The force on a sphere in a uniform flow with small-amplitude oscillations at finite Reynolds number

The unsteady force acting on a sphere that is held fixed in a steady uniform flow with small-amplitude oscillations is evaluated to 0 (Re) for small Reynolds number, Re. Good agreement is shown with the numerical results of Mei, Lawrence & Adrian (1991) up to Re = 0.5. The analytical result is transformed by Fourier inversion to allow for an arbitrary time-dependent motion which is small relative to the steady uniform flow. This yields a history-dependent force which has an integration kernel that decays exponentially for large time.