Speaking from the other side of the fence (a possible advisor...), I receive about 50 emails per day. 30 can be deleted straight away (newsletters, invitation to bogus conferences...). 20 need reading, out of which 10 need no further action (or perhaps a brief acknowledgment such as "sure, here is the photo"), leaving me with 10 that require anywhere between 5 mn and 5 days to answer (look up a reference, set an appointment, do a calculation, read a document...).
If for some reason I skip a day's worth of email (teaching, travelling, in the field, on leave...) that's then 10 + 10 = 20 emails waiting for me to process.
So - for something for which the reply is not immediate, surely a few days lag, up to a week in busy times, is not unexpected. In your case, your request probably needs some thinking (is it administratively feasible, do I have other applicants, do we need to discuss the project together etc) so your mail will go to the "reply latter" pile to give me time for an articulate answer. The delay is a good sign in a way: if it was a straight "no", it would be in the "process immediately and get rid of it" pile.
In which case I'd be actually glad to see you popping in - discussion in person, unless particular circumstances (language issues...) is so much more efficient than waiting for an email conversation! And it will be one less email to answer :-)