flux n.

1580s, "flowing freely" (of water), also, of speakers, "able and nimble in the use of words," from Latin fluentem (nominative fluens) "lax, relaxed," figuratively "flowing, fluent," present participle of fluere "to flow, stream, run, melt," from extended form of PIE *bhleu- "to swell, well up, overflow" (source also of Latin flumen "river;" Greek phluein "to boil over, bubble up," phlein "to abound"), an extended form of root *bhel- (2) "to blow, swell." Used interchangeably with fluid (adj.) in 17c. in the sense "changeable, not rigid." Related: Fluently.

Proto-Indo-European root meaning "to blow, swell," "with derivatives referring to various round objects and to the notion of tumescent masculinity" [Watkins].

It forms all or part of: bale (n.) "large bundle or package of merchandise prepared for transportation;" baleen; ball (n.1) "round object, compact spherical body;" balloon; ballot; bawd; bold; bole; boll; bollocks; bollix; boulder; boulevard; bowl (n.) "round pot or cup;" bulk; bull (n.1) "bovine male animal;" bullock; bulwark; follicle; folly; fool; foosball; full (v.) "to tread or beat cloth to cleanse or thicken it;" ithyphallic; pall-mall; phallus.

It is the hypothetical source of/evidence for its existence is provided by: Greek phyllon "leaf," phallos "swollen penis;" Latin flos "flower," florere "to blossom, flourish," folium "leaf;" Old Prussian balsinis "cushion;" Old Norse belgr "bag, bellows;" Old English bolla "pot, cup, bowl;" Old Irish bolgaim "I swell," blath "blossom, flower," bolach "pimple," bolg "bag;" Breton bolc'h "flax pod;" Serbian buljiti "to stare, be bug-eyed;" Serbo-Croatian blazina "pillow."

An extended form of the root, *bhelgh- "to swell," forms all or part of: bellows; belly; bilge; billow; bolster; budget; bulge; Excalibur; Firbolgs.

An extended form of the root, *bhleu- "to swell, well up, overflow," forms all or part of: affluent; bloat; confluence; effluent; effluvium; efflux; fluctuate; fluent; fluid; flume; fluor; fluorescence; fluoride; fluoro-; flush (v.1) "spurt, rush out suddenly, flow with force;" fluvial; flux; influence; influenza; influx; mellifluous; phloem; reflux; superfluous.

The section of entries for the various flushes in Century Dictionary opens with a caveat:

The several words spelled flush, being mostly dialectal, colloquial, or technical, and scantily recorded in early literature, have become partly confused with one another, and cannot now be entirely disentangled. Words originally different have acquired some meanings very nearly identical, while on the other hand there are some meanings not obviously related which are, nevertheless, to be referred to one original.

Weekley calls it "A very puzzling word." Sense of "a rush of water" in a stream (1520s), is probably from flush (v.1). From this likely come the extended senses "rush of emotion or passion" (1610s); "a sudden shooting up" (1773); "act of cleansing (a drain) by flushing" (1883); "glow of light or color" (especially sudden redness in the face), 1620s. Independently from the verb, probably, is the noun sense of "a flight of birds suddenly started up" (1590s).

The meaning "hand of cards all of one suit" (1520s) is of uncertain origin, perhaps formed on the model of French flus (15c.), from Old French flux, flus "a flowing, rolling" (see flux), which, in common with its Italian cognate flusso, is said to have once had a sense of "a run" of cards. The form in English probably was influenced by flush (v.1).