Unlike other standard equations in introductory classical mechanics, the Bernoulli equation is not Galilean invariant. The explanation is that, in a reference frame moving with respect to constrictions or obstacles, those surfaces do work on the fluid, constituting an extra term that needs to be included in the work-energy calculation. A quantitative example is presented here for a horizontal tapered pipe. A frame-independent expression for the pressure drop in the pipe is obtained. The concepts discussed in this paper are accessible to introductory undergraduate physics majors. (Contains 2 figures.)