Nearly spherical particles vibrating in viscous fluids: A second order asymptotic theory

1. Understanding the motion of non-spherical particles is critical to many applications, including control of the rheological properties of particulate suspensions through to membrane filtration and biotechnology. In the context of small amplitude vibrational systems, the motion of such particles is often studied using the unsteady Stokes equations. Zhang & Stone, J. Fluid Mech. 367, 329–358 (1998) reported the leading order effect of non-sphericity, which occurs at O(ε), where ε quantifies the shape deviation from a sphere. Importantly, some key physical phenomena are absent at O(ε), including coupling between the torque experienced by the particle and its linear translation, the force it experiences and its rotation, and the effect of non-sphericity on the orientation-averages of these forces and torques. We develop an explicit asymptotic theory to second-order in particle non-sphericity, i.e., O(ε²) , for the force and torque acting on a particle performing arbitrary, small-amplitude, vibrations. The derived analytical formulae apply to nearly-spherical particles of arbitrary shape, providing the leading-order asymptotic theory for the above-mentioned phenomena. The theory is demonstrated for several example nearly-spherical particles including a spheroid and a ‘pear-shaped’ particle, which exhibits force-torque coupling.