Summary: We present a general, accurate, rigorous upper bound for the time required for the temperature distribution in a solid body suddenly immersed in a flowing fluid stream to reach the steady state. The bound provides a uniform-in-Biot-number estimate that should prove of interest to the heat transfer student and heat transfer practitioner alike: to the former as a succinct summary of the relevant physical phenomena; to the latter as a rough predictive preface to, verification of, and correlation framework for, more precise numerical calculation or experiment.