The Stokes resistance of a spherical cap to translational and rotational motions in a linear shear flow

Abstract

Brenner's general results for the force and torque experienced by an arbitrary particle simultaneously translating and rotating in a linear shear flow are applied to the spherical cap. The cap's five fundamental resistance tensors are determined and the corresponding tensors for the sphere and circular disk are recovered as special cases. The problem of a freely moving cap in a linear shear is considered and the resulting translational and rotational motions are analysed. The cap's centre of free rotation is found and trajectories of this point are plotted in a few instances. The cap is also found to possess a ‘point of planar motion’ which always moves in a plane perpendicular to the vorticity vector of the undisturbed shear regardless of the initial orientation of the cap. It is shown that the motion of the cap actually serves as a model for the motions of all ‘oblate’ asymmetric bodies of revolution which are moving freely in a linear shear.