On the lower bound of net driving power in controlled duct flows

We examine mathematically the lower bound of the net driving power (i.e., the summation of pumping and actuation powers) of a controlled duct flow under a constant flow rate. The net power in a duct with arbitrary cross-section in the presence of the inertial term, blowing/suction from the wall, and arbitrary body forces can be decomposed into four terms: (1) dissipation due to the velocity profile of the Stokes flow (in other words, pumping power for the Stokes flow); (2) dissipation due to deviation of the mean velocity from the Stokes flow profile; (3) dissipation due to velocity fluctuations; and (4) correlation between the wall-pressure of the Stokes flow and the time-averaged blowing/suction velocity. Of these, the first three terms are shown to be non-negative, while the sign of the fourth term is indefinite. The fourth term vanishes in cases where the duct has a constant-shape cross-section, such as cylindrical pipes and plane channels. Namely, in such cases, the lower bound of net power is exactly given by the dissipation rate (pumping power) of the Stokes flow at the same flow rate.