Linear path instability of permeable buoyancy-driven disks

The prediction of trajectories of falling or rising objects in a viscous fluid is a relevant problem in fluid dynamics. Falling disks have been widely investigated since they exhibit diverse trajectories, ranging from zig-zag to tumbling and chaotic motions. Yet, similar studies are lacking when the object is permeable. We perform a linear stability analysis of the steady vertical path of a thin microstructured, permeable, disk modeled via a stress-jump model which stems from homogenization theory. The flow associated with the steady vertical path presents a recirculation region detached from the disk, which shrinks and disappears as the disk becomes more permeable. In analogy with the impervious disk, one non-oscillatory and several oscillatory unstable modes are identified. Permeability profoundly modifies the destabilization picture. For sufficiently large permeabilities, the primary bifurcation at low disk inertia is nonoscillatory and leads to a steady tilting of the disk. A further increase of permeability reduces the unstable regions in the parameters space until all linear instabilities are damped.