# Calculating power in fluid systems with $pQ.$

I was wondering: say we have a fan which is being used to provide lift to some mass, like in a simple air cushion device, where the air has to flow through a chamber beneath the fan. Let's say that $u_A$ is the speed of the air through the fan, and $u_B$ is the speed upon exit of this chamber. If $R$ is the radius of the fan, then $Q=\pi R^2u_A$ is the flow rate through the fan, and we could calculate the power of the fan as $pQ,$ where $p$ is the pressure due to the fan, and the pressure might be found through a little work with Bernoulli's principle.

However, if we aren't assuming that the kinetic energy of the air inside the chamber is negligible, is this formula still valid? In this case, the pressure should at the least vary with radius as we move along the chamber under the fan, so do we take the difference of the pressures at the fan and at the exit of the chamber, or do we use another expression for power, like $Fv$? This seems to suggest we integrate over the surface of whatever the fan is pushing air down on to get $F$, but I'm not sure what speed is then appropriate to use in the calculation.