Sperm number ($\mathrm{Sp}$)

The Sperm number compares viscous forcing or oscillation with filament bending stiffness. For a steadily cross-flowed filament of half-length $L$, $\mathrm{Sp}={\zeta }_{\perp }{L}^{3}U/A$.

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