Bounds on heat transfer for Bénard-Marangoni convection at infinite Prandtl number - Fantuzzi, Giovanni; Pershin, Anton; Wynn, Andrew (Journal of Fluid Mechanics, 2018)
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 as a function of the Marangoni number . 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 . In this work we extend previous background method analysis to include balance parameters and derive a variational principle for the bound on , expressed in terms of a scaled background field, that yields a better bound than Hagstrom & Doering's formulation at a given . Using a piecewise-linear, monotonically decreasing profile we then show that , lowering the previous prefactor by 4.2Â %. 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 over all admissible background fields, as well as over two smaller families of profiles constrained by monotonicity and convexity. The results show that when the background field has a non-monotonic boundary layer near the surface, while a power-law bound with exponent 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.
CRIS-Team Scopus Importer:2022-09-26
Keywords: Marangoni convection; turbulent convection; variational methods