Internal heating driven convection at infinite Prandtl number

TL;DR

This paper derives a new rigorous lower bound on the average temperature in an infinite Prandtl number fluid with internal heating, using a novel stratification approach and Hardy-Rellich inequality.

Contribution

It introduces a novel method involving singular stable stratification and a generalized Hardy-Rellich inequality to improve bounds on temperature in internally heated convection.

Findings

Established a lower bound: <T> ≥ 0.419[R log(R)]^{-1/4}

Developed a new perturbation approach with stable stratification

Proved a generalized Hardy-Rellich inequality in the appendix

Abstract

We derive an improved rigorous bound on the space and time averaged temperature $<T>$ of an infinite Prandtl number Boussinesq fluid contained between isothermal no-slip boundaries thermally driven by uniform internal heating. A novel approach is used wherein a singular stable stratification is introduced as a perturbation to a non-singular background profile, yielding the estimate $<T>\geq 0.419[R\log(R)]^{-1/4}$ where $R$ is the heat Rayleigh number. The analysis relies on a generalized Hardy-Rellich inequality that is proved in the appendix.