Convection driven by a spatially non-uniform internal heat source between two horizontal isothermal walls is studied by theoret-ical 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 analyt-ical 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(qb)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.
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