Laminar boundary layers in convective heat transport

TL;DR

This paper rigorously analyzes the temperature profile and boundary layer oscillations in high-Rayleigh and high-Prandtl number Rayleigh-Benard convection, revealing linear temperature profiles near the plates and introducing new mathematical estimates.

Contribution

It provides a rigorous mathematical investigation of boundary layer temperature profiles in convective heat transport, including new Hardy-type estimates for velocity control.

Findings

Temperature profile is essentially linear near the plates.

Boundary layer oscillations are characterized and bounded.

Results are uniform across system parameters with logarithmic corrections.

Abstract

We study Rayleigh-Benard convection in the high-Rayleigh-number and high-Prandtl-number regime, i.e., we consider a fluid in a container that is exposed to strong heating of the bottom and cooling of the top plate in the absence of inertia effects. While the dynamics in the bulk are characterized by a chaotic convective heat flow, the boundary layers at the horizontal container plates are essentially conducting and thus the fluid is motionless. Consequently, the average temperature exhibits a linear profile in the boundary layers. In this article, we rigorously investigate the average temperature and oscillations in the boundary layer via local bounds on the temperature field. Moreover, we deduce that the temperature profile is indeed essentially linear close to the horizontal container plates. Our results are uniform in the system parameters (e.g. the Rayleigh number) up to logarithmic correction terms. An important tool in our analysis is a new Hardy-type estimate for the convecting velocity field, which can be used to control the fluid motion in the layer. The bounds on the temperature field are derived with the help of local maximal regularity estimates for convection-diffusion equations.