New variational bounds on convective transport. I. Formulation and analysis

We study the maximal rate of scalar transport between parallel walls separated by distance $h$, by an incompressible fluid with scalar diffusion coefficient $\kappa$. Given velocity vector field ${\bf u}$ with intensity measured by the P\'eclet number $Pe = h^2 \langle | \nabla {\bf u} |^2 \rangle^{1/2}/\kappa$ (where $\langle\cdot\rangle$ is space-time average) the challenge is to determine the largest enhancement of wall-to-wall scalar flux over purely diffusive transport, i.e., the Nusselt number $Nu$. Variational formulations of the problem are presented and it is determined that $Nu \le c Pe^{2/3}$, where $c$ is an absolute constant, as $Pe \rightarrow \infty$. Moreover, this scaling for optimal transport---possibly modulo logarithmic corrections---is asymptotically sharp: admissible steady flows with $Nu \ge c' Pe^{2/3}/[\log{Pe}]^2$ are constructed. The structure of (nearly) maximally transporting flow fields is discussed.