Inertial effects on fibre motion in simple shear flow

Abstract

The motion of a torque-free slender axisymmetric fibre in simple shear flow is examined theoretically for small but finite ${\textit{Re}}$, where ${\textit{Re}}$ is the Reynolds number based on the fibre length, and is a measure of the inertial forces in the fluid. In the limit of zero inertia, an axisymmetric particle in simple shear is known to rotate indefinitely in any of an infinite single-parameter family of periodic orbits, originally found by Jeffery (1922) – a degenerate situation wherein the particular choice of orbit is dictated by the initial orientation of the particle. We use a generalization of the well-known reciprocal theorem for Stokes flow to derive the orbit equations, to $O({\textit{Re}})$, for the slender fibre. The structure of the equations bears some resemblance to those previously derived by Leal (1975) for a neutrally buoyant fibre in a viscoelastic (second-order) fluid. It is thereby shown that fluid inertia, for small ${\textit{Re}}$, leads to a slow $O({\textit{Re}})$ drift of the rotating fibre toward the shearing plane, thereby eliminating the aforementioned degeneracy. For Reynolds numbers above a critical value, ${\textit{Re}}_c\,{=}\, ({15}/{4 \pi})(\ln \kappa/\beta \kappa)\sin^{-2}\theta$, the fibre ceases to rotate, however, instead drifting monotonically towards the shearing plane. The limiting stationary orientation in the flow–gradient plane makes an angle $\phi_f$ with the flow direction, where $\phi_f \,{=}\, 4\pi {\textit{Re}}/(15 \ln \kappa) + \{16 \pi {\textit{Re}}^2/(225(\ln \kappa)^2-1/(\beta^2\kappa^2) \}^{1/2}$ is an increasing function of ${\textit{Re}}$. Here, $\kappa$ is the fibre aspect ratio, $\theta$ is the angle made by the fibre with the vorticity axis, and $\beta$ is an $O(1)$ coefficient related to the Jeffery period of the rotating fibre.