Compression Ratio of Spheres in a Curved Tube

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This Demonstration determines the optimal packing for spheres in a curved tube. Spheres of radius are packed in a tube of radius , where , with a bend of radius . We seek the minimum value of the compression ratio. For small ratios , the spheres are in contact with the outside wall. For larger , optimal compression is attained by arranging the spheres in a zigzag pattern parallel to the axis of curvature along the tube centerline.

Contributed by: Aaron T. Becker, Haoran Zhao and Li Huang (March 2017)
Open content licensed under CC BY-NC-SA


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Details

Inside a bend, the spheres arrange themselves so as to minimize the compression ratio; they are all either in contact with the outer wall of the tube or in a zigzag pattern parallel to the axis of curvature along the tube centerline.

If the spheres are centered in the tube, the angular separation between successive spheres is

.

When the spheres are all flush with the outer wall of the tubing, the angle is

,

and the compression ratio is

.

For the zigzag pattern, the angle is

,

and the compression ratio is

.

The transition point occurs when

.



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