The two-dimensional fluid dynamics of the different phases of a fatty acid monolayer (docosanoic acid) were examined. Using Brewster angle microscopy, we studied the polydomain structure of two liquid condensed phases (the L-2 and L＇(2) phases) and the "solid", S, phase in situ during the application of extensional flow and simple shear flow. We found that only the L-2 phase deformed nearly reversibly with a liquid-like response. Nonsymmetric domain deformations were, however, found for that phase at the lowest rate of strain studied with a four-roll mill. At higher strain rates, plots of the evolution of strain against time collapsed onto a single curve so that the rate of strain was independent of roller speed. Furthermore, a critical strain gamma(c) exists above which the rate of strain shows a stepwise increase despite the constant velocity of the rollers. The two other phases, the L＇(2) and S phases, experienced flow-induced reorientation of the lattice onto which the molecules are arranged. The reorientation process was accompanied by the appearance of shear bands in the monolayers at +/-45 degrees to the extension axis of both types of flow. The shear bands observed in the L＇(2) phase were modeled as a plastic flow accompanying the molecular tilt reorientation developed within elastic regions of the monolayer. This model describes the time evolution of the bandwidth quite well and provides strong evidence of the existence of an additional phase within the conventional L＇(2) phase region. In simple shear flow, the velocity profile for the L-2 phase across the gap of the shearing cell showed a nonlinear distribution of shear rates, which were highest at the center of the gap. Flow-induced breakup of domains was observed in the L＇(2) phase subject to simple shear.