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On the dynamics of buoyant and heavy particles in a periodic Stuart vortex flow

Published online by Cambridge University Press:  26 April 2006

Kek-Kiong Tio ,
Amable Liñán ,
Juan C. Lasheras  and
Alfonso M. Gañán-Calvo
Kek-Kiong Tio
Affiliation:
Department of Applied Mechanics and Engineering Sciences, University of California, La Jolla, CA 92093-0411, USA
Amable Liñán
Affiliation:
E.T.S. Ingenieros Aeronauticos, Plaza del Cardenal Cisneros 3, Universidad Politécnica de Madrid, 28040 Madrid, Spain
Juan C. Lasheras
Affiliation:
Department of Applied Mechanics and Engineering Sciences, University of California, La Jolla, CA 92093-0411, USA
Alfonso M. Gañán-Calvo
Affiliation:
Departamento de Ingeniería Energética y Fluidomecénica, E.T.S. Ingenieros Industriales, Universidad de Sevilla, 41012 Sevilla, Spain
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Abstract

In this paper, we study the dynamics of small, spherical, rigid particles in a spatially periodic array of Stuart vortices given by a steady-state solution to the two-dimensional incompressible Euler equation. In the limiting case of dominant viscous drag forces, the motion of the particles is studied analytically by using a perturbation scheme. This approach consists of the analysis of the leading-order term in the expansion of the ‘particle path function’ Φ, which is equal to the stream function evaluated at the instantaneous particle position. It is shown that heavy particles which remain suspended against gravity all move in a periodic asymptotic trajectory located above the vortices, while buoyant particles may be trapped by the stable equilibrium points located within the vortices. In addition, a linear map for Φ is derived to describe the short-term evolution of particles moving near the boundary of a vortex. Next, the assumption of dominant viscous drag forces is relaxed, and linear stability analyses are carried out to investigate the equilibrium points of the five-parameter dynamical system governing the motion of the particles. The five parameters are the free-stream Reynolds number, the Stokes number, the fluid-to-particle mass density ratio, the distribution of vorticity in the flow, and a gravitational parameter. For heavy particles, the equilibrium points, when they exist, are found to be unstable. In the case of buoyant particles, a pair of stable and unstable equilibrium points exist simultaneously, and undergo a saddle-node bifurcation when a certain parameter of the dynamical system is varied. Finally, a computational study is also carried out by integrating the dynamical system numerically. It is found that the analytical and computational results are in agreement, provided the viscous drag forces are large. The computational study covers a more general regime in which the viscous drag forces are not necessarily dominant, and the effects of the various parametric inputs on the dynamics of buoyant particles are investigated.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 254 , September 1993 , pp. 671 - 699
Copyright
© 1993 Cambridge University Press

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