Chaotic sedimentation of spheroidal particles in a cellular flow field

Abstract

Chaotic mixing of Lagrangian fluid elements has been shown to be a feature of several unsteady two-dimensional flows or steady three-dimensional flows. There is also an interesting group of problems relating to the sedimentation of discrete particles, and for nonspherical particles which respond to the local flow conditions we have found the particle motion to be chaotic even in simple, steady two-dimensional flows. Specifically, we have studied the motion of rigid spheroidal particles settling under gravity in a spatially periodic, cellular flow field. The particles are small compared to the cell size and sufficiently small that their motion relative to the fluid satisfies conditions for Stokes flow. The position of the particle X(t) and the symmetry axis m are determined by (dX/dt)=u[X(t),t]+W1 (ĝ⋅m)m+W2 (ĝ−ĝ⋅mm), (dm/dt)=( (1)/(2) ω+Dm×E⋅m)×m. The flow field considered is u=(sin πx cos πy, −cos πx sin πy, 0) and ω and E are the corresponding local vorticity and rate of strain; W1 and W2 are the terminal fall speeds parallel and normal to the symmetry axis m, while ĝ is (0, 1, 0). The ratio W2/W1, and the parameter D are set by the aspect ratio of the particle, but for the purpose of analysis may be varied separately. Previous work1 has dealt with the restricted problem of planar motion, where m3=0, and given general numerical results on settling rates and particle suspension by the flow. Clear evidence was found from the trajectories of selected particles of chaotic motion in certain parts of the flow. More recently we have investigated in more detail the transition from regular to chaotic motion in this restricted problem. Based on perturbation analysis of periodic solutions, Poincaré sections, and determination of Lyapunov exponents we conclude that regions of regular motion where particles are suspended by the flow persist but the extent of these regions is greatly reduced for increasingly nonspherical shapes. Chaotic tumbling motion occurs outside these regions, as the particle turns in response to the local vorticity and rate of strain. Interestingly, chaotic motion is found even if the particle turns only in response to the local vorticity. Besides summarizing these results for the restricted problem, we present results for the unrestricted problem of three-dimensional motion which could be used for comparison with experiments. We find that the motion is qualitatively similar with both regular and chaotic regions.