Sperm number ($\mathrm{Sp}$)

The sperm number compares filament length with an elastohydrodynamic penetration length. For periodic forcing at angular frequency $\omega$, the usual convention isIts fourth power is a viscous-to-bending ratio on the full filament length. In steady cross-flow problems a different convention also occurs: $\mathrm{Sp}={\zeta }_{\perp }{L}^{3}U/A$, where $U$ is flow speed and $A$ the filament bending modulus. The defining formula distinguishes these conventions.

The elastohydrodynamic penetration length balances the transverse viscous force $c_{\perp }\omega y$ against the bending force $By/{\ell }_{\omega }^4$ in a periodically driven filament. The sperm number $L/{\ell }_{\omega }$ distinguishes nearly rigid, weakly deformed filaments from long filaments whose response is localized near the actuator.

When viscous drag creates a large axial tension $T$, a clamped filament turns toward its outer direction in a boundary layer of length $\sqrt{A/T}$.