Slender body method for slender prolate spheroids and hemispheroids on planes in linearized oscillatory flow

This paper extends the slender body theory of Stokes flow [G. K. Batchelor, J. Fluid Mech. 44, 419 (1970)] to slender prolate hemispheroids on infinite planes that are undergoing rotational or translational oscillatory motion. Unsteady stokeslets and doublets are placed along the focal length axis to represent the hydrodynamics, with strengths or weightings that are determined numerically from the boundary conditions. The hydrodynamic torque is determined using two different methods. Method 1 uses Green’s functions for torque that are derived from unsteady stokeslets and doublets. Method 2 uses the stokeslet weightings, the boundary velocity, and the spheroidal shape to compute the torque, in a similar way as was done [C. Pozrikidis, Phys. Fluids A 1, 1508 (1989)] in computing the force on a prolate spheroid undergoing translational oscillatory motion. The results agree with those of the singularity- and boundary element methods.