Numerical finite-difference results of the full axisymmetric incompressible Navier–Stokes equations are presented for the problem of the slow axial motion of a disk particle in an incompressible, rotating fluid in a cylindrical container. The governing parameters are the Ekman number, E, the Rossby number, Ro, and the dimensionless height of the container, H (with respect to the diameter of the particle). The study concerns small values of E, Ro, and HE−1/2 and compares the numerical results with predictions of previous analytical (mostly approximate) studies. Special attention is focused on the drag force. First, developed (quasi-steady state) cases are considered. Excellent agreement with the exact linear (Ro = 0) solution of Ungarish & Vedensky (1995) is obtained when the computational Ro = 10−4. The effects of the nonlinear momentum advection terms are analysed and shown to be proportional to RoE−1/2. Next, the time-development for both (a) impulsive start and (b) start under a constant axial force are considered, and good qualitative agreement with previous analytical results (including the appearance of oscillations in case (b)) is indicated.