stream (v.) Look up stream at Dictionary.com
early 13c., "to flow copiously," from stream (n.). Transitive sense "discharge in a stream" is from late 14c. Related: Streamed; streaming. Compare German strömen, Dutch stroomen, Danish strömme, all verbs from nouns.
stream (n.) Look up stream at Dictionary.com
Old English stream "a course of water," from Proto-Germanic *straumaz (cognates: Old Saxon strom, Old Norse straumr, Danish strøm, Swedish ström, Norwegian straum, Old Frisian stram, Dutch stroom, Old High German stroum, German Strom "current, river"), from PIE root *sreu- "to flow" (see rheum).

From early 12c. as "anything issuing from a source and flowing continuously." Meaning "current in the sea" (as in Gulf Stream) is recorded from late 14c., as is the sense of "steady current in a river." Stream of consciousness in lit crit first recorded 1930, originally in psychology (1855). Stream of thought is from 1890.