Fetching details from the dictionary...

to pierce; to penetrate; to soak in; to permeate
to sting (wound or sensitive area, etc.); to smart; to twinge
to be infected (with vice); to be steeped (with prejudice); to be influenced
to feel keenly; to make a deep impression
真冬の冷たい北風が頬にしみる朝だった。
It was a morning when the cold north wind of midwinter bit into my cheeks.
先輩が卒業式で述べた言葉が胸に滲みた。
The words my senior gave at the graduation ceremony touched me deeply.
大量の汗が目に滲みて視界がぼやけてきた。
A large amount of sweat stung my eyes and blurred my vision.