The laminar and turbulent helical flows of non-Newtonian fluids in concentric and fully eccentric annuli with a rotating inner pipe are numerically simulated using computational fluid dynamics. Pipe rotation speeds are varied from 0 to 400 rpm at different annular flow velocities (0.005-10.05 ft/s). The predicted frictional pressure losses are validated using experimental data obtained from the literature. Then, the effects of pipe rotation on frictional pressure loss and tangential velocity and velocity profile of non-Newtonian fluids in both concentric and fully eccentric annuli are analyzed in detail. It can be observed that as the pipe rotation increases, frictional pressure losses inside the concentric annulus drastically decrease for laminar flow, especially, at a low axial Reynolds number Moreover, pipe rotation effects on pressure loss in concentric annuli for turbulent flow are negligible. Flow regime directly influences change in the pressure gradient due to pipe rotation for the concentric annulus. Pipe rotation increases pressure loss for both the laminar and turbulent regime for fully eccentric annuli. Turbulence significantly raises the pressure gradient for the fully eccentric annulus. Rotating pipe causes a considerable increase in tangential velocity of non-Newtonian fluid inside concentric and fully eccentric annuli. Additionally, eccentricity and turbulent flow exaggerate pipe rotation effects on tangential velocity.