In this work, we investigate Core Annular Flow (CAF) using computational techniques for the case in which the core fluid is non-Newtonian and the annular fluid is Newtonian. The CAF is simulated using the CFD software ANSYS FLUENT 14.5 in a horizontal pipe. The core fluid is a highly viscous shear-thickening oil while water, the annular fluid, is injected along the thin annular region. Volume of fluid (VOF) modelling is used to simulate the immiscible liquid pair in the limit of low Reynolds number (Re). We analyse data related to the pressure drop along the pipe and the hydrodynamics of flow. Grid convergence analysis is performed before selecting a suitable mesh for simulation, and the effect of changing the interfacial tension between the oil and water phases is also studied. It is observed that the lower the interfacial tension, the more stable is the CAF. We also observe that, for lower interfacial tension, the pressure drop is reduced to a value close to that for a pure water flow-demonstrating that a highly viscous non-Newtonian fluid can be effectively transported using the CAF lubrication method. A theoretical analysis is also reported for a fully developed flow, in which both fluids are non-Newtonian. Good agreement is seen between the mean velocity profiles predicted by theory and simulation.