An exact solution of the time-dependent Navier-Stokes equations is obtained for the flow due to noncoaxial rotations of a disk, executing oscillations in its own plane, and a fluid rotating at infinity. An analytical solution describing the flow at large and small times after the start is obtained by the Laplace transform method. The velocity field is given in terms of the tabulated functions. The solutions for the three cases when the angular velocity is greater than, smaller than or equal to the frequency of oscillations an given and discussed. The time required to attain steady flows for the cosine and the sine oscillations is obtained. (C) 1999 Elsevier Science Ltd. All rights reserved.