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.

A stretched elastic elastic filament can turn from a prescribed end direction toward its outer shape within a narrow boundary layer. This length results from balancing bending energy against the work of axial tension, and differs from the frequency-dependent elastohydrodynamic penetration length.

For a small transverse displacement $y(x)$, constant positive filament bending modulus $A$ and locally constant positive tension $T$, the elastic energy isIts interior Euler-Lagrange equation, away from distributed transverse loading, isThe decaying end correction has $y_{\mathrm{bl}}\propto e^{-x/{\ell }_T}$, whereIt can enforce the imposed slope at a clamped end while matching a slowly varying outer solution.

A distinct boundary layer requires ${\ell }_T\ll L$, or $T{L}^2/A\gg 1$, where $L$ is the elastic filament length. In a driven fluid, viscous drag can generate the tension $T$; this local calculation treats that tension as given. Negative $T$ represents compression and changes the equation toward buckling, so the positive-tension decay length must not be used unchanged.