The indirect boundary element method was used to study the hydrodynamics of longitudinal shear flow and cross flow with a longitudinal rate of shear over prolate and oblate hemispheroidal protuberances projecting from a plane wall. Analytic techniques such as Fourier analysis, spheroidal co-ordinates, and the method of images were used to make the numerical methods more efficient. A novel method for computing the hydrodynamic torque was used-instead of directly calculating the torque from the weightings of the Green's functions (a method that is only valid when the weightings have physical significance) the hydrodynamic torque was computed indirectly using a Green's function for torque that derived here. As a test of this method, the present scheme was applied to determine the hydrodynamic torque of full spheroids, where exact solutions are known, and excellent results were obtained. Our results for hemispheroids projecting from plane walls were, except for extremely wide oblate hemispheroids, within a factor of two of those of full spheroids. Our results also agreed with those of previous study of oblate hemispheroidal protuberances.