We study the flow induced by eye rotations in the anterior chamber (AC) of the eye, the region between the cornea and the iris. We model the geometry of the AC as a thin domain sitting on the surface of a sphere, and study both the simpler case of a constant-height domain as well as a more realistic AC shape. We model eye rotations as harmonic in time with prescribed frequency ωf and amplitude β , and use lubrication theory to simplify the governing equations. We write the equations in a reference frame moving with the domain and show that fluid motion is governed by three dimensionless parameters: the aspect ratio ϵ of the AC, the angular amplitude β and the Womersley number α . We simplify the equations under the physiologically realistic assumptions that ϵ is small and α large, leading to a linear system that can be decomposed into three harmonics: a dominant frequency component, with frequency ωf , and a steady streaming component and a third component with frequency 2ωf . We solve the problem analytically for the constant-height domain and numerically as the solution of ordinary differential equations in the more realistic geometry. Both the primary flow and the steady streaming are shown to have a highly three-dimensional structure, which has not been highlighted in previous numerical works. We show that the steady streaming is particularly relevant from the clinical point of view, as it induces fluid mixing in the AC. Furthermore, the steady flow component is the dominant mixing mechanism during the night, when the thermal flow induced by temperature variations across the AC is suppressed.