Pacific Institute for the Mathematical Sciences - PIMS

We are interested in the transport of heat through a layer of viscous
fluid which is heated from below and cooled from above. Two mechanisms
are at work: Heat is transported by simple diffusion and by advection
through the flow. The transport by advection is triggered by buoyancy
(hotter parts have lower density) but is hindered by the no-slip
boundary condition for the fluid velocity at the bottom and top
surfaces.

Neglecting inertia, the equations contain a single dimensionless
parameter, the Rayleigh number $Ra$. It measures the relative strength
of advection with respect to diffusion. For $Ra gg 1$, the flow is
aperiodic and the heat transport is mediated by plumes. As a
consequence, the horizontally averaged temperature displays boundary
layers.

Inspired by the work of Constantin and Doering, we are interested in
rigorous bounds on the average heat transport (the Nusselt number $Nu$)
in terms of $Ra$. By PDE methods, Constantin and Doering prove
$Nustackrel{le}{sim} Ra^{1/3}log^{2/3}Ra$.

We use the conceptually intriguing method of the background
(temperature) field, introduced by Hopf for the Navier--Stokes equation
and used by Teman et. al. for the Kuramoto--Sivashinski equation. We
propose a background temperature field with non--monotone boundary
layers; direct numerical simulations show an average temperature field
with the same qualitative behavior. We obtain the slightly improved
bound $Nustackrel{le}{sim} Ra^{1/3}log^{1/3}Ra$. The crucial ingredient
is a maximal regularity statement for the Stokes operator in suitably
weighted $L^2$--spaces.

This is joint work with Charles Doering and Maria Reznikoff.

PIMS Distinguished Chair Lectures 2007