Bernoulli's equation and a stream of water

[ttk3]

Homework Statement

A continuous stream of water is flowing out of a faucet and falling into a sink below. Explain why this stream of water is narrower at the bottom (near the sink) than at the top (near the faucet). Hint: Think about the change in velocity and the change in pressure as the water falls.

Homework Equations

P₁+.5pv²₁+pgy₁=P₂+.5pv²₂+pgy₂

The Attempt at a Solution

As the water falls the velocity should increase so the pressure should decrease. Wouldn't a decrease in pressure cause the water to spread out near the bottom?

 

[Gear300]

Note that in Bernoulli's equations you have K/V and U/V on both sides, K being kinetic energy, U being potential energy, and V being volume. delta(K) = -delta(U); taking that into account, P1 = P2.

There is a continuous volume flow for ideal fluid flow. So A1v1 = A2v2. At the bottom, v2 > v1 and so A2 < A1.

 

[baba89]

This phenomenon can be explained using Bernoulli's equation, which states that the sum of pressure, kinetic energy, and potential energy remains constant along a streamline in a fluid flow. In this case, as the water falls from the faucet to the sink, it experiences a decrease in potential energy due to the change in height. This decrease in potential energy is converted into kinetic energy, causing the water to increase in velocity.

According to Bernoulli's equation, as the velocity of the water increases, the pressure decreases. This is because the total energy of the water remains constant, so as kinetic energy increases, pressure must decrease to maintain this balance. This decrease in pressure causes the water to spread out as it falls, resulting in a narrower stream near the bottom.

Additionally, as the water falls, it also experiences a decrease in velocity due to friction and air resistance. This decrease in velocity leads to an increase in pressure, which helps to maintain the balance of energy along the streamline. However, the increase in pressure is not enough to counteract the decrease caused by the increase in velocity, resulting in a net decrease in pressure and a narrower stream near the bottom.

In summary, the decrease in pressure caused by the increase in velocity and the decrease in potential energy leads to a narrower stream of water near the bottom. This phenomenon is a result of the principles outlined in Bernoulli's equation and can be observed in various fluid flow situations.