= Sperm number
{title2=$\operatorname{Sp}$}
{wiki}
The <sperm number> compares filament <length> with an <elastohydrodynamic penetration length>. For periodic forcing at angular <frequency> $\omega$, the usual convention is
$$
\operatorname{Sp}=L\left(\frac{c_\perp\omega}{B}\right)^{1/4}.
$$
Its fourth power is a viscous-to-bending ratio on the full filament <length>. In steady cross-flow problems a different convention also occurs: $\operatorname{Sp}=\zeta_\perp L^3U/A$, where $U$ is flow speed and $A$ the <filament bending modulus>. The defining formula distinguishes these conventions.
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