Flagellated bacteria on nutrient-rich substrates can differentiate into a swarming state and move in dense swarms across surfaces. A recent experiment measured the flow in the fluid around an Escherichia coli swarm (Wu, Hosu and Berg, 2011 Proc. Natl. Acad. Sci. USA 108 4147). A systematic chiral flow was observed in the clockwise direction (when viewed from above) ahead of the swarm with flow speeds of about 10 µm/s, about 3 times greater than the radial velocity at the edge of the swarm. The working hypothesis is that this flow is due to the action of cells stalled at the edge of a colony which extend their flagellar filaments outwards, moving fluid over the virgin agar. In this work we quantitatively test his hypothesis. We first build an analytical model of the flow induced by a single flagellum in a thin film and then use the model, and its extension to multiple flagella, to compare with experimental measurements. The results we obtain are in agreement with the flagellar hypothesis. The model provides further quantitative insight on the flagella orientations and their spatial distributions as well as the tangential speed profile. In particular, the model suggests that flagella are on average pointing radially out of the swarm and are not wrapped tangentially.