On a stainless steel barrel, it is particularly important to use a bore guide and to have quality one piece cleaning rod - like a Dewey. It is also more important to wipe off the cleaning rod during the process frequently, so that dirty solvent isn't rubbing between the rod and the barrel as the rod flexes. For some reason, stainless steel barrels can be ruined a lot quicker than chrome moly barrels if the cleaning process is lax. NEVER use a stainless steel brush on a stainless steel barrel.

When you are finished, you should be able to leavethe barrelwet with shooter's choiceovernight without any blue showing up on the patch. I usually store the barrel with a light coating of Break Free on it - hang a tag on the barrel that says "Oiled Bore" so I don't forget to run a couple patches wet with Shooter's Choice through the barrel before taking it out to shoot again.