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On my site I have a search bar like this:
The user types something in and presses enter. There is nothing they can click on to submit the search. I could make the icon on the right clickable but it certainly doesn't look like a button now and I quite like the way it looks. I could add something to the placeholder, e.g. "Search and press Enter" but again I like how minimal it is now and this will only help if they think to read the placeholder before typing something in.
Can I assume that the vast majority of people on the Internet have figured out when pressing Enter is likely to work?
The site aggregates search results from multiple online stores (linking directly to them) so the users already need some web proficiency to be able to accomplish anything.