  We investigate the two-dimensional transport of heat through viscous flow
between two parallel rough interfaces with a given fractal geometry. The flow
and heat transport equations are solved through direct numerical simulations,
and for different conduction-convection conditions. Compared with the behavior
of a channel with smooth interfaces, the results for the rough channel at low
and moderate values of the Peclet number indicate that the effect of roughness
is almost negligible on the efficiency of the heat transport system. This is
explained here in terms of the Makarov's theorem, using the notion of active
zone in Laplacian transport. At sufficiently high Peclet numbers, where
convection becomes the dominant mechanism of heat transport, the role of the
interface roughness is to generally increase both the heat flux across the wall
as well as the active length of heat exchange, when compared with the smooth
channel. Finally, we show that this last behavior is closely related with the
presence of recirculation zones in the reentrant regions of the fractal
geometry.
