Mathematics Colloquia and Seminars

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We discuss the problem of formulating an upper bound principle for the heat transport in a convecting fluid with fixed heat flux through the layer. The heat transport is shown to be inversely proportional to the temperature drop across the plates, and is bounded above according to Nu< c R^(1/3), where c is an absolute constant and R a non-dimensional forcing scale. The relation between the parameter R and the Rayleigh number of the flow, Ra, is discussed and this relation is used to recast the bound in terms of Ra, yielding Nu< c Ra^(1/2). The `full' convection problem, consisting of a fluid bound by two plates with some thickness and finite conductivity, is also discussed.