Bounds on heat transfer for Bénard-Marangoni convection at infinite Prandtl number
Author(s)
Fantuzzi, G
Pershin, A
Wynn, A
Type
Journal Article
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 proved by Hagstrom & Doering (Phys. Rev. E, vol. 81, 2010, art. 047301) that 𝑁𝑢⩽0.838𝑀𝑎2/7 . 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⩽0.803𝑀𝑎2/7 , 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⩽𝑂(𝑀𝑎2/7(ln𝑀𝑎)−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.
Date Issued
2018-02-25
Date Acceptance
2017-11-21
Citation
Journal of Fluid Mechanics, 2018, 837 (1), pp.562-596
ISSN
0022-1120
Publisher
Cambridge University Press
Start Page
562
End Page
596
Journal / Book Title
Journal of Fluid Mechanics
Volume
837
Issue
1
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.
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.
License URL
Sponsor
Engineering and Physical Sciences Research Council
Engineering & Physical Science Research Council (EPSRC)
Identifier
https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/bounds-on-heat-transfer-for-benardmarangoni-convection-at-infinite-prandtl-number/DCD362E227D7562848C042EB7AAC1F7E
Grant Number
1864077
EP/K503381/1
Subjects
physics.flu-dyn
physics.flu-dyn
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
Date Publish Online
2018-01-05