Compression Ratio of Spheres in a Curved Tube

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.

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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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