Elastohydrodynamic boundary layer of a filament

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.