2021/10/20 by Anuj Kumar, Ali Arslan, Giovanni Fantuzzi +2
Engineering · Mathematics · Physics and Astronomy · #Boundary (topology) #Computational Fluid Dynamics and Aerodynamics #Computer science #Constant (computer programming) #Constraint (computer-aided design) #Convection #Fluid Dynamics and Turbulent Flows #Gas Dynamics and Kinetic Theory #Geometry #Heat flux #Heat transfer #Inverse #Mathematical analysis #Mathematics #Mechanics #Physics #Scaling #Upper and lower bounds #math-ph #math.MP #physics.flu-dyn
paper · pdf · doi:10.1017/jfm.2022.170
20 pages, 3 figures
arxiv created 2021/10/20 · openalex publication_date 2022/03/17 · arxiv updated 2022/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We obtain an analytical bound on the mean vertical convective heat flux ⟨ w T ⟩ between two parallel boundaries driven by uniform internal heating. We consider two configurations, one with both boundaries held at the same constant temperature, and the other one with a top boundary held at constant temperature and a perfectly insulating bottom boundary. For the first configuration, Arslan et al. (J. Fluid Mech. 919:A15, 2021) recently provided numerical evidence that Rayleigh-number-dependent corrections to the only known rigorous bound ⟨ w T ⟩ ≤ 1/2 may be provable if the classical background method is augmented with a minimum principle stating that the fluid's temperature is no smaller than that of the top boundary. Here, we confirm this fact rigorously for both configurations by proving bounds on ⟨ wT ⟩ that approach 1/2 exponentially from below as the Rayleigh number is increased. The key to obtaining these bounds are inner boundary layers in the background fields with a particular inverse-power scaling, which can be controlled in the spectral constraint using Hardy and Rellich inequalities. These allow for qualitative improvements in the analysis not available to standard constructions.