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

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 Title: Bounds on heat transfer for Bénard-Marangoni convection at infinite Prandtl number Authors: Fantuzzi, GPershin, AWynn, A Item Type: Journal Article Abstract: The vertical heat transfer in B\'enard-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 $Nu\leq 0.838\,Ma^{2/7}$. 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 $Nu \leq 0.803\,Ma^{2/7}$, 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 $Nu \leq O(Ma^{2/7}(\ln Ma)^{-1/2})$ 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. Issue Date: 5-Jan-2018 Date of Acceptance: 20-Nov-2017 URI: http://hdl.handle.net/10044/1/54669 DOI: https://dx.doi.org/10.1017/jfm.2017.858 ISSN: 0022-1120 Publisher: Cambridge University Press (CUP) Start Page: 562 End Page: 596 Journal / Book Title: Journal of Fluid Mechanics Volume: 837 Copyright Statement: © 2018 Cambridge University Press This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. Sponsor/Funder: Engineering and Physical Sciences Research Council Funder's Grant Number: 1864077 Keywords: Science & TechnologyTechnologyPhysical SciencesMechanicsPhysics, Fluids & PlasmasPhysicsMarangoni convectionturbulent convectionvariational methodsENERGY-DISSIPATIONVARIATIONAL BOUNDSINCOMPRESSIBLE FLOWSCONIC OPTIMIZATIONSURFACE-TENSIONSHEAR-FLOWTRANSPORTSIMULATIONSBOUNDARIESSYSTEMSphysics.flu-dyn01 Mathematical Sciences09 EngineeringFluids & Plasmas Notes: 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 Publication Status: Published Appears in Collections: Faculty of EngineeringAeronautics