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1. To exert pressure on and move in continuous contact with something for some extended period of time. A noun or pronoun can be used between "rub" and "away." Often followed by "at something." He rubbed away at the rare coin with a polishing cloth to make it shine like new. I've been rubbing away for nearly 10 minutes, but I just can't seem to get this stain out. Something must be rubbing away at the tire, because it shouldn't be wearing down this quickly.
2. To clean, clear, or remove something with a rubbing motion. A noun or pronoun can be used between "rub" and "away." Let me just rub some of this dirt and grime away so I can make out the inscription on the headstone. This new phone has a really slick design, but I feel like I'm constantly rubbing away fingerprints smudges.
rub something away
to remove something by chafing or rubbing. See if you can rub some of the dirt away. Rub away the dirt if you can.