Scaling law for a buckled elastic filament in a shear flow

We analyze the three-dimensional buckling of an elastic filament in a shear flow of a viscous fluid at low Reynolds number and high Peclet number. We apply the Euler-Bernoulli beam (elastica) theoretical model. We show the universal character of the full 3D spectral problem for the small perturbation of the thin filament from a straight position of arbitrary orientation. We use the eigenvalues and eigenfunctions for the linearized elastica equation in the shear plane, found earlier by [Liu et al., 2024] with the Chebyshev spectral collocation method, to solve the full 3D eigenproblem. We provide a simple analytic approximation to the eigenfunctions, represented as Gaussian wavepackets. As the main result of the paper, we derive square-root dependence of the eigenfunction wavenumber on the parameter $\tilde{\chi}=-\eta \sin 2\phi \sin^2\theta$, where $\eta$ is the elastoviscous number, and the filament orientation is determined by the zenith angle $\theta$ with respect to the vorticity direction and the azimuthal angle $\phi$ relative to the flow direction. We also compare the eigenfunctions with shapes of slightly buckled elastic filaments with a non-negligible thickness with the same Young's modulus, using the bead model and performing numerical simulations with the precise Hydromultipole numerical codes.