Three numerical solutions of the Leslie–Ericksen equations for low‐molar‐mass rod‐like nematic liquid crystals are obtained for pressure‐driven radial outflow between concentric parallel disks. This radial geometry promotes the director escape from the shear plane; two dissipatively equivalent out‐of‐plane modes appear. The bulk of the flow consist of three distinct regions separated by sharp boundaries. The centerline, elongation‐dominated, region is characterized by complete azimuthal director alignment. The two neighboring regions are shear dominated, and characterized by an in‐plane flow‐aligned director field. The high elastic twist strains are relieved by the nucleation and growth of a twist disclination loop of strength ±1/2 that nucleates in the entrance hole region of the cell. The loop growth transforms the highly twisted out‐of‐plane modes into a twist‐free, flow‐aligned, in‐plane mode.