2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/176976I use a simple analytical model to show that simple heat conduction models cannot significantly suppress cluster cooling flows. I build a static medium where heat conduction globally balances radiative cooling, and then perturb it. I show that a perturbation extending over a large fraction of the cooling flow region will grow to the non-linear regime within a Hubble time. Such perturbations are reasonable in clusters which frequently experience mergers and/or AGN activity. This result strengthens previous findings which show that a steady solution does not exist for a constant heat conduction coefficient.MNRAS, in pressAstrophysicstext