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, G
Pershin, A
Wynn, 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
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 (, 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 & Technology
Physical Sciences
Physics, Fluids & Plasmas
Marangoni convection
turbulent convection
variational methods
01 Mathematical Sciences
09 Engineering
Fluids & 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 Engineering

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