The stream of water falling from a faucet decreases in diame

The stream of water falling from a faucet decreases in diameter as it falls (see picture). Derive an equation for the diameter of the stream as a function of distance y below the faucet, given that the water has a speed v_0 when it leaves the faucet, which has diameter d.

Solution

volume flow speed = Av = ( pi d^2 / 4 )( v0 )

= pi v0 d^2 / 4

now using bernoulli\'s equation for faucket leaving point and

at distance y below.

P + rho*g*h + rho*v^2/2 = constant

Patm + rho * g * y + rho * v0^2/2 = Patm + rho*g*0 + rho*v^2/2

v^2 = v0^2 + 2gy

v = sqrt[ v0^2 + 2gy]

and Volume flow will not change hence.

pi v0 d^2 / 4 = (pi d\'^2/4) (sqrt[ v0^2 + 2gy] )

d\'^2 = vo d^2 / sqrt [v0^2 + 2gy]

d\' = sqrt(v0)d / (v0 + 2gy)^(1/4)