Simulation of the sedimentation of a falling oblate ellipsoid

We present a numerical study of the dynamics of one falling oblate ellipsoid particle in a viscous fluid, in three dimensions, using a constrained-force technique (Doctoral Thesis, Stuttgart University, 2000, Phys. Rev. E 61 (2000) 7146, J. Eng. Math. 41 (2001) 221). We study the dynamical behavior for a typical downward motion. Three types of falling motions are established: steady-falling, periodic oscillations and chaotic oscillations. For the periodic and steady-falling regime we find a similarity law derived from the invariance of the Reynolds and Froude number. In the chaotic regime the trajectory of the oblate ellipsoid is characterized by a high sensitivity to tiny variations in the initial orientation. The Lyapunov exponent is λ=0.052±0.005. A phase diagram is presented and compared to the results of Field et al. (Nature (London) 388 (1997) 252). The transition from oscillatory to steady-falling occurs at Rec=355, where the transient time of oscillation in the steady-falling regime tends to infinity, beyond this value the system is oscillatory. The transient time has a power law divergence at Rec with an exponent of 0.5. The transition from steady-falling to chaotic regime becomes abrupt, for an aspect-ratio Δrc≈=0.22.