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 ($R_{L}$), and the energy analysis yields a Rayleigh number below which it is globally stable ($R_{E}$). For various boundary conditions on the velocity, exact expressions for $R_{L}$ and $R_{E}$ are found using long-wavelength asymptotics. Each $R_{E}$ is strictly smaller than the corresponding $R_{L}$ 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 $H^{2/3}$.
Long Wavelength Instability for Uniform Shear Flow
