, Volume 3, Issue 2, pp 183-211

Translational Steady Fall of Symmetric Bodies in a Navier-Stokes Liquid, with Application to Particle Sedimentation

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Abstract.

Let \( \cal B \) ; be a homogeneous body of revolution around an axis a, with fore-and-aft symmetry. Typical examples are bodies of constant density having the shape of cylinders of circular cross-section, of prolate and oblate spheroids, etc. In this paper we prove that, provided a certain geometric condition is satisfied, the only possible orientations that \( \cal B \) ; can eventually achieve when dropped in a Navier-Stokes fluid under the action of the acceleration of gravity g and at a small and nonzero Reynolds number, is with a either parallel or perpendicular to g. This result is obtained by a rigorous calculation of the torque exerted by the fluid on the body. We also show that the above geometric condition is certainly satisfied if \( \cal B \) ; is a prolate spheroid. Moreover, in this case, we prove, by a "quasi-steady" argument, that, at first order in λ, the configuration with a perpendicular to g is stable to small disorientation, while the other is unstable, in accordance with experiments.

Accepted: January 19, 2001