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We analyze the three-dimensional buckling of an elastic filament in a shearflow of a viscous fluid at low Reynolds number and high Peclet number. We applythe Euler-Bernoulli beam (elastica) theoretical model. We use the eigenvaluesand eigenfunctions for the linearized elastica equation, found earlier by [Liuet al., 2022] with the Chebyshev spectral collocation method. We show theuniversal character of the full 3D spectral problem for the small perturbationof the thin filament from a straight position of arbitrary orientation. Weprovide a simple analytic approximation to the eigenfunctions, represented asGaussian wavepackets. As the main result of the paper, we derive square-rootdependence of the eigenfunction wavenumber on the parameter $\tilde{\chi}=-\eta\sin 2\phi \sin^2\theta$, where $\eta$ is the elastoviscous number, and thefilament orientation is determined by the zenith angle $\theta$ with respect tothe vorticity direction and the azimuthal angle $\phi$ relative to the flowdirection. We also perform numerical simulations with the preciseHydromultipole numerical codes, that take into account non-negligible filamentthickness, and are based on the bead model and the multipole expansion of theStokes equations, corrected for lubrication. We show that at early times, thefilament shapes determined numerically are similar to the eigenfunctions ofelastica with the same Young modulus.

Scaling law for a buckled elastic filament in a shear flow