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 over-densities, irrespective of their origin. Our solutions describe the slow and steady radiative cooling-driven local gas inflow in the saturated regime of nonlinear thermal instability in clouds, sheets and filaments. 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 that deviations from steady state, 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 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.
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