Perrin Friction Factors - Rotation Friction Factor

Rotation Friction Factor

There are two rotational friction factors for a general spheroid, one for a rotation about the axial semiaxis (denoted ) and other for a rotation about one of the equatorial semiaxes (denoted ). Perrin showed that

$F_{ax} \ \stackrel{\mathrm{def}}{=}\ \left( \frac{4}{3} \right) \frac{\xi^{2}}{2 - (S/p^{2})}$

$F_{eq} \ \stackrel{\mathrm{def}}{=}\ \left( \frac{4}{3} \right) \frac{(1/p)^{2} - p^{2}}{2 - S \left}$

for both prolate and oblate spheroids. For spheres, as may be seen by taking the limit .

These formulae may be numerically unstable when, since the numerator and denominator both go to zero into the limit. In such cases, it may be better to expand in a series, e.g.,

$\frac{1}{F_{ax}} = 1.0 + \left(\frac{4}{5}\right) \left( \frac{\xi^{2}}{1 + \xi^{2}}\right) + \left(\frac{4 \cdot 6}{5 \cdot 7}\right) \left( \frac{\xi^{2}}{1 + \xi^{2}}\right)^{2} + \left(\frac{4 \cdot 6 \cdot 8}{5 \cdot 7 \cdot 9}\right) \left( \frac{\xi^{2}}{1 + \xi^{2}}\right)^{3} + \ldots$

for oblate spheroids.

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