady state profilesgenerated by our model. Complex geometries and turbulence all add complexitybeyond our analytic solutions. We derive an exact relation between the masscooling rate ($\dot{\rm M}_{\rm cool}$) and the radiative cooling rate($\dot{\rm E}_{\rm cool}$) for a steady cooling flow. A comparison with theTNG50 clouds shows that this cooling flow relation only applies in a narrowtemperature range around $\rm \sim 10^{4.5}$ K where the isobaric cooling timeis the shortest. In general, turbulence and mixing, instead of radiativecooling, may dominate the transition of gas between different temperaturephases.

Cooling flows around cold clouds in the circumgalactic medium: steady-state models & comparison with TNG50