AbstractThis paper describes a second‐order method to calculate approximate solutions to flow of viscous incompressible fluid between rotating concentric spheres. The governing partial differential equations are presented in the stream–vorticity formulation and are written as a series of second‐order equations. The technique employed makes use of second‐order approximations for all terms in the governing equations and is dependent upon the direction of flow at a given point. This upwind technique has allowed us to generate approximate solutions with larger Reynolds numbers than has generally been possible for second and higher‐order techniques. Solutions have been obtained with Reynolds numbers as large as 3000 and with grids as fine as a 40 × 40 mesh. Results are displayed in the form of level curves for both the stream and vorticity functions. A dimensionless quantity related to the torque acting on both spheres has been calculated from the approximate solution and compared with other results. Results with smaller Reynolds numbers such as 100 and 1000 are in excellent agreement with other publish