My daughter had clogged tear ducts. I would get a warm, wet wash cloth and really gently press on the tear duct. But I'm no expert.
Below is quotes from ehow.com:
Clogged tear ducts are a common problem at all ages, although they are far more common in young children. While surgical intervention is sometimes required, most blocked tear ducts can be treated at home with common household products.
.Diagnosis
Diagnosing a tear-duct blockage is simple. Symptoms include excessive tearing, pain, and redness or swelling in the inner corner of the eye and lower lid. The swelling may be warm to the touch, and painful when pressed. A thick discharge, the same consistency as pus, is also common in a blocked tear duct.
Massage Treatment
To treat a blocked tear duct, massage from the inner corner of the eye down and across to the bridge of the nose. Alternate between massaging and pulsing the area gently. The eye will begin to flood and tear--the effect you are going for. The theory is that by stimulating the duct, the blockage will wash out.
Hot Compresses
After massaging the area, place a hot compress over the affected eye for 10 to 15 minutes at a time. The compress will help with the pain and swelling, and loosen any residual blockage. Reapply the compress every two to three hours, as long as the swelling persists.
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