New bounds on the vertical heat transport for B\'enard-Marangoni convection at infinite Prandtl number

TL;DR

This paper establishes a new rigorous upper bound on the vertical heat transport in Bénard-Marangoni convection at infinite Prandtl number, revealing how heat transfer scales with the Marangoni number.

Contribution

It introduces a novel mathematical bound on heat transport for Bénard-Marangoni convection, using a background temperature profile and new coupling estimates.

Findings

Nu rac{Ma^{2/7}}{(\ln Ma)^{1/7}} asymptotic bound

Key role of hyperbolic temperature profile near surface

New estimates for temperature-velocity coupling

Abstract

We prove a new rigorous upper bound on the vertical heat transport for B\'enard-Marangoni convection of a two- or three-dimensional fluid layer with infinite Prandtl number. Precisely, for Marangoni number $Ma \gg 1$ the Nusselt number $Nu$ is bounded asymptotically by $Nu \lesssim Ma^{2/7}(\ln Ma)^{-1/7}$. Key to our proof are a background temperature field with a hyperbolic profile near the fluid's surface, and new estimates for the coupling between temperature and vertical velocity.