The Poiseuille capillary flow of discotic nematic liquid crystals (DNLCs) is analyzed using the Ericksen-Leslie theory, for three families of static patterns known as escaped and defect core onion textures, defined by the orientation at the capillary' centerline. The escaped textures have low-elastic energy and high dissipation cores. The defect core texture has a high elastic energy-low dissipation core. The Poiseuille flow of these textures exhibits non-parabolic velocity profiles and their apparent viscosities display shear thickening, with exponents close to 1/5 +/- 0.07. Up to an Ericksen number of a thousand, core conditions have a strong effect on the non-Newtonian rheology of discotic nematics displaying the onion texture. (C) 2004 Elsevier B.V. All rights reserved.