The effect of shape on the hydrodynamics of a hemispheroid projecting from a plate in irrotational fluid

In this paper, we examine the problem of hemispheroid that is compliantly hinged to a plate oscillating with infinitesimal motions in fluid that can be modeled as irrotational. We use this model to understand the effect of the size and shape of hair bundles of sensory cells in the inner ear on hair bundle hydrodynamics at high frequencies. In response to the oscillating plate, the hemispheroid translates and rotates in the plane of the oscillation. Since the equations of motion are linear, the solution can be described as the superposition of the response due to translation of the hemispheroid and the plate, and the response due to rotation of the hemispheroid. The solution can be found by solving Laplace’s equation in prolate and oblate spheroidal coordinates. The response due to translational motion is the same as that derived in Hydrodynamics, 6th ed. (Dover, New York, 1945) for a full spheroid undergoing translational motion. The response due to rotational motion of the hemispheroid about its hinge ha...