This binding method involves appying thin strips of glue to one side of every sheet of paper to make a well-glued hinge. The fan refers to the way in which the sheets are fanned-out to expose the edge plus a tiny portion of one face of each sheet.
This technique can be compared to 'No Fan' binding  ~ or 'Double Fan' binding .
The flexibility and effectivenes of the binding-hinge depends on the thickness of glue, and the extent to which it is spread and penetrates into and beyond the edges ~ and also on its flexibility when dry. When carefully done single fan padding can result in long-lasting books that compare favourably with any other binding. The double fan adds greater certainty of glue sticking to adjacent faces of every sheet, for little extra trouble.
In all fan-binding cases the longevity of the binding depends on the qualty of the glue. PVA is most commonly used these days, but it does come in different formulations. Woodworkers PVA dries very hard, which spoils the opening purpose of the hinge ~and may lead to cracking. It is important ~ as so aften in craftwork to balance the properties of all the media in use. There is no point in having a flexible medium like paper co-existing with a rock-hard substance like solidified glue.
Continuing the previous theme ~ whatever glue is being used it needs to be used sparingly. A huge thickness will not flex. A thin layer of hard-setting glue will still flex sufficiently ~ but may still crack, or tear itself away from the paper ~ with time.
I am reluctant to advise on products,although there is much good advice relating to the old fashioned glues. For example the once standard 'hot' pearl glue sets quite hard, but is flexible when applied thinly ~ and as such it can does its job well ~ as testified by the millions of books bound with it.. As an improvement ~ at greater expense ~ rabbit's foot glue was used. I have found whichever PVA was the cheapest from any hobby shop worked quite well enough for me. Recently I have been using EVA which seems to have a number of useful ~ and advanatageous ~ properties. It comes in a very runny form, and may be difficult to obatin in small quantity.