Oscillations of free cylinders at low Reynolds numbers inside a Hele-Shaw cell*

We study two instabilities of a horizontal free cylinder in a vertical viscous Hele-Shaw flow: they are shown experimentally to depend critically on the transverse and lateral confinement of the flow characterized respectively by the ratios $D/H$ (resp. $L/W$) of the diameter (resp. the length) of the cylinder to the gap (resp. the width) of the cell. The onset of the instabilities depends essentially on $D/H$. For $0.4 \le D/H \le 0.6$, we observe transverse horizontal oscillations of the cylinder perpendicular to the walls: their frequency is constant with $D/H$ and $L/W$ at a given vertical cylinder velocity $V_c$. This instability is locally $2D$ along the length of the cylinder and controlled by the local relative velocity $V_r^{loc}$ of the cylinder and the fluid: it occurs down to Reynolds numbers $Re^{loc} = V_r^{loc}H/\nu \simeq 15 $, i.e. below the vortex shedding threshold ($150-250$) for fixed cylinders between parallel planes. These results are compared to $2D$ numerical simulations. For $D/H \ge 0.55$, we observe a fluttering motion with periodic oscillations of the tilt angle of the cylinder from the horizontal and of its horizontal position: their frequency decreases as $L/W$ increases and is independent of $D/H$ and $V_c$.