Internally heated convection beneath a poor conductor

We consider convection in an internally heated (IH) layer of fluid that is bounded below by a perfect insulator and above by a poor conductor. The poorly conducting boundary is modelled by a fixed heat flux. Using solely analytical methods, we find linear and energy stability thresholds for the static state, and we construct a lower bound on the mean temperature that applies to all flows. The linear stability analysis yields a Rayleigh number above which the static state is linearly unstable (RL), and the energy analysis yields a Rayleigh number below which it is globally stable (RE). For various boundary conditions on the velocity, exact expressions for RL and RE are found using long-wavelength asymptotics. Each RE is strictly smaller than the corresponding RL but is within 1 %. The lower bound on the mean temperature is proven for no-slip velocity boundary conditions using the background method. The bound guarantees that the mean temperature of the fluid, relative to that of the top boundary, grows with the heating rate (H) no slower than H2/3.