Bounds for Rayleigh-Benard convection between free-slip boundaries with an imposed heat flux

We prove the first rigorous bound on the heat transfer for three-dimensional RayleighBenard convection of finite-Prandtl-number fluids between free-slip boundaries with an imposed heat flux. Using the auxiliary functional method with a quadratic functional, which is equivalent to the background method, we prove that the Nusselt number Nu is bounded by Nu <= 0.5999R(1/3) uniformly in the Prandtl number, where R is the Rayleigh number based on the imposed heat flux. In terms of the Rayleigh number based on the mean vertical temperature drop, Ra, we obtain Nu <= 0.4646Ra(1/2). The scaling with Rayleigh number is the same as that of bounds obtained with no-slip isothermal, free-slip isothermal and no-slip fixed-flux boundaries, and numerical optimisation of the bound suggests that it cannot be improved within our bounding framework. Contrary to the two-dimensional case, therefore, the Ra-dependence of rigorous upper bounds on the heat transfer obtained with the background method for three-dimensional Rayleigh-Benard convection is insensitive to both the thermal and the velocity boundary conditions.