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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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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