Flow of a Herschel-Bulkley (H-B) fluid in tubes of non circular cross-section in investigated analytically. This study complements results presented in [1] where the equation of motion was solved in tubes of arbitrary cross-section for Bingham type of fluids, and the shapes of plug zones centered on the tube axis and stagnant zones attached to the corners were predicted when the cross-section is triangular and square. In this paper we investigate the effect of the power index in the H-B model on the flow for values greater and lesser than unity, considering thus the shear-thinning and shear-thickening effects, which could not be accounted for with the Bingham model. The equation of motion is solved when the cross-section is an equilateral triangle or a square by means of the shape factor method previously introduced in [2]. Thus, shear thickening and shear-thinning effects are accounted for and related to the tube geometry in predicting the existence and the extent of undeformed regions in the flow field.