From Longman Dictionary of Contemporary Englishdrink up phrasal verbDFDto drink all of somethingdrink something ↔ upDrink up your milk. →drink→ See Verb table
Examples from the Corpus
drink up• Come on, drink your milk up.• And it makes a map you want to stare at and drink up.• Now, go on, drink up.• First, he told Simpleton to find a man who could drink up a wholecellar full of wine.• They drank up and began to walk out.• She drank up her brandy and signalled to the waiter to bring another.• But he had drunk up her feeling with her blood.• Come on, drink up. I want to go home.• So come and see me, drink up my DutyFree.• So drink up, then get your coat.• Inside they are drinking up: two hundred and fifty blackfellersnursing their last long drinks.