Internal heating profiles for which downward conduction is impossible

We consider an internally heated fluid between parallel plates with fixed thermal fluxes. For a large class of heat sources that vary in the direction of gravity, we prove that $\langle\delta T \rangle_h \geq \sigma R^{-1/3} - \mu$, where $\langle\delta T \rangle_h$ is the average temperature difference between the bottom and top plates, $R$ is a `flux' Rayleigh number and the constants $\sigma,\mu >0$ depend on the geometric properties of the internal heating. This result implies that mean downward conduction (for which $\langle\delta T \rangle_h< 0$) is impossible for a range of Rayleigh numbers smaller than a critical value $R_0$. The bound demonstrates that $R_0$ depends on the heating distribution and can be made arbitrarily large by concentrating the heating near the bottom plate. However, for any given fixed heating profile of the class we consider, the corresponding value of $R_0$ is always finite. This points to a fundamental difference between internally heated convection and its limiting case of Rayleigh-B\'enard convection with fixed flux boundary conditions, for which $\langle\delta T \rangle_h$ is known to be positive for all $R$.