In a manner full of curves or bends; in a way that winds and turns smoothly.
From flexuous (full of bends) + -ly (adverbial suffix that converts adjectives to adverbs indicating manner). The -ly suffix is one of English's most productive affixes.
When you add -ly to an adjective, you're essentially asking 'in what way or manner?'—so flexuously means 'in a bendy, curving way,' which perfectly describes how water flows downstream!
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