The small-amplitude oscillatory capillary Poiseuille flow of uniaxial incompressible discotic nematic liquid crystals, representative of discotic mesophases, is characterized using analytical, computational, and scaling methods. Linear viscoelastic material functions are calculated and discussed in terms of fundamental anisotropic viscoelastic processes. The role of orientation to generate flow and store elastic energy is discussed. Viscoelastic behavior is found only at frequencies of similar magnitude to the single orientation relaxation time. In the terminal small-frequency zone the storage modulus scale as G' similar to omega(2), and the loss modulus as G" similar to omega, typical of viscous fluids. Comparisons between steady and oscillatory Poiseuille flow shows that the Cox-Merz rule is not obeyed, but that as expected the steady and complex viscosities in the terminal zone are identical. A remarkable and useful correspondence between the stored elastic energy under steady flow and the storage modulus G' has been discovered. (C) 2004 Elsevier Ltd. All rights reserved.