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

Abstract

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 proven by Hagstrom & Doering that $Nu\le 0.838M{a}^{2/7}$. 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 $Nu\le 0.803M{a}^{2/7}$, 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 $Nu\le O(M{a}^{2/7}(\ln Ma{)}^{-1/2})$ 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.

Revised version: 33 pages, 9 figures. Extended discussion in Sections 6 and 7. Fixed mistakes in bibliography. Fixed typos. JFM style with patch for author-year references with hyperref

References in corpus (2)

• Optimal wall-to-wall transport by incompressible flows
• Exact energy stability of Bénard-Marangoni convection at infinite Prandtl number

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