I'd recommend using a joint chain and set driven keys.
Model your party whistle in its unrolled state. Set up a joint chain along the length of the whistle in a straight line. You want the root of the chain to be at the base of the whistle, and end of the chain at the end of the whistle.
Then use driven keys to make it so that the chain rolls up one joint at a time, starting with the child at the end and moving up all the way to the root joint.
There is, incidentally, a precise ratio for the distance between the joints, if you want a proper spiral.
If the distance between the Joint One and Joint Two is 1.0 , then the distance between Joint Two and Three should be 0.618033988. And so on. The "golden ratio."