Bounds on heat flux for Rayleigh-Bénard convection between Navier-slip fixed-temperature boundaries

Theodore D Drivas; Huy Q Nguyen; Camilla Nobili

doi:10.1098/rsta.2021.0025

Bounds on heat flux for Rayleigh-Bénard convection between Navier-slip fixed-temperature boundaries

Philos Trans A Math Phys Eng Sci. 2022 Jun 13;380(2225):20210025. doi: 10.1098/rsta.2021.0025. Epub 2022 Apr 25.

Authors

Theodore D Drivas¹, Huy Q Nguyen², Camilla Nobili³

Affiliations

¹ Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA.
² Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
³ Department of Mathematics, University of Hamburg, Hamburg, Germany.

PMID: 35465719
DOI: 10.1098/rsta.2021.0025

Abstract

We study two-dimensional Rayleigh-Bénard convection with Navier-slip, fixed temperature boundary conditions and establish bounds on the Nusselt number. As the slip-length varies with Rayleigh number [Formula: see text], this estimate interpolates between the Whitehead-Doering bound by [Formula: see text] for free-slip conditions (Whitehead & Doering. 2011 Ultimate state of two-dimensional Rayleigh-Bénard convection between free-slip fixed-temperature boundaries. Phys. Rev. Lett. 106, 244501) and the classical Doering-Constantin [Formula: see text] bound (Doering & Constantin. 1996 Variational bounds on energy dissipation in incompressible flows. III. Convection. Phys. Rev. E 53, 5957-5981). This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Keywords: Navier-slip boundary conditions; Nusselt number; Rayleigh–Bénard convection; background field method; scaling laws.