Heat transport in fixed flux non-uniform internally heated convection

Rigorous scaling laws for turbulence driven by thermal convection can be obtained by use of the auxiliary functional method. By use of a quadratic auxiliary functional since the 1990s upper bounds on heat transport for Rayleigh-Benard convection have been known. If the boundary conditions are fixed flux bounds can be proven albeit with greater effort, however the scaling laws remain unchanged as compared to fixed temperature boundary conditions. This is not the case for internally heated convection. Further still for internal heating with fixed flux boundaries, the rigorous upper bounds cannot rule out conduction from the top to the bottom for all Rayleigh numbers. One idea for the proof of a "physical" result is to consider a non-uniformly heated plane layer. We can demonstrate that a class of non-uniform heating profiles exist, which decreases the unphysical bound on mean inverse conduction. This inverse mean conduction is zero when the non-uniform heating is localised entirely at the lower boundary.