# The Science Guys

### June 2001

Why does a stream of water from a faucet become smaller at it falls?

Everyone has seen this phenomenon in their home. Turn on the water and adjust it so that the water flows in a steady, smooth manner (called laminar flow). You will observe that the stream narrows as it falls toward the sink. Why does this happen? Water does have a cohesiveness that holds it together but that is not why the stream gets smaller. At first thought it appears there is less water at the bottom of the stream than at the top but this is not the case.

First we are talking about a smooth, steady flow. By smooth we mean non-turbulent and by steady we mean the stream stays the same from one moment to the next. That is, it does not change in time. This being the case, the amount of water in any section of the stream stays the same. Pick any inch of the stream and the amount of water in that section remains constant over time.

For this to be valid, then the amount of water flowing into that section must equal the amount flowing out of that section. Or phrased a more general way, the amount of water flowing through any cross-section of the stream per second (the flow rate) at any point must be the same. How can we represent the amount of water flowing through any cross-section of the stream?

Let’s imagine a special highway and let’s follow a group of cars as they travel down the highway. There won’t be turnoffs, exit, or entrance ramps, and furthermore you tell the drivers that the same number of cars must pass any given point on the highway every second. To maintain this constant flow rate, when the highway is broad, the drivers know they must slow down because the road can accommodate more cars. But when the highway narrows the drivers must speed up to maintain the constant flow rate because fewer cars can pass abreast down a narrow highway. Therefore the flow rate for the cars is proportional to both the cross-section size of the highway and the speed of the cars.

Now consider two points along the highway. At point one, the flow is proportional to the cross-section (A1) and the speed (v1) at that point. And at point two, the flow is proportional to the cross-section at point two (A2) and the speed at point two (v2). Since the same number of cars must pass both points, then A1 times v1 must equal A2 times v2 . Although some people do not appreciate mathematical expressions, this fact is probably best represented in that manner,

(A1) x (v1) = (A2) x (v2) . In fluid physics this equation is called the equation of continuity, which simply says "what flows in must equal what flows out."

The water that emerges from the faucet is falling. What happens to any object that falls under the influence of gravity? It travels faster the further it falls (at least over short distances). From the above mathematical expression, one can understand that if we have a higher velocity (larger v2) at the bottom of the stream, then the cross-section (A2) is going to have to be smaller in order for the flow rate to remain the same. Thus the size of the stream (A2) gets smaller the further (and faster) the water falls. If the stream falls far enough, the water reaches a terminal speed and the size of the stream will stop decreasing in size or becoming smaller as it falls.