We construct solutions for the Navier–Stokes equations in three dimensions with a time periodic force which is of compact support in a frame that moves at constant speed. These solutions are related to solutions of the problem of a body which moves within an incompressible fluid at constant speed and rotates around an axis which is aligned with the motion. In contrast to other authors who analyze stationary solutions in a frame of reference attached to the body, the analysis for the present problem is done in a frame which is moving at constant speed but is not rotating. This avoids the unpleasant unbounded linear terms which are present in a description with respect to a rotating frame.