Bounds on heat transfer for Bénard–Marangoni convection at infinite Prandtl number

The vertical heat transfer in Bénard–Marangoni convection of a fluid layer with infinite Prandtl number is studied by means of upper bounds on the Nusselt number $Nu$ as a function of the Marangoni number $Ma$. Using the background method for the temperature field, it has recently been proved by Hagstrom & Doering (Phys. Rev. E, vol. 81, 2010, art. 047301) that $Nu\leqslant 0.838Ma^{2/7}$. In this work we extend previous background method analysis to include balance parameters and derive a variational principle for the bound on $Nu$, expressed in terms of a scaled background field, that yields a better bound than Hagstrom & Doering’s formulation at a given $Ma$. Using a piecewise-linear, monotonically decreasing profile we then show that $Nu\leqslant 0.803Ma^{2/7}$, lowering the previous prefactor by 4.2 %. However, we also demonstrate that optimisation of the balance parameters does not affect the asymptotic scaling of the optimal bound achievable with Hagstrom & Doering’s original formulation. We subsequently utilise convex optimisation to optimise the bound on $Nu$ over all admissible background fields, as well as over two smaller families of profiles constrained by monotonicity and convexity. The results show that $Nu\leqslant O(Ma^{2/7}(\ln Ma)^{-1/2})$ when the background field has a non-monotonic boundary layer near the surface, while a power-law bound with exponent $2/7$ is optimal within the class of monotonic background fields. Further analysis of our upper-bounding principle reveals the role of non-monotonicity, and how it may be exploited in a rigorous mathematical argument.