Inertial settling of an arbitrarily oriented cylinder in a quiescent flow : from short-time to quasi-steady motion

In this article, we investigate the inertial settling of an arbitrarily oriented cylinder settling under gravity. We focus on two regimes: the very short-time and long-time dynamic. By using the generalized Kirchhoff equations to describe the particle motion, we demonstrate that during the very short dynamic regime, a cylinder starting from rest behaves with sedimenting velocities and angular velocity proportional to $t$ and $t^3$, respectively. We then explore the long-time behaviour and evaluate the validity of the quasi-steady assumption under which the fluid unsteady term can be neglected. Using a dimensional analysis, we establish that the quasi-steady assumption is only applicable to Reynolds numbers much smaller than one. However, by comparing the results of quasi-steady models to recent experiments and direct numerical simulations, we demonstrate that this assumption is valid for a broader range of Reynolds numbers, particularly for long fibres. We also analyze the effect of particle inertia. We show particle inertia plays no significant role in the magnitude of the sedimenting velocities and angular velocity. However, for sufficiently large inertia we reveal that the quasi-steady model takes the form of a damped oscillator when the particle approaches its equilibrium position, which is broadside on to its direction of motion. We discuss the relevance of this solution in light of direct numerical simulations.