Orientation dynamics of two-dimensional concavo-convex bodies

Abstract

We study the orientation dynamics of two-dimensional concavo-convex solid bodies more dense than the fluid through which they fall under gravity. We show that the orientation dynamics of the body, quantified in terms of the angle $\varphi$ relative to the horizontal, undergoes a transcritical bifurcation at a Reynolds number $R{e}_c^{(1)}$, and a subcritical pitchfork bifurcation at a Reynolds number $R{e}_c^{(2)}$. For $Re<R{e}_c^{(1)}$, the concave-downwards orientation of $\varphi =0$ is unstable and bodies overturn into the $\varphi =\pi$ orientation. For $R{e}_c^{(1)}<Re<R{e}_c^{(2)}$, the falling body has two stable equilibria at $\varphi =0\text{ and }\varphi =\pi$ for steady descent. For $Re>R{e}_c^{(2)}$, the concave-downwards orientation of $\varphi =0$ is again unstable, and bodies that start concave-downwards exhibit overstable oscillations about the unstable fixed point, eventually tumbling into the stable $\varphi =\pi$ orientation. The $R{e}_c^{(2)}\approx 15$ at which the subcritical pitchfork bifurcation occurs is distinct from the $Re$ for the onset of vortex shedding, which causes the $\varphi =\pi$ equilibrium to also become unstable, with bodies fluttering about $\varphi =\pi$. The complex orientation dynamics of irregularly shaped bodies evidenced here are relevant in a wide range of settings, from the tumbling of hydrometeors to settling of mollusk shells.

6 pages, 6 figures

References in corpus (1)

• Levitation of heavy particles against gravity in asymptotically downward flows

Cited by in corpus (4)

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• Torques on curved atmospheric fibres
• Smart navigation of a gravity-driven glider with adjustable centre-of-mass
• Sedimenting rigid particles of certain shapes approach a stationary orientation