ALMA observations in the 12CO line of the interacting galaxy pair IC 2163 and NGC 2207 at 2″ × 1.″5 resolution reveal how the encounter drives gas to pile up in narrow, ∼1 kpc wide, "eyelids" in IC 2163. IC 2163 and NGC 2207 are involved in a grazing encounter, which has led to the development in IC 2163 of an eye-shaped (ocular) structure at mid-radius and two tidal arms. The CO data show that there are large velocity gradients across the width of each eyelid, with a mixture of radial and azimuthal streaming of gas at the outer edge of the eyelid relative to its inner edge. The sense of the radial streaming in the eyelids is consistent with the idea that gas from the outer part of IC 2163 flows inward until its radial streaming slows down abruptly and the gas piles up in the eyelids. The radial compression at the eyelids causes an increase in the gas column density by direct radial impact and also leads to a high rate of shear. A linear regression of the molecular column density on the magnitude of across the width of the eyelid at fixed values of azimuth finds a strong correlation between and . Substantial portions of the eyelids have high velocity dispersion in CO, indicative of elevated turbulence there.