= 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> is
$$
E[y]=\frac12\int\left[A(y'')^2+T(y')^2\right]\,dx.
$$
Its interior <Euler-Lagrange equation>, away from distributed transverse loading, is
$$
Ay^{(4)}-Ty''=0.
$$
The decaying end correction has $y_{\rm bl}\propto e^{-x/\ell_T}$, where
$$
\ell_T=\sqrt{\frac AT}.
$$
It 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 $TL^2/A\gg1$, 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.
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