Rayleigh-Bénard convection is the buoyancy-driven flow of a fluid heated from below and cooled from above and is a paradigm for nonlinear dynamics with important applications to meteorology, oceanography and engineering. We are interested in obtaining quantitative bounds on the Nusselt number, the vertical heat transport enhancement factor. The Nusselt number, besides being an interesting quantity for engineering applications, it is the natural quantity to measure the intensity and effectiveness of the motion. For this reason, we are interested in proving (upper) bounds which catch the relation between the Nusselt number and the (nondimensional control parameter) Rayleigh number, in turbulent regimes. Despite great scientific developments in this field in the last 30 years, it is still not clear what role the boundary conditions play in the scaling laws for the Nusselt number. In this talk we address this problem, establishing rigorous bounds for the Rayleigh-Bénard convection problem with Navier-slip boundary conditions for the velocity. We employ the background field method and deal with a careful PDE analysis, given the production of vorticity at the walls. In conclusion, we relate this result to other bounds derived for no-slip and stress-free boundary conditions and discuss open problems. This seminar is based on a joint work with T. Drivas and H. Nguyen.
- Séminaire Analyse non linéaire et EDP