Much respect to those who've chimed in, but you do NOT need an airstone anywhere on this system.
It gets more than enough dissolved oxygen without it. The water is circulated and replaced plenty. They don't need much dissolved oxygen if temps are good, and his are.
Excess D.O. can be bad as it can produce a welcoming environment for bad organisms. There
is a limit to how much D.O. plants can benefit from.
I still believe that your gunk is mostly a normal by-product of the nutrients you are using and the plants expiring their waste back into the mix. I doubt there is a problem here. Some gunk is just going to happen. Not being there, I can't say for certain, but this is my opinion.
With all due respect to those weighing in, this is different than DWC which definitely needs airstones. Also, this is just my opinion, but I feel pretty confident in my knowledge sources and my own experience. I get this gunk sometimes, too, but so does almost everyone with this type of system I think. Maybe your gunk is worse than I think, but I was pretty freaked by it at first, too.
I just try to get what I can during each weekly res. change and not worry about the rest until I'm between grows when everything gets a heavy bleach cleaning.
If I'm being honest, some of my hoses I never could get the gunk out of after the last grow, so this grow started with some gunky sludgey stuff already in the hoses (I did run bleach through first, but it's stubborn stuff). From day one of this grow and last, things were really healthy and normal at all stages of both grows. Any and all issues I've had were due to my own faults, not the system. Minor gunk has proven to me so far to be no more than a minor annoyance and byproduct of the process. Not all sludge/residue/gunk is bad or due to problems. Some weeks, I have hardly any, some weeks a little, some weeks it's more. Heavier nutrient solutions seem to produce sludgier byproduct.
This is just my take and opinion, please nobody take this as a slight. I think airstones would be a waste of money and effort - just not necessary here.
EDIT - I wanted to note, too, that Mr Smith has also been very clear in his opinion on not needing airstones with this type of system. Not trying to name drop here, but he does know his stuff, and has a lot of experience with this type of system. I trust his opinion above nearly anybody's on this site as it relates to hydroponics, personally, and much of how I do things I've gleaned from his experience. I know this is third hand, but I've heard his opinion on this topic so wanted to share that as well...
suppose to be 1450. I was hoping I didn't have to dump the complete 55 gallon res and start over. He told me to remove 15 gallons and refill.......I SWEAR, came right out to 1450 after deductions for my water PPM. 
