Perrin Friction Factors - Perrin S Factor

For brevity in the equations below, we define the Perrin S factor. For prolate spheroids (i.e., cigar-shaped spheroids with two short axes and one long axis)

$S \ \stackrel{\mathrm{def}}{=}\ 2 \frac{\mathrm{atanh} \ \xi}{\xi}$

where the parameter is defined

$\xi \ \stackrel{\mathrm{def}}{=}\ \frac{\sqrt{\left| p^{2} - 1 \right|}}{p}$

Similarly, for oblate spheroids (i.e., discus-shaped spheroids with two long axes and one short axis)

$S \ \stackrel{\mathrm{def}}{=}\ 2 \frac{\mathrm{atan} \ \xi}{\xi}$

For spheres, as may be shown by taking the limit for the prolate or oblate spheroids.

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