^{1}, Scott D. Kelly

^{2}, Stuart Smith

^{2}and Jeff D. Eldredge

^{1,a)}

### Abstract

The motion of an inertial particle in a viscous streaming flow of Reynolds number order 10 is investigated theoretically and numerically. The streaming flow created by a circular cylinder undergoing rectilinear oscillation with small amplitude is obtained by asymptotic expansion from previous work, and the resulting velocity field is used to integrate the Maxey–Riley equation with the Saffman lift for the motion of an inertial spherical particle immersed in this flow. It is found that inertial particles spiral inward and become trapped inside one of the four streaming cells established by the cylinder oscillation, regardless of the particle size, density and flow Reynolds number. It is shown that the Faxén correction terms divert the particles from the fluid particle trajectories, and once diverted, the Saffman lift force is most responsible for effecting the inward motion and trapping. The speed of this trapping increases with increasing particle size, decreasing particle density, and increasing oscillation Reynolds number. The effects of Reynolds number on the streaming cell topology and the boundaries of particle attraction are also explored. It is found that particles initially outside the streaming cell are repelled by the flow rather than trapped.

Support for this work by the National Science Foundation, under Award Nos. CMMI-0969869 and CMMI-1000656, is gratefully acknowledged. The authors would also like to thank Dr. Phanindra Tallapragada, for many helpful discussions.

I. INTRODUCTION

II. PROBLEM STATEMENT AND METHODOLOGY

A. A note on the change of reference frame

B. Calculation of inertial particle trajectories

III. RESULTS

A. Inertial particle trapping speed

B. Inner and outer streaming

IV. CONCLUSIONS

### Key Topics

- Reynolds stress modeling
- 40.0
- Stokes flows
- 17.0
- Poiseuille flow
- 14.0
- Microscale flows
- 13.0
- Particle trajectory
- 13.0

## Figures

(a) Lagrangian streamlines (top half) and instantaneous Stokes-layer vorticity (bottom half) of canonical streaming pattern for and (here, ). (b) Streaming regimes (adapted from Wang 20 ).

(a) Lagrangian streamlines (top half) and instantaneous Stokes-layer vorticity (bottom half) of canonical streaming pattern for and (here, ). (b) Streaming regimes (adapted from Wang 20 ).

Inertial particle trajectory (solid line) and trajectory sampled once per cycle (shown as dots) for Re = 40, a/R = 0.175, and ρ p /ρ f = 1.

Inertial particle trajectory (solid line) and trajectory sampled once per cycle (shown as dots) for Re = 40, a/R = 0.175, and ρ p /ρ f = 1.

Panels (a)–(c) Inertial particles trajectories (in dark gray, or blue), sampled once per cycle, for Re = 40, a/R = 0.175. Initial locations depicted with circles. Lagrangian streamlines are depicted in light gray. (a) ρ p /ρ f = 1.05, (b) 1, (c) 0.95. In (b), square symbols denote the instants t/T = 290 and 350. (d) Inertial (dark gray, or red) and fluid (light gray, or green) particle trajectories over one oscillation cycle, t/T ∈ [317, 318] for ρ p /ρ f = 1. Arrows depict the total hydrodynamic force vectors in Eq. (14) . Dots denote inertial particle trajectory sampled once per cycle.

Panels (a)–(c) Inertial particles trajectories (in dark gray, or blue), sampled once per cycle, for Re = 40, a/R = 0.175. Initial locations depicted with circles. Lagrangian streamlines are depicted in light gray. (a) ρ p /ρ f = 1.05, (b) 1, (c) 0.95. In (b), square symbols denote the instants t/T = 290 and 350. (d) Inertial (dark gray, or red) and fluid (light gray, or green) particle trajectories over one oscillation cycle, t/T ∈ [317, 318] for ρ p /ρ f = 1. Arrows depict the total hydrodynamic force vectors in Eq. (14) . Dots denote inertial particle trajectory sampled once per cycle.

Comparison of trapping position from current results (squares) and experiments (circles) of Lutz, Chen, and Schwartz. 11 Dashed and solid lines depict, respectively, inner streaming cell size δ DC /R and inner streaming cell center location.

Comparison of trapping position from current results (squares) and experiments (circles) of Lutz, Chen, and Schwartz. 11 Dashed and solid lines depict, respectively, inner streaming cell size δ DC /R and inner streaming cell center location.

Illustration of the tendency of a inertial particle to deviate from a Lagrangian streamline. Note that α < 0 in the upper right portion of the streamline, and α > 0 in the lower left portion.

Illustration of the tendency of a inertial particle to deviate from a Lagrangian streamline. Note that α < 0 in the upper right portion of the streamline, and α > 0 in the lower left portion.

(a) History of α during the cycle t/T ∈ [290, 350] for Re = 40, a/R = 0.175, ρ p /ρ f = 1. (b) Contributions to dα/dt from each term in Eq. (21) : Stokes drag (solid black); convective term (dashed, cyan); Basset history (dashed-dotted, magenta); Saffman lift (dashed-dotted-dotted, blue); Faxén corrections (dotted, green); (solid light gray, or yellow).

(a) History of α during the cycle t/T ∈ [290, 350] for Re = 40, a/R = 0.175, ρ p /ρ f = 1. (b) Contributions to dα/dt from each term in Eq. (21) : Stokes drag (solid black); convective term (dashed, cyan); Basset history (dashed-dotted, magenta); Saffman lift (dashed-dotted-dotted, blue); Faxén corrections (dotted, green); (solid light gray, or yellow).

Hydrodynamic force (arrows) at sampled times along the inertial particle trajectory (dark solid line) during the intervals t/T ∈ [0, 485] and t/T ∈ [1300, 1370] for Re = 40, a/R = 0.175 and ρ p /ρ f = 1. All portions of the inertial particle trajectory not in these intervals are shown as a light gray line.

Hydrodynamic force (arrows) at sampled times along the inertial particle trajectory (dark solid line) during the intervals t/T ∈ [0, 485] and t/T ∈ [1300, 1370] for Re = 40, a/R = 0.175 and ρ p /ρ f = 1. All portions of the inertial particle trajectory not in these intervals are shown as a light gray line.

Saffman lift force (black arrow), relative particle velocity, w (light gray arrow, or green), and local profile of fluid velocity, u , perpendicular to w (gray arrows, or blue) at t/T = 317.5 and t/T = 318. Velocity vectors are plotted with the same scale. Solid line (red) denotes the inertial particle trajectory during the oscillation cycle t/T ∈ [317, 318]. Local fluid particle trajectory, sampled once per period, shown with dots.

Saffman lift force (black arrow), relative particle velocity, w (light gray arrow, or green), and local profile of fluid velocity, u , perpendicular to w (gray arrows, or blue) at t/T = 317.5 and t/T = 318. Velocity vectors are plotted with the same scale. Solid line (red) denotes the inertial particle trajectory during the oscillation cycle t/T ∈ [317, 318]. Local fluid particle trajectory, sampled once per period, shown with dots.

The final limit cycle for inertial particles of various sizes: a/R = 0.1 (blue); a/R = 0.115 (red); a/R = 0.13 (green); a/R = 0.145 (cyan); a/R = 0.16 (black); a/R = 0.175 (magenta), each plotted over one cycle for , ρ p /ρ f = 1. Mean Lagrangian streamlines shown in light gray for reference.

The final limit cycle for inertial particles of various sizes: a/R = 0.1 (blue); a/R = 0.115 (red); a/R = 0.13 (green); a/R = 0.145 (cyan); a/R = 0.16 (black); a/R = 0.175 (magenta), each plotted over one cycle for , ρ p /ρ f = 1. Mean Lagrangian streamlines shown in light gray for reference.

(a) The sampled history of the x position of an inertial particle (- - -) and an exponential fit to the envelope (—), for , ρ p /ρ f = 1, and a/R = 0.175. (b) Trapping timescale dependence on a/R, for , ρ p /ρ f = 1. This figure contains results for (green triangles), 0.2 (blue circles), 0.1 (red squares). (c) Trapping timescale dependence on ρ p /ρ f for , a/R = 0.175. (d) Trapping timescale dependence on for ρ p /ρ f = 1, with a/R varied so that Ωa 2/ν is fixed at 1.2.

(a) The sampled history of the x position of an inertial particle (- - -) and an exponential fit to the envelope (—), for , ρ p /ρ f = 1, and a/R = 0.175. (b) Trapping timescale dependence on a/R, for , ρ p /ρ f = 1. This figure contains results for (green triangles), 0.2 (blue circles), 0.1 (red squares). (c) Trapping timescale dependence on ρ p /ρ f for , a/R = 0.175. (d) Trapping timescale dependence on for ρ p /ρ f = 1, with a/R varied so that Ωa 2/ν is fixed at 1.2.

Inertial particle trajectories at for particle of density ρ p /ρ f = 1 and radius a/R = 0.175, initially located at (1.2R, 1.2R) (circle), (1.8R, 1.8R) (square), (2R, 2R) (triangle), (2.2R, 2.2R) (diamond). Lagrangian streamlines are in light gray.

Inertial particle trajectories at for particle of density ρ p /ρ f = 1 and radius a/R = 0.175, initially located at (1.2R, 1.2R) (circle), (1.8R, 1.8R) (square), (2R, 2R) (triangle), (2.2R, 2.2R) (diamond). Lagrangian streamlines are in light gray.

(a) Mean Eulerian streamlines, and (b) mean Lagrangian streamlines, for .

(a) Mean Eulerian streamlines, and (b) mean Lagrangian streamlines, for .

## Tables

Time averaged forces applied on the inertial particle during the interval t/T ∈ [4200, 4700] for Re = 40, a/R = 0.175, ρ p /ρ f = 1.

Time averaged forces applied on the inertial particle during the interval t/T ∈ [4200, 4700] for Re = 40, a/R = 0.175, ρ p /ρ f = 1.

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