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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
「染みる」 has another reading with different meanings.
真冬の寒さが骨の髄まで染みる日だった。
It was a day when the midwinter cold pierced to the bone.
疲れた体に温泉の湯が染み渡る心地よさ。
The comfort of hot spring water soaking into my tired body.
恩師の言葉が今になって心に染みてくる。
My mentor's words are now sinking into my heart.