I’m trying to figure out if there is a way to have particles travel along a surface that is not flat.  In the attached image, I have particles starting at the bottom and I want them to travel up the cone then follow the bend and fan out on to part of the surface.  I keep trying Speed by surface but can’t seem to get it to work.

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It should work with the speed by surface operator. Make sure to check control speed continuously and direction: parallel to surface.
And I think it would be easier to spawn the particles directly on the surface of you mesh with the position object operator.