**Title:** Heat transport bounds for Rayleigh-Benard convection in an infinite

**Speaker:** Professor Charlie Doering

**Speaker Info:** University of Michigan

**Brief Description:**

**Special Note**: **This is a joint analysis/probability and PDE seminar**

**Abstract:**

Heat transport bounds for Rayleigh-B\'enard convection in an infinite Prandtl number fluidCharlie Doering Department of Mathematics and Michigan Center for Theoretical Physics University of Michigan, Ann Arbor, MI 48109-1043 USA

Rayleigh-B\'enard convection, the buoyancy-driven flow resulting from a fluid being heated from below and cooled from above, is an important physical process for many areas of engineering and the applied sciences. In recent decades it has also come to serve as one of the fundamental paradigms of complex nonlinear dynamics and pattern formation in continuum mechanics. A key feature of convection flow is the enhancement of heat transport over simple condution. The enhancement factor is the Nusselt number $Nu$. A central challenge for mathematical analysis is to rigorously estimate $Nu$ as a function of the applied temperature gradient (measured in terms of the nondimensional Rayleigh number $Ra$) directly from the equations of motion --- even when the flow is turbulent.

In this work we show that for the infinite Prandtl number limit of the Boussinesq equations of motion, the Nusselt number $Nu$ is bounded from above in terms of the Rayleigh number $Ra$ according to $Nu \le .644 \times Ra^{\frac{1}{3}} [\log{Ra}]^{\frac{1}{3}}$ as $Ra \rightarrow \infty$. This result follows from the utilization of a novel logarithmic profile in the background method for producing bounds on bulk transport together with new estimates for the bi-Laplacian in a weighted $L^{2}$ space. It is a quantitative improvement over the best previously available analytic result, and it comes within the logarithmic factor of the pure $1/3$ scaling anticipated by both the classical marginally stable boundary layer argument and the most recent high-resolution numerical computations of the optimal bound on $Nu$ using the background method.

This is joint work with Felix Otto (Bonn) and Maria Reznikoff (Bonn/Princeton/GaTech) and is the content of a paper in press in Journal of Fluid Mechanics.

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