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The motion of a single and multiple neutrally buoyant elliptical cylinders in plane Poiseuille flow

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10.1063/1.4757387

### Abstract

In this article, we investigate the motion of neutrally buoyant elliptical cylinders in plane Poiseuille flow of a Newtonian fluid. The method of distributed Lagrange multiplier/fictitious domain was used to solve the Navier-Stokes equations as well as for the motion of elliptical cylinders. The motion of a single elliptical cylinder is shown to be dependent on the channel Reynolds number*Re*, the particle size ratio *K* = *a* ^{ * }/*H* ^{ * }, and the aspect ratio *A* = *a* ^{ * }/*b* ^{ * } of the cylinder, where *H* ^{ * } is the half height of the channel, *a* ^{ * } and *b* ^{ * } are the lengths of the semi-major axis and semi-minor axis of the cylinder, respectively. It is found that there is a critical Reynolds number,*Re* _{ c } ∼ 3, which distinguishes the lateral migration of a single elliptical cylinder below and above it. As *Re* is increased, the equilibrium position of the elliptical cylinder shifts towards the wall when *Re* ≤ *Re* _{ c } or shifts closer to the central axis when *Re* ≥ *Re* _{ c }. Moreover, there are interesting correlations between the center-of-mass trajectories and the orientation dynamics, which depend on the ranges of *K* and *Re*. The motion of multiple elliptical cylinders is also affected by the total solid area fraction ϕ_{ T }, which is defined to be the proportion of the area occupied by the cylinders in the domain of computation. For a few elliptical cylinders (the number of cylinders *ND* = 16 and the corresponding ϕ_{ T } = 3.77%), the cylinders may scatter into several groups at lower *Re* (≤ 100), and each group fluctuates about an averaged position. At the higher *Re* (= 1000), the cylinders may converge to an equilibrium position on each side of the channel center. For a larger number of cylinders (*ND* = 36, 54, 72, 108, and the corresponding ϕ_{ T } = 8.48%–25.45%), we observed a significant rheological behavior in the velocity profiles. In addition, there exists a particle-free layer next to each wall, and the thickness of the particle-free layers is increased as *A* (or *K*) or *Re* is increased.

© 2012 American Institute of Physics

Received 16 January 2012
Accepted 10 September 2012
Published online 12 October 2012

Acknowledgments: The work was supported in part by the National Science Council (Taiwan) under Contract Nos. NSC97-2221-E-002-223-MY3, NSC99-2628-M-002-003, and NSC100-2221-E-002-152-MY3. T.-W. Pan acknowledges the support of the US NSF (Grant No. DMS-0914788).

Article outline:

I. INTRODUCTION

II. DESCRIPTION OF THE PROBLEM

III. NUMERICAL METHOD

IV. RESULTS AND DISCUSSION

A. Validation of an elliptical cylinder in plane Couette flow

B. The motion of a single elliptical cylinder in plane Poiseuille flow

1. The effect of the shape of the cylinder (*A*, *K*)

2. The effect of Reynolds number (*Re*)

C. The cases of a few elliptical cylinders (*ND* = 16, ϕ_{ T } = 3.77%)

1. The effect of the Reynolds number (*Re*)

2. The effect of the initial distributions of cylinders

D. The cases for many cylinders (*ND* = 72, ϕ_{ T } = 16.9%)

1. The effect of the aspect ratio (*A*)

2. The effect of the Reynolds number (*Re*)

E. The effect of the total solid area fraction (ϕ_{ T } = 0.94%–25.45%, *ND* = 4–108)

V. CONCLUDING REMARKS

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2012-10-12

2014-04-23

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