The flow of an incompressible fluid contained between two infinitely long circular cylinders is considered when the inner cylinder performs small harmonic oscillations about the centre of the larger (fixed) cylinder. Numerical solutions are presented for the Navier–Stokes equations governing the steady-streaming component of the motion. Special attention is then paid to this flow when the Reynolds number of the steady streaming is large, and when the radius of the outer cylinder is much larger than that of the inner. Results obtained show an improved correlation with experimental results and indicate strongly that the finiteness of the domain is the major cause of the discrepancies between experiment and previous theoretical studies.