Temperature and heat flux bounds of convection driven by non-uniform internal heating

Convection driven by a spatially non-uniform internal heat source between two horizontal isothermal walls is studied by theoretical analysis and numerical simulation, in order to explore the bounds of the temperature and the vertical heat flux. Specifically, the rigorous lower bound of the weighted average temperature ⟨QT⟩ is derived analytically, by decomposing the temperature field into a background profile and a fluctuation part. This bound obtained for the first time to consider non-uniform heat sources is found to be compatible with the existing bound obtained in uniform internal heat convection. Of physical importance, an analytical relationship is derived as an inequality connecting ⟨QT⟩ and the average vertical heat flux ⟨wT⟩, by employing the average heat flux on the bottom wall (q_b) as an intermediary variable. It clarifies the intrinsic relation between the lower bound of ⟨QT⟩ and the upper bound of ⟨wT⟩, namely, these two bounds are essentially equivalent providing an easy way to obtain one from another. Furthermore, the analytical bounds are extensively demonstrated through a comprehensive series of direct numerical simulations.