Convective flows driven by internal sources of heat are encountered in numerous fields, ranging from engineering to astrophysics, but their properties remain poorly understood. In particular, the mean vertical heat transport ‹wT› measured in experiments and numerical simulations displays nontrivial and confounding behaviour as the heating rate is varied, while the only rigorous scaling result available to date is the (nondimensional) uniform bound ‹wT› = ½. This talk will summarize an ongoing attempt to derive new upper bounds on ‹wT› that depend explicitly on the heating rate, measured by a suitably defined Rayleigh number R, for an idealized model of internally-heated convection. Using a modern interpretation of the classical "background method", the search for such upper bounds is posed as a convex variational problem. Numerical solution of this problem and suboptimal analytical constructions provide R-dependent improvements on the uniform bound ‹wT› = ½ for a finite range of Rayleigh numbers. However, they fail to do so asymptotically due to a violation of a "minimum principle", according to which physically realistic temperature fields in the variational problem should be nonnegative pointwise. When this constraint is imposed using a Lagrange multiplier, numerically optimized upper bounds on ‹wT› appear to asymptote to ½ from below, but confirming this analytically remains an open challenge. Obstacles to analytical constructions based on standard elementary estimates will be discussed. This work is joint with Ali Arslan, John Craske and Andrew Wynn.
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