Scaling law for a buckled elastic filament in a shear flow

We analyze the three-dimensional (3D) buckling of an elastic filament in a shear flow of a viscous fluid at low Reynolds number and high Péclet number. We apply the Euler-Bernoulli beam (elastica) theoretical model. We show the universal character of the full 3D spectral problem for a small perturbation of a 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. Phys. Rev. Fluids 9, 014101 (2024) 2469-990X10. 1103/PhysRevFluids. 9. 014101 with the Chebyshev spectral collocation method, to solve the full 3D eigenproblem. We provide a simple analytic approximation of the eigenfunctions, represented as Gaussian wave packets. As the main result of the paper, we derive the square-root dependence of the eigenfunction wave number on the parameter χover ̃=-ηsin2ϕsin^2θ, where η is the elastoviscous number and the filament orientation is determined by the zenith angle θ with respect to the vorticity direction and the azimuthal angle ϕ 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.