1. According to equation of continuity, the velocity of the flow of any fluid is inversely proportional to the area. However, when we close the the tap, the velocity of the water decreases though the area is minimized when the tap is being closed. Why?

2.

3. Try to cover a tap with your finger, then see if the velocity of the water increases or not...

4. Prafulla is right. If you start turning off a faucet, the pressure coming from the tap decreases and is just a slight drip before total "Off."

The reason is that home water pressure is regulated.

Let's start at the street: Your incoming city water line has a pressure regulator on it. This keeps the pressure around 70psi.
Some areas are a bit lower and some a bit higher... I've seen spikes of over 120psi but they were abnormal and not intended. The pressure does fluctuate quite a bit though at the city line, which is why there is a pressure regulator to keep it reasonably close to a constant pressure.
Once inside the house, each major plumbing area has a manifold and reducer point. Pressure reduction is necessary because you're going from a 3/4 or 1 inch pipe to a 1/2 inch pipe.
At the faucets, you have a mixing chamber within- This way when you turn on the cold and hot at the same time to get a warm tempered flow, it doesn't come out with hot and cold on each side of the stream, but rather, comes out well mixed. The other effect of this as that the pressure is held within the pipe at the point of entry into the mixing chamber. Beyond that point, it's basic feed flow which is why the pressure decreases as you turn the tap, increasing the pressure within the chamber.

This can be observed by some Outside Hose Spigots that are NOT on a pressure reducing manifold. As you turn off the spigot, the pressure shooting out the spigot tap increases. Not common, but not entirely uncommon, this happens when the builder either cuts corners or has a change in plans and must add an outside hose spigot last minute and they don't tie it into a main manifold.

5. The equation of continuity is velocity тιмєѕ the area. It holds Ƒor pipes having different areas. In other words J=A1V1=A2V2, where J is discharge rate,A is area,v is velocity(the above is an equation Ƒor continuity Ƒor two different pipes with a fluid flowing through them). The discharge rate at the 1st pipe must equal that of the second pipe. Its the velocity that will increase or decrease, the rate of flow must be true Ƒor 2nd pipe as it is Ƒor 1st.

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