We have observed the surface pressure-driven flow of langmuir monolayers through a narrow channel by means of Brewster angle microscopy (BAM). The monolayers, composed of long chain carboxylic acids, were studied in a "hexatic" liquid crystalline mesophase (L-2 phase). The velocity profile across the channel demonstrated transient behavior; i.e., it evolved from the expected parabolic shape to triangular as a function of flow rate, location along the channel, and time elapsed after start-up of flow. These transient effects appear to be related to the total strain in the system and suggest the importance of elasticity in the monolayer flow response. For situations in which the velocity profile was triangular, the distribution of domain widths across the channel indicated that the degree of domain stretching was systematically less than expected from the amount of strain the monolayer had experienced. This implied that elasticity of domain boundaries and/or slippage between domains resists domain stretching and may contribute to the observed non-Newtonian behavior of the monolayer.