Rigid bounds on heat transport by a fluid between slippery boundaries

• Published in: Journal of fluid mechanics 2012-09, Vol.707, p.241-259
• Main Authors: Whitehead, Jared P., Doering, Charles R.
• Format: Article
• Language: English
• Subjects: Boundaries; Buoyancy-driven instability; Convection; Convective and constrained heat transfer; Exact sciences and technology; Fluid dynamics; Fluid flow; Fluid mechanics; Fundamental areas of phenomenology (including applications); Heat; Heat transfer; Heat transport; Hydrodynamic stability; Physics; Rayleigh number; Three dimensional; Transport; Turbulence
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2012.274

Rigorous bounds on heat transport are derived for thermal convection between stress-free horizontal plates. For three-dimensional Rayleigh–Bénard convection at infinite Prandtl number ( $\mathit{Pr}$ ), the Nusselt number ( $\mathit{Nu}$ ) is bounded according to $\mathit{Nu}\leq 0. 28764{\mathit{Ra}}^{5/ 12} $ where $\mathit{Ra}$ is the standard Rayleigh number. For convection driven by a uniform steady internal heat source between isothermal boundaries, the spatially and temporally averaged (non-dimensional) temperature is bounded from below by $\langle T\rangle \geq 0. 6910{\mathit{R}}^{\ensuremath{-} 5/ 17} $ in three dimensions at infinite $\mathit{Pr}$ and by $\langle T\rangle \geq 0. 8473{\mathit{R}}^{\ensuremath{-} 5/ 17} $ in two dimensions at arbitrary $\mathit{Pr}$ , where $\mathit{R}$ is the heat Rayleigh number proportional to the injected flux.