Rheum is an old-fashioned word for the watery discharge that drips from your nose and eyes when you have a cold or allergies.
You may come across the word rheum in 19th-century novels, or your doctor might use the term to describe your cold symptoms. It’s not the word you’d use in everyday speech, though. You wouldn't say, "My allergies are really acting up. Please excuse the rheum streaming down my face." But you might use the adjective rheumy to describe your dog's runny eyes or even your own. The word comes from the Greek rheuma, meaning "a stream, or current."