The stream is instantaneous, whilst the acid lasts for the spell's duration.
The description of Tasha's Caustic Brew strongly suggests that the stream occurs just once, when the spell is cast, and then the acid covering a creature which failed their saving throw lasts for the spell's duration or until it is scrapped away. For reference:
A stream of acid emanates from you in a line 30 feet long and 5 feet wide in a direction you choose. Each creature in the line must succeed on a Dexterity saving throw or be covered in acid for the spell’s duration or until a creature uses its action to scrape or wash the acid off itself or another creature. [...]
If the stream had last continuously for the entire duration, the description would have stated explicitly: indeed, compare Tasha's Caustic Brew with Gust of Wind (emphasis mine):
A line of strong wind 60 feet long and 10 feet wide blasts from you in a direction you choose for the spell's duration. Each creature that starts its turn in the line must succeed on a Strength saving throw or be pushed 15 feet away from you in a direction following the line.
If the stream had been available once (or more) per turn, the description would have stated something similar to Dragon's Breath (emphasis mine):
[...] Until the spell ends, the creature can use an action to exhale energy of the chosen type in a 15-foot cone. [...]