At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
The stream of water from faucet becomes narrower as it falls because the velocity of the water increases from acceleration due to gravity and that water also has to pass equal amount through the cross section at any given point in time. Due to the adhesiveness of water the water will draw a line that gradually decreases in the diameter (assuming a tube form) through which the water passing at any given point in time is the same despite ever decreasing diameter. To counteract this, the water will again more velocity assisted by acceleration due to gravity. This is the reason why water falling from faucet will become narrower and narrower yet the velocity at which that water is passing through imaginary cross section at any given point in time increases to hold the equation of continuity principle
*to hold the equation of continuity principle true