mid-15c., "action of flowing," from flow (v.). Meaning "amount that flows" is from 1807. Sense of "any strong, progressive movement comparable to the flow of a river" is from 1640s. Flow chart attested from 1920 (flow-sheet in same sense from 1912). To go with the flow is by 1977, apparently originally in skiing jargon.
Go with the flow, enjoy the forces, let ankles, knees, hips and waist move subtly to soak up potential disturbances of acceleration and deceleration. [Ski magazine, November 1980]
"smoke channel in a chimney," 1580s, of uncertain origin, perhaps related to Middle English flue, flewe "mouthpiece of a hunting horn" (early 15c.), which is perhaps from Old French fluie "stream;" or the modern word is perhaps from Middle Dutch vluwe, from Germanic *flowan "to flow" (see flow (v.)). Originally a small chimney in a furnace connected to the main chimney.