Faucet Streams

The stream of water tends to maintain it's rate of flow of water, $\frac {dV}{dt}$ , where $V$ is the volume. Thus, it obeys the continuity equation.

$\frac {dV}{dt}=constant$

$A\cdot\frac {dx}{dt}=constant$

$A_1v_1=A_2v_2$ , where A is the area of cross section of the stream and $v=\frac {dx}{dt}$ = velocity of the water.

As the water falls, it accelerates, thus gaining speed. According to the above equation, the area of cross section should decrease, and hence the stream becomes narrower.

The stream does not narrow indefinitely.

It gets narrower, until it starts forming separate drops because of surface tension, which then continues to fall as water drops without becoming narrower, even as they continue towards accelerate to terminal velocity. Instead of narrowing, the drops get more spaced out as they fall, so that the flow rate remains uniform with height.

we could imagine that a group of people gathering in front of gate. Once gate open , queue all people run through gate and rush into Narrow passage with the regular acceleration.So some people ahead of group move a bit earlier than people following them, then distance between them will increase, people need not shoulder by shoulder, width of queue is narrow

As the water falls down it moves faster than the water above, due acceleration due to gravity ,hence stretching the stream and becoming narrower .

The answer to this question is quite practical actually. As we know that because of the pull of gravity, the water is accelerating towards the earth, we know that the velocity is increasing as well.

Therefore, the equation will look like this: A1*V1= rate of flow, as the water is maintaining a constant rate of flow. Where A is the cross-sectional area and V is the Velocity

Therefore, if A1 V1= Constant Then, A2 V2= Constant

And, as we know that V is an increasing quantity, then, naturally, A has to become a decreasing quantity.

So the answer is: Yes, the stream will become narrower.

We already know that the stream of water will narrow according to the continuity equation, but it does not narrow indefinitely, as we tend to observe the phenomenon known as Plateau-Rayleigh instability beyond a certain height, where the narrowed neck of water collapses into separate droplets due to a property of surface tension being the desire to acquire the least surface area possible. The distance between the droplets, however, increases over time as a necessity of the flow rate remaining constant.

To maintain the volume flow rate which is constant for a streamlined flow the stream narrows as its speed increases due to gravity.

See, you don’t need much math for this one. Go turn on your sink (and make sure the stream isn’t bubbly) and WATCH! :D

The equation of continuity of stream-lined flow of fluid // Density(p) * Cross section area (A) * Velocity(v)= constant // so the liquid won't compress indefinitely and it will do so in proportion.

Just from a layman's perspective, air resistance alone would cause the stream to narrow

The water coming out of the faucet is flowing at a constant rate, but as it continues accelerating, it starts to "spread out". Surface tension, however, keeps it from forming into drops just yet, so it just becomes a tighter column.