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Analysis of the trajectory of a sphere moving through a geometric constriction
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10.1063/1.4809729
/content/aip/journal/pof2/25/6/10.1063/1.4809729
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/6/10.1063/1.4809729
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

(a) System, simulation box, and boundary conditions. The schematic is to scale, i.e., the box is 16 obstacle/particle radii in length. The top wall creates the pinching gap. In the absence of walls, we use the periodic boundaries for the box, and the obstacle is at the center of the box. (b) The part of the simulation box indicated with dashed lines in (a) is enlarged here. The initial offset , the final offset , and the aperture of the pinching gap are indicated. The dashed circle depicts the position of the particle when the particle-obstacle separation is minimum (in the absence of inertia). (c) Enlarged view of the particle (dashed) and the obstacle (solid) surfaces showing the minimum separation ξ between them.

Image of FIG. 2.
FIG. 2.

Trajectories corresponding to the three possible types of particle-obstacle “collisions,” adapted from Ref. . The particle travels from left (incoming half of the trajectory) to right (outgoing half of the trajectory). The dashed circle represents the excluded volume inaccessible to the particle centers.

Image of FIG. 3.
FIG. 3.

In presence of significant inertia (in this particular case, particle inertia, = 1.11), the trajectories are asymmetric even in the absence of non-hydrodynamic interactions. Such asymmetry leads to two critical offsets, viz., and as shown.

Image of FIG. 4.
FIG. 4.

Various trajectories with = 2.0, exhibiting the effect of pinching in the Stokes regime ( = 5 × 10 and = 1.11 × 10). The vertical dashed lines at ≈ ±7 represent the initial and final -coordinates of the particle, respectively.

Image of FIG. 5.
FIG. 5.

Various trajectories with = 2 exhibiting the effect of inertia in the absence of pinching. The vertical dashed lines at ≈ ±7 represent the initial and final -coordinates of the particle, respectively.

Image of FIG. 6.
FIG. 6.

The dimensionless surface-to-surface separation between the particle and the obstacle as a function of the dimensionless -coordinate of the particle. The plot shows the combined effect of particle and fluid inertia on the dimensionless separation along particle trajectories for = 2 in the absence of pinching. The vertical dashed line depicts the symmetry plane of the system at = 0.

Image of FIG. 7.
FIG. 7.

Effect of reducing the aperture of the pinching gap (/ ↑) on the minimum surface-to-surface separation between the particle and the obstacle for different magnitudes of inertia and a fixed initial offset, = 2.

Image of FIG. 8.
FIG. 8.

The final offset (corresponding to = 2) as a function of Stokes (top) and Reynolds (bottom) numbers for different apertures of the pinching gap.

Image of FIG. 9.
FIG. 9.

The critical final-offset corresponding to ε = 0.3, as a function of Reynolds and Stokes numbers in the presence and the absence of a pinching wall.

Image of FIG. 10.
FIG. 10.

The minimum separation ξ as a function of the initial offset , and are negligible.

Image of FIG. 11.
FIG. 11.

The minimum separation ξ as a function of the initial offset in the absence of pinching. (Left) Effect of particle inertia when fluid inertia is negligible and (right) effect of fluid inertia when particle inertia is negligible.

Image of FIG. 12.
FIG. 12.

The minimum separation ξ as a function of the initial offset showing the effect of significant fluid as well as particle inertia in the absence of pinching.

Image of FIG. 13.
FIG. 13.

The dimensionless surface-to-surface separation ξ as a function of the dimensionless -coordinate of the particle. The -coordinate at which the minimum separation is attained along a trajectory varies with inertia. (Left) Effect of particle inertia and (right) effect of fluid inertia. The minimum separation is attained before the = 0 symmetry-plane in both cases.

Image of FIG. 14.
FIG. 14.

The effect of inertia on the ξ relationship in the presence of a pinching wall. The aperture of the pinching gap is 3 (i.e., / = 0.333). (Left) The effect of particle inertia and (right) the effect of fluid inertia.

Image of FIG. 15.
FIG. 15.

The effect of inertia on the ξ relationship in the presence of pinching. The aperture of the pinching gap is 1.7 (i.e., / = 0.588). (Left) The effect of particle inertia and (right) the effect of fluid inertia.

Image of FIG. 16.
FIG. 16.

Combined effect of the pinching wall, particle and fluid inertia on the ξ relationship. (Left) The aperture of the pinching gap is 3 (i.e., / = 0.333) and (right) the aperture is 1.7 (i.e., / = 0.588).

Image of FIG. 17.
FIG. 17.

The outgoing offset as a function of the incoming offset . (Left) Effect of particle inertia and (right) effect of fluid inertia.

Image of FIG. 18.
FIG. 18.

The minimum separation as a function of the final offset. (Left) Effect of particle inertia and (right) effect of fluid inertia.

Image of FIG. 19.
FIG. 19.

Effect of both significant fluid as well as particle inertia on the ξ relationship in the absence of pinching.

Image of FIG. 20.
FIG. 20.

The individual effect of particle and fluid inertia on the ξ relationship in the presence of pinching. The aperture of the pinching gap is 3.0 (i.e., / = 0.333). (Left) The effect of particle inertia and (right) the effect of fluid inertia.

Image of FIG. 21.
FIG. 21.

The individual effect of particle and fluid inertia on the ξ relationship in the presence of pinching. The aperture of the pinching gap is 1.7 (i.e., / = 0.588). (Left) The effect of particle inertia and (right) the effect of fluid inertia.

Image of FIG. 22.
FIG. 22.

The effect of inertia on the ξ relationship in the presence of pinching for two different apertures of the pinching gap. (Left) The aperture of the pinching gap is 3.0 (i.e., / = 0.333) and (right) the aperture is 1.7 (i.e., / = 0.588).

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/content/aip/journal/pof2/25/6/10.1063/1.4809729
2013-06-11
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Analysis of the trajectory of a sphere moving through a geometric constriction
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/6/10.1063/1.4809729
10.1063/1.4809729
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