Infinite-Prandtl-number convection. Part 2. A singular limit of upper bound theory

An upper bound on the heat flux for infinite-Prandtl-number convection between two parallel plates is determined for the cases of no-slip and free-slip boundary conditions. For no-slip the large-Rayleigh-number (Ra) scaling for the Nusselt number is consistent with Nu <c Ra-1/3, as predicted by Chan (1971). However, his commonly accepted picture of an infinite hierarchy of multiple boundary layer solutions smoothly approaching this scaling is incorrect. Instead, we find a novel terminating sequence in which the optimal asymptotic scaling is achieved with a three-boundary-layer solution. In the case of free-slip, we find an asymptotic scaling of Nu <c Ra-5/12 corroborating the conservative estimate obtained in Plasting & lerley (2005). Here the infinite hierarchy of multiple-boundary-layer solutions obtains, albeit with anomalous features not previously encountered. Thus for neither boundary condition does the optimal solution conform to the well-established models of finite-Prandtl-number convection (Busse 1969 b), plane Couette flow, and plane or circular Poiseuille flow (Busse 1970). We reconcile these findings with a suitable continuation from no-slip to free-slip, discovering that the key distinction - finite versus geometric saturation is entirely determined by the singularity, or not, of the initial, single-boundary-layer, solution. It is proposed that this selection principle applies to all upper bound problems.