To ensure an effective drilling operation of an explored well, the associated hydraulics program should be established carefully based on the correct prediction of a drilling fluid’s pressure drop and velocity field. For that, the impact of the drill string orbital motion should be considered by drilling engineers since it has an important influence on the flow of drilling fluid and cuttings transport process. In the present investigation, the finite volume method coupled with the sliding mesh approach is used to analyze the influence of the inner cylinder orbital motion on the flow of a power-law fluid (Ostwald-de Waele) in an annular geometry. The findings indicate that the orbital motion positively affects the homogeneity of the power-law axial velocity through the entire eccentric annulus; however, this impact diminishes as the diameter ratio increases. In addition, higher torque is induced when the orbital motion occurs, especially for high values of eccentricity and diameter ratio; nonetheless, a slight decrease in torque is recorded when the fluid velocity increases.