Emailing professor for PhD supervision When you email a professor to inquire about an available PhD position is it a good idea to ask if they have any available projects?
You should do basic research on the topics the professors is researching himself. Afterwards, you may either see whether there is a project around where you could supply your workpower, or you have an idea on a concrete project where he could extend research. (Keep in mind, he might need to get a funding for that).

If you do have an own project with independent funding, it might be easier - however, it should fit in the reseach area of the professor.

If you just come around, most professors I know simply will skip your application, as the normally have sufficient requests and work to do.