We simulate the evolution of cluster galaxies hot interstellar medium (ISM) gas due to ram pressure and thermal conduction in the intracluster medium (ICM). At the density and temperature of the ICM, the mean free paths of ICM electrons are comparable to the sizes of galaxies, therefore electrons can efficiently transport heat due to thermal conduction from the hot ICM to the cooler ISM. Galaxies consisting of dark matter halos and hot gas coronae are embedded in an ICM-like `wind tunnel in our simulations. In this paper, we assume that thermal conduction is isotropic and include the effects of saturation. We find that as heat is transferred from the ICM to the ISM, the cooler denser ISM expands and evaporates. This process is significantly faster than gas loss due to ram pressure stripping; for our standard model galaxy the evaporation time is $160$ Myr while the ram pressure stripping timescale is $2.5$ Gyr. Thermal conduction also suppresses the formation of shear instabilities, and there are no stripped ISM tails since the ISM evaporates before tails can form. Observations of long-lived X-ray emitting coronae and ram pressure stripped X-ray tails in galaxies in group and cluster environments therefore require that thermal conduction is suppressed or offset by some additional physical process. The most likely process is anisotropic thermal conduction due to magnetic fields in the ISM and ICM, which we simulate and study in the next paper in this series.
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