Internal heating driven convection at infinite Prandtl number

Jared P. Whitehead; Charles R. Doering

doi:10.1063/1.3637032

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arXiv · 1104.2792

Internal heating driven convection at infinite Prandtl number

Jared P. Whitehead·Charles R. Doering

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Abstract

We derive an improved rigorous bound on the space and time averaged temperature $ $ 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 $ \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.

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Jared P. Whitehead, Charles R. Doering. 2011-04-14. Internal heating driven convection at infinite Prandtl number. https://doi.org/10.1063/1.3637032

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