Skip to main content
×
Home
    • Aa
    • Aa

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

  • Kek-Kiong Tio (a1), Amable Liñán (a2), Juan C. Lasheras (a1) and Alfonso M. Gañán-Calvo (a3)
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.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

Auton, T. R., Hunt, J. C. R. & Prud'homme, M. 1988 The force exerted on a body in inviscid unsteady non-uniform rotational flow. J. Fluid Mech. 197, 241257.

Chein, R. & Chung, J. N. 1987 Effects of vortex pairing on particle dispersion in turbulent shear flows Intl J. Multiphase Flow 13, 785802.

Chein, R. & Chung, J. N. 1988 Simulation of particle dispersion in a two-dimensional mixing layer. AIChE J. 34, 946954.

Chung, J. N. & Troutt, T. R. 1988 Simulation of particle dispersion in an axisymmetric jet. J. Fluid Mech. 186, 199222.

Crowe, C. T., Gore, R. A. & Troutt, T. R. 1985 Particle dispersion by coherent structures in free shear flows. Particulate Sci. Technol. 3, 149158.

Dandy, D. S. & Dwyer, H. A. 1990 A sphere in shear flow at finite Reynolds number: effect of shear on particle lift, drag, and heat transfer. J. Fluid Mech. 216, 381410.

Manton, M. J. 1974 On the motion of a small particle in the atmosphere. Boundary-Layer Met. 6, 487504.

Maxey, M. R. 1987 The motion of small spherical particles in a cellular flow field. Phys. Fluids 30, 19151928.

Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26, 883889.

Mclaughlin, J. B. 1991 Inertial migration of a small sphere in linear shear flows. J. Fluid Mech. 224, 261274.

Mei, R. 1992 An approximate expression for the shear lift force on a spherical particle at finite Reynolds number. Intl J. Multiphase Flow 18, 145147.

Mei, R., Lawrence, C. J. & Adrian, R. J. 1991 Unsteady drag on a sphere at finite Reynolds number with small fluctuations in the free-stream velocity. J. Fluid Mech. 233, 613631.

Nielsen, P. 1984 On the motion of suspended sand particles. J. Geophys. Res. 89, 616626.

Oliver, D. L. R. & Chung, J. N. 1987 Flow about a fluid sphere at low to moderate Reynolds numbers. J. Fluid Mech. 177, 118.

Reeks, M. W. & Mckee, S. 1984 The dispersive effects of Basset history forces on particle motion in a turbulent flow. Phys. Fluids 27, 15731582.

Saffman, P. G. 1965 The lift on a small sphere in a slow flow. J. Fluid Mech. 22, 385400. Also Corrigendum, 31(1968), 624.

Sommeria, J., Staquet, C. & Robert, R. 1991 Final equilibrium state of a two-dimensional shear layer. J. Fluid Mech. 233, 661689.

Stuart, J. T. 1967 On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29, 417440.

Tio, K.-K., Lasheras, J. C., Gañán-Calvo, A. M. & Liñán, A. 1993b The dynamics of bubbles in periodic vortex flows. Appl. Sci. Res. (in press).

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 5 *
Loading metrics...

Abstract views

Total abstract views: 60 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th May 2017. This data will be updated every 24 hours.