Heat transfer in convective turbulence at large Prandtl number

At first we will consider the Boussinesq equations for Rayleigh-Bénard convection in the limit of infinite Prandtl number and recall some well known upper bounds on the vertical heat transport, i.e the Nusselt number Nu, in terms of the Rayleigh number Ra.

We will then introduce the Boussinesq system for Rayleigh-Bénard convection at large Prandtl number and we will prove a new upper bound for the Nusselt number of the form $Ra^{1/3}$ (modulo logarithmic correction) under the assumption that the the ratio of Prandtl number and Rayleigh number is greater than c_0, where c_0 is a non-dimensional constant depending on the aspect ratio of the domain only.

The method applied to derive the result is to view the Boussinesq system at large Prandtl number as a small perturbation of the infinite Prandtl number model for convection.

The seminar is based on the following papers:
"Infinite Prandtl number convection" by P. Constantin and C.Doering (1999),
"Bounds on vertical heat transport for infinite-Prandtl-number Rayleigh-Bénard convection" by C. Doering, F. Otto and M.G. Reznikoff (2006),
"Bound on vertical heat transport at large Prandtl number" by X. Wang (2007).