Bounds on Rayleigh-Benard convection with an imposed heat flux

We formulate a bounding. principle for the heat transport in Rayleigh-Benard convection with fixed heat flux through the boundaries. The heat transport, as measured by a conventional Nusselt number, is inversely proportional to the temperature drop across the layer and is bounded above according to Nu less than or equal to c (R) over cap (1/3), where c < 0.42 is an absolute constant and (R) over cap = alphagbetah(4) /(nukappa) is the 'effective' Rayleigh number, the non-dimensional forcing scale set by the imposed heat flux kappabeta. The relation among the parameter (R) over cap, the Nusselt number, and the conventional Rayleigh number defined in terms of the temperature drop across the layer, is NuRa = (R) over cap, yielding the bound Nu less than or equal to c(3/2) Ra-1/2.